Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2019
Volume
Vol. 39
Number
No. 3
Description
Journal Volume
Opuscula Mathematica
Vol. 39 (2019)
Projects
Pages
Articles
Oscillations of equations caused by several deviating arguments
(Wydawnictwa AGH, 2019) Chatzarakis, George E.
Linear delay or advanced differential equations with variable coefficients and several not necessarily monotone arguments are considered, and some new oscillation criteria are given. More precisely, sufficient conditions, involving $\lim\sup$ and $\lim\inf$, are obtained, which essentially improve several known criteria existing in the literature. Examples illustrating the results are also given, numerically solved in MATLAB.
A partial refining of the Erdős-Kelly regulation
(Wydawnictwa AGH, 2019) Górska, Joanna; Skupień, Zdzisław
The aim of this note is to advance the refining of the Erdős-Kelly result on graphical inducing regularization. The operation of inducing regulation (on graphs or multigraphs) with prescribed maximum vertex degree is originated by D. König in 1916. As is shown by Chartrand and Lesniak in their textbook Graphs & Digraphs (1996), an iterated construction for graphs can result in a regularization with many new vertices. Erdős and Kelly have presented (1963, 1967) a simple and elegant numerical method of determining for any simple $n$-vertex graph $G$ with maximum vertex degree $\Delta$, the exact minimum number, say $\theta =\theta(G)$, of new vertices in a $\Delta$-regular graph $H$ which includes $G$ as an induced subgraph. The number $\theta(G)$, which we call the cost of regulation of $G$, has been upper-bounded by the order of $G$, the bound being attained for each $n \geq 4$, e.g. then the edge-deleted complete graph $K_{n}-e$ has $\theta=n$. For $n \geq 4$, we present all factors of $K_n$ with $\theta=n$ and next $\theta=n-1$. Therein in case $\theta=n-1$ and $n$ odd only, we show that a specific extra structure, non-matching, is required.
On the zeros of the Macdonald functions
(Wydawnictwa AGH, 2019) Hamana, Yuji; Matsumoto, Hiroyuki; Shirai, Tomoyuki
We are concerned with the zeros of the Macdonald functions or the modified Bessel functions of the second kind with real index. By using the explicit expressions for the algebraic equations satisfied by the zeros, we describe the behavior of the zeros when the index moves. Results by numerical computations are also presented.
Decomposing complete 3-uniform hypergraph Kn(3) into 7-cycles
(Wydawnictwa AGH, 2019) Meihua; Guan, Meiling; Jirimutu
We use the Katona-Kierstead definition of a Hamiltonian cycle in a uniform hypergraph. A decomposition of complete $k$-uniform hypergraph $K^{(k)}_{n}$ into Hamiltonian cycles was studied by Bailey-Stevens and Meszka-Rosa. For $n\equiv 2,4,5\pmod 6$, we design an algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of $K^{(3)}_{n}$ into 5-cycles has been presented for all admissible $n \leq 17$, and for all $n=4^{m}+1$ when $m$ is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we show if $42~|~(n-1)(n-2)$ and if there exist $\lambda=\frac{(n-1)(n-2)}{42}$ sequences $(k_{i_{0}},k_{i_{1}},\ldots,k_{i_{6}})$ on $D_{all}(n)$, then $K^{(3)}_{n}$ can be decomposed into 7-cycles. We use the method of edge-partition and cycle sequence. We find a decomposition of $K^{(3)}_{37}$ and $K^{(3)}_{43}$ into 7-cycles.
Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term
(Wydawnictwa AGH, 2019) Nakao, Mitsuhiro
We give an existence theorem of global solution to the initial-boundary value problem for $u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)$ under some smallness conditions on the initial data, where $\sigma (v^2)$ is a positive function of $v^{2}\neq 0$ admitting the degeneracy property $\sigma(0)=0$. We are interested in the case where $\sigma(v^{2})$ has no exponent $m \geq 0$ such that $\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0$. A typical example is $\sigma(v^2)=\operatorname{log}(1+v^2)$. $f(u)$ is a function like $f=|u|^{\alpha} u$. A decay estimate for $\|\nabla u(t)\|_{\infty}$ is also given.