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Articles 1 - 10 of 10
Full-Text Articles in Discrete Mathematics and Combinatorics
Two Quick Combinatorial Proofs, Arthur T. Benjamin, Michael E. Orrison
Two Quick Combinatorial Proofs, Arthur T. Benjamin, Michael E. Orrison
All HMC Faculty Publications and Research
Presentation of two simple combinatorial proofs.
Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden
Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden
Mathematical Sciences Technical Reports (MSTR)
We explore some questions related to one of Brizolis: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to ask about not only fixed points but also two-cycles. Campbell and Pomerance have not only answered the fixed point question for sufficiently large p but have also rigorously estimated the number of such pairs given certain conditions on g and h. We attempt to give heuristics for similar estimates given other conditions on g and h and also in the case …
Totally Magic Graphs, Geoffrey Exoo, Alan C. H. Ling, John P. Mcsorley, Nicholas C. Phillips, Walter D. Wallis
Totally Magic Graphs, Geoffrey Exoo, Alan C. H. Ling, John P. Mcsorley, Nicholas C. Phillips, Walter D. Wallis
Articles and Preprints
A total labeling of a graph with v vertices and e edges is defined as a one-to-one map taking the vertices and edges onto the integers 1, 2, · · · , v+e. Such a labeling is vertex magic if the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex, and edge magic if the sum of an edge label and the labels of the endpoints of the edge is constant. In this paper we examine graphs possessing a labeling that is simultaneously …
A Stirling Encounter With Harmonic Numbers, Arthur T. Benjamin, Gregory O. Preston '01, Jennifer J. Quinn
A Stirling Encounter With Harmonic Numbers, Arthur T. Benjamin, Gregory O. Preston '01, Jennifer J. Quinn
All HMC Faculty Publications and Research
No abstract provided in this article.
Orthogonal Arrays Of Strength Three From Regular 3-Wise Balanced Designs, Charles J. Colbourn, D. L. Kreher, John P. Mcsorley, D. R. Stinson
Orthogonal Arrays Of Strength Three From Regular 3-Wise Balanced Designs, Charles J. Colbourn, D. L. Kreher, John P. Mcsorley, D. R. Stinson
Articles and Preprints
The construction given in Kreher, J Combin Des 4 (1996) 67 is extended to obtain new infinite families of orthogonal arrays of strength 3. Regular 3-wise balanced designs play a central role in this construction.
Applications Of Graph Theory To Separability, Stephen Young
Applications Of Graph Theory To Separability, Stephen Young
Mathematical Sciences Technical Reports (MSTR)
Let S be a surface with a triangular tiling T. Let R be a reflection a side of one of the triangles; so that R is an orientation reversing isometry of the surface. Define M = {s in S |S : Rs = s}. We then say that the surface S separates along the reflection R if S-R has two components. This paper considers the applications of graph theoretic methods to determining whether a reflection is separating or not and compares the algorithmic efficiency of these methods to the current known methods.
Enumeration Of Matchings In The Incidence Graphs Of Complete And Complete Bipartite Graphs, Nicholas Pippenger
Enumeration Of Matchings In The Incidence Graphs Of Complete And Complete Bipartite Graphs, Nicholas Pippenger
All HMC Faculty Publications and Research
If G = (V, E) is a graph, the incidence graphI(G) is the graph with vertices I ∪ E and an edge joining v ∈ V and e ∈ E when and only when v is incident with e in G. For G equal to Kn (the complete graph on n vertices) or Kn,n (the complete bipartite graph on n + n vertices), we enumerate the matchings (sets of edges, no two having a vertex in common) in I(G), both exactly (in terms of generating …
A Note On Interpolation In The Generalized Schur Class. I. Applications Of Realization Theory, Daniel Alpay, T. Constantinescu, A. Dijksma, J. Rovnyak, A. Dijksma
A Note On Interpolation In The Generalized Schur Class. I. Applications Of Realization Theory, Daniel Alpay, T. Constantinescu, A. Dijksma, J. Rovnyak, A. Dijksma
Mathematics, Physics, and Computer Science Faculty Articles and Research
Realization theory for operator colligations on Pontryagin spaces is used to study interpolation and factorization in generalized Schur classes. Several criteria are derived which imply that a given function is almost the restriction of a generalized Schur function. The role of realization theory in coefficient problems is also discussed; a solution of an indefinite Carathéodory-Fejér problem is obtained, as well as a result that relates the number of negative (positive) squares of the reproducing kernels associated with the canonical coisometric, isometric, and unitary realizations of a generalized Schur function to the number of negative (positive) eigenvalues of matrices derived from …
Some Extensions Of Loewner's Theory Of Monotone Operator Functions, Daniel Alpay, Vladimir Bolotnikov, A. Dijksma, J. Rovnyak, A. Dijksma
Some Extensions Of Loewner's Theory Of Monotone Operator Functions, Daniel Alpay, Vladimir Bolotnikov, A. Dijksma, J. Rovnyak, A. Dijksma
Mathematics, Physics, and Computer Science Faculty Articles and Research
Several extensions of Loewner’s theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar- to matrix-valued functions of an operator argument. A notion of -monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented.
On Minimal Regular Graphs With Given Constraintsm, Che Sheng Gan
On Minimal Regular Graphs With Given Constraintsm, Che Sheng Gan
Student Works (2000-2009)
A classical question in graph theory has been the search of cages. The problem of determining the smallest (r,g)-cage is unsolved for most pairs of (r,g) and is extremely hard in the general case. Finding the crossing number of a graph is another challenging problem in graph theory. The crossing numbers of very few classes of graphs arc known. In this thesis, we would like to (a) investigate those smallest regular graphs with given girths and having small crossing numbers. In particular, we investigate those smallest regular graphs with given girths and crossing number c, for c ∈ {0, 1, …