The Delta Epsilon, Issue 5

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The fifth issue of The Delta Epsilon. McGill Undergraduate Mathematics Journal

Transcript of The Delta Epsilon, Issue 5




    Perfectly Matched 3Cyndie Cottrell

    Constructing Cryptographic Hash Functions 10Francois Seguin

    Interview with Axel Hundemer 16Cathryn Supko

    A One-Parameter Family of Elliptic Curves Over the Rational Numbers 17Dieter Fishbein

    Singularly Perturbed Multiple State Dependent Delay Differential Equations 23Daniel Bernucci

    Prime Numbers: Solitary Yet Ubiquitous 28Cathryn Supko

    On the Generalization of a Hardy-Ramanujan Theorem and the Erdos Multiplication Table Problem 29Crystel Bujold

    Validating a Statistical Model 35Julie Novak

    Interview with Heekyoung Hahn 37Cathryn Supko

    Random Walks on a Graph 39Maria Eberg

    Differentes approches a` la loi de composition de Gauss 45Francois Seguin

    Crossword Puzzle 51

  • 2 Letters


    Five is a unique number for many reasons, such as the fact that it is the only prime with two distinct twins. Similarly,the fifth issue of the Delta-Epsilon is unique. Not only did we try to put together a balanced and interesting issue, butwe also attempted to standardize the editing process. One of the great things about mathematics and science is that it isnot necessary to reinvent the wheel; as Newton once said, If I have seen further it is by standing on the shoulders ofgiants. By facilitating communication between editorial teams from year to year, we ensure a successful elta-psilon inthe future.

    The dual nature of our role gave this project its challenging flavor. We sincerely hope that we have succeeded in ourtask and our efforts will be appreciated in years to come. (To next years editors: If you read this while completely dis-couraged, keep on going! The fun is just ahead of you!)

    The elta-psilon provides not only an incredible opportunity for mathematics students to be introduced to the pub-lishing process, but also to think, share, and contribute to the rest of our undergraduate mathematical community. In orderto take advantage of this wonderful opportunity, we need contributions from you, our undergraduate readers. Your help isalways integral to the existence of the elta-psilon. We strongly encourage every one of you to participate in next yearsjournal, and to make it the best issue yet. If you are interested, do not hesitate to contact us by e-mail, or to talk to us inperson. We hope you enjoy the fifth issue of the elta-psilon, and if you have questions or comments just let us know.

    Francois SeguinExecutive Editor

    The elta-psilon Editing [email protected]


    What would mathematics be without research. Mathematical tools are used on a daily basis by everyone, yet mathematicalresearch is what allowed these tools and much better ones to be developed. Research is invaluable and mathematicianshave to start doing it as early as possible. The undergraduate level is a great place to start.

    Publishing its fifth issue, the elta-psilon continues to be a high-quality journal that allows undergraduates to show-case their work, and share their love of various mathematical topics. The Society of Undergraduate Mathematics Students(SUMS) is proud to see the efforts of the mathematics undergraduates of McGill pay off in this beautiful journal.

    On behalf of SUMS, congratulations to the elta-psilon team!Sincerely,

    Cyndie CottrellSUMS President

    (On behalf of SUMS council)


  • Perfectly Matched 3

    PERFECTLY MATCHEDCyndie Cottrell

    In [2], Elkies et al. presented domino shuffling on a family of finite subgraphs of the square lattice. This shufflingprovides us with an easy way to count perfect matchings on the family of subgraphs. Here, we develop a similar shufflingon the hexagonal lattice superimposed on its dual graph. We define two families of finite subgraphs and count perfectmatchings on these subgraphs. We will be reproving Ciucus results from [1] regarding one family of subgraphs. Theresults regarding the second family of subgraphs follow from this new shuffling.


    A mathematical article unequivocally begins with defini-tions. The topic of enumerating perfect matchings, ex-plored here, combines the field of graph theory with thatof combinatorics. Yet, what is a perfect matching? Whatis a graph?

    Definition. A graph, G, is a set of vertices and edges. Amatching is a set of vertex-disjoint edges in G. A matchingis perfect when the matching covers every single vertex inG.

    In laymans terms, a graph consists of two things: dots,and lines connecting these dots. When we choose someof these lines and they do not share endpoints, we have amatching. This matching is perfect when it includes all ofthe dots on the graph. See Figure 1.

    Figure 1: A matching on the left, and a perfect matchingon the right

    We are interested in counting the number of differentperfect matchings in a graph. In 1992, Elkies et al. intro-duced a procedure called domino shuffling on the squarelattice to count the number of perfect matchings on a fam-ily of graphs called Aztec Diamonds [2]. This shuffling in-spired us to develop a similar shuffling algorithm to countthe number of perfect matchings on certain subgraphs onthe following more complicated lattice: the dP3 lattice isa hexagonal lattice superimposed on its dual graph (seeFigure 2).

    Figure 2: The dP3 Lattice

    We will study two different families of finite sub-graphs of the lattice: whole diamonds and half diamonds.Let Dm be a diamond in one of these families with orderm, for m 12N. If m N, then Dm is a whole diamond.If m+ 12 N, then Dm is a half diamond. We will give aprecise definition of both families of graphs in sections 4and 5, but here we only give an intuitive definition. Wecover the lattice with diamonds of order 1, as in Figure 3.

    Figure 3: The dP3 Lattice

    We then select one diamond and call it the center. Letthe ring of order m be the set of diamonds of order 1 di-rectly surrounding the ring of order m1, where the ringof order 1 is the center. Then, a whole diamond of order mis the set of rings of order n where n ranges from 1 to m.

    The half diamonds of order m, defined as Dm+ 12, are

    more complicated. Here, we fix two diamonds of order 1


  • 4 Cyndie Cottrell

    such that one is directly north of the other and call this thecenter, or the ring of order 1. Then, Dm+ 12

    is the set ofrings of order n where n ranges from 1 to m, as well as afew extra edges and vertices.

    Now, our purpose here is to show the following theo-rem about the diamonds:

    Theorem 1. The number of perfect matchings on a dia-mond of order m, where n = bmc, is{

    2n(n+1) m N2(n+1)

    2(m+ 12 ) N.

    The first part of the theorem related to the whole dia-monds was proven by Ciucu in 2003 [1], while the secondpart is new. The proof is based on the application of ourshuffling algorithm to a perfect matching on a diamond oforder n. This will allow us to count the perfect matchingson a diamond of order n+ 12 . Hence, by a simple induc-tion we will show the theorem. Due to the tediousness ofcertain technical details, a mere sketch of the proof will bepresented here. First, we give some more definitions.


    Definition. We define a square to be a face whose bound-ary has 4 edges. We also define the tail as the edge shownin the figure below; the tail depends on the orientation ofthe square. We define a kite to be a square and its tail. Theessential vertex of a kite K is the unique vertex of K thatis of degree 3.

    Figure 4: Orientations

    This figure displays a N square. There are 6 possibleorientations: N, S, NE, SE, NW, and SW. Further, whentwo squares have parallel tails, they have the same pair oforientations. There are three such pairs of orientations:

    N/S, NE/SW, and NW/SE. We will now abuse notation byusing the term orientation in the place of the term pairs oforientations, unless clarity is absolutely necessary.

    We define a strip to be a series of adjacent diamondswhich share the same pair of orientations. Diamonds areadjacent if they share a common edge.

    Figure 5: A strip


    Now, our shuffling is based on two graph transformations:urban renewal and double-edge contraction. Urban re-newal is described by Propp in [3] and essentially con-sists of transforming the graph so that it has a predictablechange in weight and number of perfect matchings. Wedescribe double-edge contraction here. We consider agraph G, and will apply this transformation to a vertex bof degree 2. We can only apply double-edge contractionto vertices of degree 2 and obtain a new graph G. We con-sider the two edges which share b as their endpoints andcall them ab and bc. We will transform the graph by con-tracting the edges ab and bc into one vertex c. We claimthat G and G have the same number of perfect matchings.

    Figure 6: Double-Edge Contraction

    We now show that our claim is true. Consider a per-fect matching M on G. Then, since M is perfect, b mustbe matched. Hence, ab M or bc M. Without loss ofgenerality, let ab M. There must be some edge c Madjacent to c. When we apply double-edge contraction tothe graph, a, b, and c become one vertex c. Yet, all otheredges are the same. We build a matching M on the newgraph G. Then, let M\ab M, so that every vertex ofG remains matched. In particular, we know that a and bbecame a part of the vertex c and hence c is in M. So, Mis perfect, and for each matching M on G, we have exactly


  • Perfectly Matched 5

    one matching of M on G.