We examine series of the form $$\sum_{n=0}^\infty {2n \choose n}^{-1} \frac{(4x^2)^n}{2n+2m+1} \ \ \ \textrm{and}\ \ \ \sum_{n=0}^\infty {2n \choose n}^{-1} \frac{(-4x^2)^n}{2n+2m+1} .$$ In each case, there is an evaluation of the form $(p_m (x) f(x) - q_m (x))/x^{2m}$ where $f(x)$ is a transcendental function and $p_m (x)$ and $q_m (x)$ are polynomials with rational coefficients. We prove that for $\vert x \vert \lt 1$, $$\lim_{m \to \infty} \frac{q_m (x)}{p_m (x)} = f(x) .$$ From this result, we derive recurrences for $\pi$ and for various logarithms.

Sobolev descent has long been established as an efficient method for numerically solving boundary value problems, ordinary differential equations and partial differential equations in a small number of iterations. We demonstrate that for any linear ordinary differential equation with initial value conditions sufficient to assure a unique solution, there exists a Hilbert space in which gradient descent will converge to the solution in one iteration. We provide two elementary examples, one initial value problem and one boundary value problem, demonstrating the effectiveness of the theory in numerical settings. As there are ample efficient numerical methods for solving such problems, the significance of the paper is in the approach and the question it raises. Namely, do such spaces exist for wider classes of differential equations?

Let $p$ be a prime number with $p\neq 2$. We consider second order linear recurrence relations of the form $S_n=aS_{n-1}+bS_{n-2}$ over the finite field $Z_p$ (we assume $b\neq 0$). Results regarding the period and distribution of elements in the sequence $\{ S_0, S_1, \ldots \}$ are well-known (see works by Kuipers, Niederreiter, Wall, and Webb). We examine these second order recurrences using matrices, groups, and $G$-sets.

We derive interesting combinatorial identities by employing the instantaneous centralized moments obtained from the Lévy measure of the Kou jump-diffusion market.

In the study of various objects indexed by permutations, a natural notion of minimal excluded structure, now known as a permutation pattern, has emerged and found diverse applications. One of the earliest results from the study of permutation pattern avoidance in enumerative combinatorics is that the Catalan numbers $c_n$ count the permutations of size $n$ that avoid any fixed pattern of size three. We refine this result by enumerating the permutations that avoid a given pattern of size three and have a given letter in the first position of their one-line notation. Since there are two parameters, we obtain triangles of numbers rather than sequences. Our main result is that there are two essentially different triangles for any of the patterns of size three, and each of these triangles generalizes the Catalan sequence in a natural way. All of our proofs are bijective, and relate the permutations being counted to recursive formulas for the triangles.

The chromatic index of a graph $G$ is most often defined to be the minimum size of a partition of the edge set of $G$ into matchings. An equivalent but different definition is the minimum size of a cover of the edge set of $G$ by matchings. We consider the analogous problem of covering the edge set of $G$ by subgraphs that are vertex-disjoint unions of cliques, known as equivalence graphs. The minimum size of such a cover is the equivalence number of $G$. We compute the equivalence number of the line graph of a clique on at most 12 vertices. We also construct a particular type of cover to show that, for all graphs $G$ on at most n vertices, the equivalence number of the line graph of $G$ has an upper bound on the order of log n. Finally, we show that if $G$ is a clique on 13 vertices then the minimum size of this particular cover is 5.

The design and implementation analysis of a $k$-step linear multistep method for direct integration of second order initial value problems of ordinary differential equations without reformulation into first order systems is discussed. The derivation of the method and analysis of its basic properties are adopted from the Taylor series expansion and Dahlquist stability model test methods. The result when examined with step-number $k=6$ shows that the scheme is symmetric, consistent, zero-stable, and convergent.

This paper deals with functionally Hausdorff spaces. Some separation axioms which are introduced recently are studied in this paper. An interesting categorical properties of functionally Hausdorff spaces are given. Finally, we characterize topological spaces for which the functionally Hausdorff-reflection is a spectral space.

It is well-known that solutions to integer programming problems usually cannot be obtained by simply solving the corresponding linear programming relaxation. There are, however, examples of integer programming problems whose solutions can be obtained by simply solving the linear program and ignoring the integer constraints. Proving that these particular models have this trait is generally beyond the scope of a beginning course in operations research. In this paper an integer programming model, with only two constraints, is presented whose solution can be directly obtained using the standard simplex method. A proof is provided that makes a connection between analysis and operations research.

In the note, two theorems are presented concerning the well-known linear differential equation $x''+a(t)x=0$. While the results are not new, the proofs presented simplify previous work since the Gronwall inequality is avoided which is the usual case.