Abstract:
We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit formulas, recurrence relations, and ordinary and exponential generating functions. A generalization to more than one fixed length is also considered.

Abstract:
We investigate tournaments with a specified score vector having additional structure: loopy tournaments in which loops are allowed, Hankel tournaments which are tournaments symmetric about the Hankel diagonal (the anti-diagonal), and combinatorially skew-Hankel tournaments which are skew-symmetric about the Hankel diagonal. In each case, we obtain necessary and sufficient conditions for existence, algorithms for construction, and switches which allow one to move from any tournament of its type to any other, always staying within the defined type.

Abstract:
In celebration of the distinguished achievements of Professor Tsuyoshi Ando in matrix analysis and operator theory, we conducted an interview with him via email. This paper presents Professor Ando's responses to several questions we gave him regarding his education and life as a mathematician.

Abstract:
In alternating sign matrices the first and last nonzero entry in each row and column is specified to be +1. Such matrices always exist. We investigate a generalization by specifying independently the sign of the first and last nonzero entry in each row and column to be either a +1 or a -1. We determine necessary and sufficient conditions for such matrices to exist.

Abstract:
For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to be the largest number of 1s in A with at most one 1 in each column and at most t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize some basic results for the term rank to the t-term rank, including a formula for the maximum term rank over a nonempty class of (0,1)-matrices with the the same row sum and column sum vectors. We also show the surprising result that in such a class there exists a matrix which realizes all of the maximum terms ranks between 1 and t.

Abstract:
We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices, we find the minimum number of nonzero entries and characterize the case of equality. We also study symmetric alternating sign matrices, in particular, those with only zeros on the main diagonal. These give rise to alternating signed graphs without loops, and we determine the maximum number of edges in such graphs. We also consider n by n alternating sign matrices whose patterns are maximal within the class of all n by n alternating sign matrices.

Abstract:
We define the cyclic matching sequencibility of a graph to be the largest integer $d$ such that there exists a cyclic ordering of its edges so that every $d$ consecutive edges in the cyclic ordering form a matching. We show that the cyclic matching sequencibility of $K_{2m}$ and $K_{2m+1}$ equal $m-1$.

Abstract:
Let ${\mathcal T}(n,m)$ and ${\mathcal F}(n,m)$ denote the classes of weighted trees and forests, respectively, of order $n$ with the positive integral weights and the fixed total weight sum $m$, respectively. In this paper, we determine the minimum energies for both the classes ${\mathcal T}(n,m)$ and ${\mathcal F}(n,m)$. We also determine the maximum energy for the class ${\mathcal F}(n,m)$. In all cases, we characterize the weighted graphs whose energies reach these extremal values. We also solve the similar maximum energy and minimum energy problems for the classes of (0,1) weighted trees and forests.

Abstract:
A novel approach for improving antenna bandwidth is described using a 6-element Yagi-Uda array as an example. The new approach applies Central Force Optimization, a deterministic metaheuristic, and Variable Z_{0} technology, a novel, proprietary design and optimization methodology, to produce an array with 33.09% fractional impedance bandwidth. This array’s performance is compared to its CFO-optimized Fixed Z_{0}counterpart, and to the performance of a 6-ele- ment Dominating Cone Line Search-optimized array. Both CFO-optimized antennas exhibit better performance than the DCLS array, especially with respect to impedance bandwidth. Although the Yagi-Uda antenna was chosen to illustrate this new approach to antenna design and optimization, the methodology is entirely general and can be applied to any antenna against any set of performance objectives.

Abstract:
Bell’s theorem,
first presented by John Bell in 1964, has been used for many years to prove that
no classical theory can ever match verified quantum mechanical predictions for entangled
particles. By relaxing the definition of entangled slightly, we have found a mathematical
solution for two entangled photons that produces the familiar quantum mechanical
counting statistics without requiring a non-local theory such as quantum mechanics.
This solution neither is claimed to be unique nor represents an accurate model
of photonic interactions. However, it is an existence proof that there are local
models of photonic emission that can reproduce quantum statistics.