Let $E^{2}_{T}$ be the group of all isometries of the $2$-dimensional taxicab space $R^{2}_{T}$. For the taxicab group $E^{2}_{T}$, the taxicab type of curves is introduced. All possible taxicab types are found. For every taxicab type, an invariant parametrization of a curve is described. The $E^{2}_{T}$-equivalence of curves is reduced to the problem of the $E^{2}_{T}$-equivalence of paths.

An interactive approach to constructing small LR(1) grammars is presented. An example involving parsing permutations is used to illustrate the approach.

Let $p$ be a prime number with $p\neq 2$. We consider sequences generated by $n$th order linear recurrence relations over the finite field $Z_p$. In the first part of this paper we generalize some of the ideas in [6] to $n$th order linear recurrences. We then consider the case where the characteristic polynomial of the recurrence has one root in $Z_p$ of multiplicity $n$. In this case, we show that the corresponding recurrence can be generated by a relatively simple matrix.

The classical Intermediate Value Theorem (IVT) states that if $f$ is a continuous real-valued function on an interval $[a,b]\subseteq\mathbb{R}$ and if $y$ is a real number strictly between $f(a)$ and $f(b)$, then there exists a real number $x\in(a,b)$ such that $f(x)=y$. The standard counterexample showing that the converse of the IVT is false is the function $f$ defined on $\mathbb{R}$ by $f(x):=\sin(\frac{1}{x})$ for $x\neq 0$ and $f(0):=0$. However, this counterexample is a bit weak as $f$ is discontinuous only at $0$. In this note, we study a class of strong counterexamples to the converse of the IVT. In particular, we present several constructions of functions $f \colon \mathbb{R}\rightarrow\mathbb{R}$ such that $f[I]=\mathbb{R}$ for every nonempty open interval $I$ of $\mathbb{R}$ ($f[I]:=\{f(x):x\in I\}$). Note that such an $f$ clearly satisfies the conclusion of the IVT on every interval $[a,b]$ (and then some), yet $f$ is necessarily nowhere continuous! This leads us to a more general study of topological spaces $X=(X,\mathcal{T})$ with the property that there exists a function $f \colon X\rightarrow X$ such that $f[O]=X$ for every nonvoid open set $O\in\mathcal{T}$.

Istvan Beck introduced the zero-divisor graph in 1988. We explore the directed and undirected zero-divisor graphs of the rings of $2 \times 2$ upper triangular matrices mod $n$, denoted by $\Gamma(T_2(n))$ and $\tilde{\Gamma}(T_2(n))$, respectively. For prime $p$, we completely characterize the graph $\Gamma(T_2(p))$ by partitioning $T_2(p)$, and prove several key properties of the graphs using this approach. We establish additional properties of $\Gamma(T_2(n))$ for arbitrary $n$. We prove that $\Gamma(T_2(n))$ is Hamiltonian if and only if $n$ is prime, and we give explicit formulas for the edge connectivity and clique number of $\Gamma(T_2(n))$ in terms of the prime factorization of $n$.

This expository note aims at illustrating weak convergence of probability measures from a broader view than a previously published paper. Though the results are standard for functional analysts, this approach is rarely known by statisticians and our presentation gives an alternative view to most standard probability textbooks. In particular, this functional approach clarifies the underlying topological structure of weak convergence. We hope this short note is helpful for those who are interested in weak convergence as well as instructors of measure theoretic probability.

There is a magnificent mathematical gem missing from most numerical analysis curricula. A literature search of numerical analysis and mathematical modeling texts indicates its absence. What is missing from the numerical analysis toolbox and why isn't it there? The missing tool is the Kalman filter. The Kalman filter requires a modest knowledge of statistics. Is this why it is missing from the toolbox? Read and reach your own conclusion whether this piece of mathematics should be part of a numerical analysis or mathematical modeling curriculum.

We study a batch arrival queueing system of phase vacation with two stages of service based on a Bernoulli schedule. A single server provides essential service to all arriving customers with service time following a general distribution. After two stages of service completion, the server leaves for phase one vacation of random length with probability $p$ or to continue staying in the system with probability $1-p$. As soon as the completion of phase one vacation, the server undergoes phase two vacation. On completion of two heterogeneous phases of vacation the server returns back to the system. The vacation times are assumed to be general. The server is interrupted and the service interruption follows an exponential distribution. The arrivals follow a Poisson distribution. Using supplementary variable technique, the Laplace transforms of time dependent probabilities of system state are derived. From this we deduce the steady state results. We also obtain the average queue size and average waiting time.