Project: 1.) Choose an application related to discrete mathematics. 2.) Write a description of your project (at least 1/2 page) in html format. E-mail the URL to me by 3.) Write an outline and first draft in html format. E-mail the URL to me by 4.) Written report should be in html format with a length of about 2 pages/person. Hence a group of 2-3 people should write at least 5 pages. E-mail the URL to me by 5.) Present your work in class during the last week of class. Presentation should be about 10 minutes/person. Hence a group of 2-3 people should give a 20-25 minute presentation. The following information should be included in your written report Title: 5 key words: Mathematics used: Mathematical Difficulty: Area of Application: Application Area Difficutly: You may choose one of the following projects or you may find your own. Extra credit will be given for finding a good project. References are given to introduce the topics below. These references are a good place to start, but you need to find additional, preferably more current references. The following are potential topics OR may be covered in class. Some require more biology background than others. 1.) Sequence Comparison Chapter 3 in Setubal, Meidanis, Introduction to Computational Molecular Biology. 2.) Fragment Assembly Chapter 4 in Setubal, Meidanis, Introduction to Computational Molecular Biology. 3.) Physical Mapping of DNA Chapter 5 in Setubal, Meidanis, Introduction to Computational Molecular Biology. 4.) Phylogentic trees Chapter 6 in Setubal, Meidanis, Introduction to Computational Molecular Biology. 5.) Genome rearrangements Chapter 7 in Setubal, Meidanis, Introduction to Computational Molecular Biology. 6.) RNA/Protein Folding Chapter 8 in Setubal, Meidanis, Introduction to Computational Molecular Biology. 7.) DNA computing 8.) Pairwise alignment using hidden markov models Biological sequence analysis by Durbin, Eddy, Korgh, Mitchenson. 9.) Genetics Inbreeding Section 2.9 in An introduction to stochastic processes with applications to biology by Allen I am not familiar with this book, but will look more at this section when we get to Markov chains. 10.) Chemical chirality 11.) 10.) Voting? Other methods of finding project material. 1.) Search the web. 1a.) Use MathSciNet 1b.) Search the Dimacs, Rutgers website. 1c.) Search the web for Research Experiences for Undergraduates (REU) and Discrete Mathematics. 1d.) Search for any area of interest. 2.) Look at Humanities Math books such as For all Practical Purposes, COMAP. 3.) Check other discrete math books such as Applied Combinatorics by Roberts. Basic Counting Rules: Product Rule: If $A$ can happen in $n_1$ ways and $B$ can happen in $n_2$ ways, then the number of ways $A$ and $B$ can happen together in $n_1n_2$ Example: Sum Rule: If $A$ can happen in $n_1$ ways and $B$ can happen in $n_2$ ways, then the number of ways $A$ or $B$ can happen together in $n_1n_2$ Permutations rule: the number of ways to permute $n$ elements (where order matters) is $n!$ $P(n, r)$ = number of ways to HW 2.1: 2, 4, 9, 11 2.2 2.1 Defn: A graph is a pair of sets $(V, E)$ where the elements of $V$ are called vertices and the elements of $E$ are unordered pairs of elements of $V$. The elements of $E$ are called edges. Defn: A directed graph or digraph is a pair of sets $(V, E)$ where the elements of $V$ are called vertices and the elements of $E$ are ordered pairs of elements of $V$. The elements of $E$ are called arcs. Defn: A digraph is said to have multiple arcs if there exists more than one arc from $u$ to $v$. Hence a digraph that contains two arcs from $u$ to $v$ has multiple arcs, but a digraph that contains exactly one arc from $u$ to $v$ and one arc from $v$ to $u$ does NOT have multiple arcs. Defn: A graph is said to have multiple edges if there exists more than one edge between $u$ and $v$. Note the above definitions mean that we are assuming (unless otherwise stated) that graphs do not contain multiple edges and digraphs do not contain multiple arcs. In chapters 2 and 3, we will also assume (unless otherwise stated) that graphs and digraphs have a finite number of vertices and do not contain loops. 2.2: Connectedness table 2.1 $v$ reachable from $u$ if there exists a path from $u$ to $v$. Thm 2.1: If $v$ is reachable from $u$, then there is a simple path from $u$ to $v$. Proof 1 (ugly version): If $v$ is reachable from $u$, there exists a path $u = u_1, u_2, ..., u_{n-1}, u_n = v$ If $u_1 = u_2$ let $p_1 =$ the path $u_2, ..., u_n$ If $u_1 \not= u_2$ let $p_1 =$ the path $u_1, u_2, ..., u_n$ Suppose the path $p_{k-1} = u_{m_1}, ..., u_{m_j}, u_{k}, u_{k+1}, ..., u_n$ is defined such that 1.) $m_i < m_r$ if $i < r$. 2.) $m_i < k $ for $i = 1, ..., j$. 3.) $u_{k} \not= u_{m_i}$ for $i = 1, ..., j$. Note $p_{k-1}$ is a path from $u$ to $v$. If $u_k = u_{m_i}$ for some $i = 1, ..., j$ let $p_k =$ the path $u_{m_1}, ..., u_{m_i}, u_{k+1}, ..., u_n$$ If $u_k \not= u_{m_i}$ for some $i = 1, ..., j$ let $p_k =$ the path $u_{m_1}, ..., u_{m_j}, u_k, u_{k+1}, ..., u_n$$ The path $p_{n-1}$ is a simple path from $u$ to $v$. $v$ joined to $u$ if there exists a semipath between $u$ and $v$. 2.2.2 Distance The length of a path is the number of arcs in the path If $v$ is reachable from $u$, $d(u, v) = $ the distance between $u$ and $v$ = the length of a shortest path from $u$ to $v$ Note 0.) $d(u, v)$ is not defined if v$ is NOT reachable from $u$. 1.) $d(u, u) = 0$. 2.) $d(u, v) \not= d(v, u)$. 3.) If $w$ is reachable from $u$ and $v$ is reachable from $w$, then $v$ is reachable from $u$ and $d(u, v) \leq d(u, w) + d(w, v)$. 2.2.4 Connectedness Categories A digraph is strongly connected or strong if for every pair of vertices $u$ and $v$, $u$ is reachable from $v$ AND $v$ is reachable from $u$. A digraph is unilaterally connected or unilateral if for every pair of vertices $u$ and $v$, $u$ is reachable from $v$ OR $v$ is reachable from $u$. A digraph is weakly connected or week if every pair of vertices $u$ and $v$ is joined. A digraph is disconnected if it is not weakly connected. A digraph is connected of degree 0 if it is disconnected degree 1 if it is weak, but not unilateral degree 2 if it is unilateral, but no strong degree 3 if it is strong 2.2.5 Criteria for Connectness Thm 2.3: A digraph is strongly connected if and only if it has a complete closed path. Lemma: In any set of vertices of a unilateral digraph D, there is a vertex which can reach all others in the set. Thm 2.4: A digraph is unilaterally connected if and only if it has a complete path. Thm 2.5: A digraph is weakly connected if and only if it has a complete semipath. \end A path is a sequence of $u_1, (u_1, u_2), u_2, (u_2, u_3), u_3, ..., u_{n-1}, (u_{n-1}, u_n), u_n$ where $u_i \in V$ and $(u_i, u_{i+1}) \in E$. Or equivalently a path is a sequence of vertices $u_1, u_2, u_3, ..., u_n$ where $(u_i, u_{i+1}) \in E$. The path $u_1, u_2, u_3, ..., u_n$ is closed if $u_1 = u_n$. Pop quizzes (if you read this, make sure you read the last sentence). We probably won't have time for more than one pop quiz. But giving exactly one pop quiz is impossible: Proof: Suppose that the pop quiz is not given until the last day of class. The pop quiz is not a pop quiz