In this paper, we study the properties of the generalized harmonic numbersHn,k,r(α, β). In particular, by means of the method of coefficients, generating functions and Riordan arrays, we establish some identities involving the numbers Hn,k,r(α, β), Cauchy numbers, generalized Stirling numbers, Genocchi numbers and higher order Bernoulli numbers. Furthermore, we obtain the asymptotic values of s...

For each natural number n, let Cn represent the set of all 2-colorings of the set {1, 2, . . . , n}. Given a natural number n and a coloring ∆ ∈ Cn, let S(∆) represent the set S(∆) = {x3 | ∃ x1, x2 s.t. x1 + x2 = x3 and ∆(x1) = ∆(x2) = ∆(x3)}. Given a natural number n, let f(n) = min ∆∈Cn S(∆). For all natural numbers n and r where n2 ≤ r ≤ n, let Cn,r represent the set of all 2-colorings of th...

For any lacunary sequence $theta = (k_{r})$, we define the concepts of $S_{theta}-$limit point and $S_{theta}-$cluster point of a sequence of fuzzy numbers $X = (X_{k})$. We introduce the new sets  $Lambda^{F}_{S_{theta}}(X)$, $Gamma^{F}_{S_{theta}}(X)$ and prove some inclusion relaions between these and the sets $Lambda^{F}_{S}(X)$, $Gamma^{F}_{S}(X)$ introduced in ~cite{Ayt:Slpsfn} by Aytar [...

Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed find asymptotic results for very large graphs, so it seems that the not suitable finding numbers. But this intuition wrong, and we will develop technique do just in paper. We new upper bounds many graph hypergraph As result, prove values $R(K_4^-,K_4^-,K_4^-)=28$, $R(...

BACKGROUND National Strategic Plans (NSPs) for HIV have become foundational documents that frame responses to HIV. Both Global Fund and PEPFAR require coordination with NSPs as a component of their operations. Despite the role of NSPs in country planning, no rigorous assessment of NSP targets and performance outcomes exists. We performed a quantitative analysis of the quality of NSP indicators ...

A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.

In this note, we provide a combinatorial proof of a generalized recurrence formula satisfied by the Stirling numbers of the second kind. We obtain two extensions of this formula, one in terms of r-Whitney numbers and another in terms of q-Stirling numbers of Carlitz. Modifying our proof yields analogous formulas satisfied by the r-Stirling numbers of the first kind and by the r-Lah numbers.

By the Von Neumann regular graph of R, we mean the graph that its vertices are all elements of R such that there is an edge between vertices x,y if and only if x+y is a von Neumann regular element of R, denoted by G_Vnr (R). For a commutative ring R with unity, x in R is called Von Neumann regular if there exists x in R such that a=a2 x. We denote the set of Von Neumann regular elements by V nr...

Recent research into analog computing has introduced new notions of real numbers. Huang, Klinge, Lathrop, Li, and Lutz defined a notion numbers in real-time with chemical reaction networks (CRNs), introducing the classes $\mathbb{R}_\text{LCRN}$ (the class all Lyapunov CRN-computable numbers) $\mathbb{R}_\text{RTCRN}$ numbers). In their paper, they show inclusion algebraic $ALG \subseteq \mathb...

Generalized estimating equation solvers in R only allow for a few pre-determined options for the link and variance functions. We provide a package, geeM, which is implemented entirely in R and allows for user specified link and variance functions. The sparse matrix representations provided in the Matrix package enable a fast implementation. To gain speed, we make use of analytic inverses of the...