The steady states of a mathematical model for the dynamics of Chagas disease, developed by Spagnuolo et al., are studied and numerically simulated. The model consists of a system of four nonlinear ordinary differential equations for the total number of domestic carrier insects, and the infected insects, infected humans, and infected domestic animals. The equation for the vector dynamics has a growth rate of the blowfly type with a delay. In the parameter range of interest, the model has two unstable disease-free equilibria and a globally asymptotically stable (GAS) endemic equilibrium. Numerical simulations, based on the fourth-order Adams–Bashforth predictor corrector scheme for ODEs, depict the various cases.

Repulsive point processes arise in models where competition forces entities to be more spread apart than if placed independently. Simulation of these types of processes can be accomplished using dominated coupling from the past with a running time that depends on the intensity of the number of points. These algorithms usually exhibit what is called an artificial phase transition, where below a critical intensity the algorithm runs in finite expected time, but above the critical intensity the expected number of steps is infinite. Here the artificial phase transition is examined. In particular, an earlier lower bound on this artificial phase transition is improved by including a new type of term in the analysis. In addition, the results of computer experiments to locate the transition are presented.

We establish that the only positive integral solutions common to the two Pell’s equations $U^2-18V^2=-119$ and $Z^2-29V^2=-196$ are $U=41$, $V=10$ and $Z=52$.

Given two points inside a circle in the hyperbolic plane, we study the problem of finding an isosceles triangle inscribed in the circle so that the two points belong to distinct congruent sides. By means of a reduction to the corresponding result in Euclidean geometry, we prove that this problem cannot generally be solved with hyperbolic ruler and compass.

Using the concepts of Bochner measurability and Bochner space, we introduce a continuous version of $\left(p,Y\right)$-operator frames for a Banach space. We also define independent Bochner $\left(p,Y\right)$-operator frames for a Banach space and discuss some properties of Bochner $\left(p,Y\right)$-operator frames.

A bifurcus semigroup is a semigroup in which every nonunit nonatom can be written as the product of exactly two atoms. We generalize this notion to $k$-furcus semigroups: every element that can be factored as the product of at least $k$ nonunits can be factored as the product of exactly $k$ atoms. We compute some factorization-theoretic invariants of $k$-furcus semigroups that generalize the bifurcus results. We then define two variations on the $k$-furcus property, one stronger (presumabaly strictly) and the other strictly weaker than the $k$-furcus property.

We report the results of a project with Six Mile Creek Winery in Ithaca, NY, in which we investigate possible climate impacts on the area wine industry. Specifically, we examine winter minimum temperatures in Ithaca since temperatures below $-15^{\circ }$F damage buds of French-American hybrid grapes and temperatures below $-5^{\circ }$F affect Vinifera grape varieties. We used the generalized extreme value distribution to model the winter minimum and adjusted this model based on climate simulation data.

A Scramble Squares puzzle is made up of nine square pieces such that each edge of each piece contains half of an image. A solution to the puzzle is obtained when the pieces are arranged in a $3\times 3$ grid so that the adjacent edges of different pieces together make up a complete image. We describe a graph-theoretical approach to solving Scramble Squares puzzles and a method for decreasing randomness in the backtracking solution algorithm.

We study the notion of $n$-diameter for sets of constant width. A convex set in the plane is said to be of constant width if the distance between two parallel support lines is constant, independent of the direction. The Reuleaux triangles are the well-known examples of sets of constant width that are not disks. The $n$-diameter of a compact set $E$ in the plane is

$$d_n\left(E\right)=max\left(\right.\prod _{1\le i<j\le n}|z_i-z_j|{\left)\right.}^{\frac{2}{n\left(n-1\right)}},$$

where the maximum is taken over all $z_k\in E$, $k=1,2,\dots ,n$. We prove that if $n=5$, then the Reuleaux $n$-gons have the largest $n$-diameter among all sets of given constant width. The proof is based on the solution of an extremal problem for $n$-diameter.

We prove that a permutation in the Bruhat order on twisted involutions is Boolean if and only if it avoids the following patterns: 4321, 3421, 4312, 4231, 32541, 52143, 351624, 456123, 426153, 321654, 561234, 345612, 3416275, 3561274, 1532746, 4517236, 34127856, 35172846, and 36712845. This result answers a question proposed by Hultman and Vorwerk. Our technique provides an application of the pictorial representation of the Bruhat order given by Incitti.

In the sport of cricket, as with any other sport, spectators and officials would like the games to be as fair as possible. To this end, we evaluate methods used to determine the winner of interrupted games using statistical accuracy. In the traditional One Day International cricket matches, the current Duckworth–Lewis (DL) method and the discounted most productive overs (DMPO) method are each used for predicting the winner. However, with the growing popularity of shorter Twenty20 matches, a new Bhattacharya–Gill–Swartz (BGS) method has also been introduced. We created both classical and Bayesian intervals to estimate the true accuracy of each. Using past game data from 2005–2010, we compared the DL, DMPO and BGS methods using the new accuracy intervals and receiver operating characteristic (ROC) curves.

We look at hexagons whose vertex triangles have equal area, and identify necessary conditions for these hexagons to also have vertex quadrilaterals with equal area. We discover a method for creating a hexagon whose vertex quadrilaterals have equal area without necessarily having vertex triangles of equal area. Finally, we generalize the process to build any polygon with an even number of sides to have certain vertex polygons with equal area.

The sum of distances between vertices of a tree has been considered from many aspects. The question of characterizing the extremal trees that maximize or minimize various such “distance-based” graph invariants has been extensively studied. Such invariants include, to name a few, the sum of distances between all pairs of vertices and the sum of distances between all pairs of leaves. With respect to the distances between internal vertices, we provide analogous results that characterize the extremal trees that minimize the value of any nonnegative and nondecreasing function of internal distances among trees with various constraints.