The well-known ``do dogs know calculus'' problem optimizes the travel time from an onshore dog to an offshore stick, given different running and swimming speeds and a straight coastline. Here, we optimize the travel time between two points on the boundary of a rectangular swimming pool, assuming that running speed along the edge differs from the swimming speed.

A hereditary class on a set $X$ is a nonempty collection of subsets of $X$ closed under the hereditary property. In this paper, we define and study the notion of Lindelöfness in generalized topological spaces with respect to a hereditary class called, $\mu\mathcal{H}$-Lindelöf spaces and discuss their properties.

A topological space $X$ is $\omega$-jointly metrizable if for every countable collection of metrizable subspaces of $X$, there exists a metric on $X$ which metrizes every member of this collection. Although the Sorgenfrey line is not jointly partially metrizable, we prove that it is $\omega$-jointly metrizable.

We show that if $X$ is a regular first countable $T_{1}$-space such that $X$ is the union of two subspaces one of which is separable and metrizable, and the other is closed and discrete, then $X$ is $\omega$-jointly metrizable.

Consider the following game: Player A chooses an integer $\alpha$ between $1$ and $n$ for some integer $n\geq1$, but does not reveal $\alpha$ to Player B. Player B then asks Player A a yes/no question about which number Player A chose, after which Player A responds truthfully with either ``yes'' or ``no.'' After a predetermined number $m$ of questions have been asked ($m\geq 1$), Player B must attempt to guess the number chosen by Player A. Player B wins if she guesses $\alpha$. The purpose of this note is to find, for every $m\geq 1$, all canonical $m$-question algorithms which maximize the probability of Player B winning the game (the notion of ``canonical algorithm'' will be made precise in Section 3).

The following is a discussion regarding a specific class of operators acting on the space of entire functions, denoted $H(\mathbb{C})$. A diagonal operator $D$ on $H(\mathbb{C})$ is defined to be a continuous linear map, sending $H(\mathbb{C})$ into $H(\mathbb{C})$, that has the monomials $z^n$ as its eigenvectors and $\{\lambda_n\}$ as the corresponding eigenvalues. A closed subspace $M$ is invariant for $D$ if $Df\in M$ for all $f\in M$. The study of invariant subspaces is a popular topic in modern operator theory. We observe that the closed linear span of the orbit, which we write $\overline{\mbox{span}}\{D^kf:k\geq0\}=\overline{\mbox{span}}\{\sum^{\infty}_{n=0}a_n\lambda_n^kz^n:k\geq0\}$, is the smallest closed invariant subspace for $D$ containing $f$. If every invariant subspace for a diagonal operator $D$ on $H(\mathbb{C})$ can be expressed as a closed linear span of some subset of the eigenvectors of $D$, we say that $D$ admits spectral synthesis on $H(\mathbb{C})$. Until recently, it was not known whether or not every diagonal operator on $H(\mathbb{C})$ admitted spectral synthesis. This article focuses on using techniques from calculus and linear algebra to construct a class of operators which fail spectral synthesis on $H(\mathbb{C})$. If the reader is not familiar with the operator theory definitions provided in the background, he or she can still appreciate the construction of an interesting infinite series relying on properties of logarithms, various convergence tests, and Cramer's Rule.

Let $\psi$ be an entire self-map of the $n$-dimensional Euclidean complex space $\mathbb{C}^n$ and $u$ be an entire function on $\mathbb{C}^n$. A weighted composition operator induced by $\psi$ with weight $u$ is given by $(uC_{\psi}f)(z)= u(z)f(\psi(z))$, for $z \in \mathbb{C}^n$ and $f$ is the entire function on $\mathbb{C}^n$. In this paper, we study weighted composition operators acting between generalized Fock-types spaces. We characterize the boundedness and compactness of these operators act between $\mathcal{F}_{\phi}^{p}(\mathbb{C}^n)$ and $\mathcal{F}_{\phi}^{q}(\mathbb{C}^n)$ for $0\lt p, q\leq\infty$. Moreover, we give estimates for the Fock-norm of $uC_{\psi}: \mathcal{F}_{\phi}^{p}\rightarrow \mathcal{F}_{\phi}^{q}$ when $0\lt p, q\lt \infty$, and also when $p=\infty$ and $0\lt q\lt \infty$.

Regression trees are an alternative to classical linear regression models that seek to fit a piecewise linear model to data. The structure of regression trees makes them well-suited to the modeling of data containing outliers. We propose an algorithm that takes advantage of this feature in order to automatically detect outliers. This new algorithm performs well on the four test datasets [7] that are considered to be necessary for a valid outlier detection algorithm in a linear regression context, even though regression trees lack the global linearity assumption. We also show the practical use of this approach in detecting outliers in an ecological dataset collected in the Shenandoah Valley.

Although the Cayley-Dickson algebras are twisted group algebras, little attention has been paid to the nature of the Cayley-Dickson twist. One reason is that the twist appears to be highly chaotic and there are other interesting things about the algebras to focus attention upon. However, if one uses a doubling product for the algebras different from yet equivalent to the ones commonly used, and if one uses a numbering of the basis vectors different from the standard basis a quite beautiful and highly periodic twist emerges. This leads easily to a simple closed form equation for the product of any two basis vectors of a Cayley-Dickson algebra.

The (outer) planar coarseness of a graph is the largest number of pairwise-edge-disjoint non-(outer)planar subgraphs. It is shown that the maximum outerplanar coarseness, over all $n$-vertex planar graphs, lies in the interval $\;\big [\lfloor (n-2)/3 \rfloor, \lfloor (n-2)/2 \rfloor \big ]$.