Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2017
Volume
Vol. 37
Number
No. 5
Description
Journal Volume
Opuscula Mathematica
Vol. 37 (2017)
Projects
Pages
Articles
Block colourings of 6-cycle systems
(Wydawnictwa AGH, 2017) Bonacini, Paola; Gionfriddo, Mario; Marino, Lucia
Let $\Sigma=(X,\mathcal{B})$ be a $6$-cycle system of order $v$, so $v\equiv 1,9\mod 12$. A $c$-colouring of type $s$ is a map $\phi\colon\mathcal {B}\rightarrow \mathcal{C}$, with $C$ set of colours, such that exactly $c$ colours are used and for every vertex $x$ all the blocks containing $x$ are coloured exactly with s colours. Let $\frac{v-1}{2}=qs+r$, with $q, r\geq 0$. $\phi$ is equitable if for every vertex x the set of the $\frac{v-1}{2}$ blocks containing $x$ is partitioned in $r$ colour classes of cardinality $q+1$ and $s-r$ colour classes of cardinality $q$. In this paper we study bicolourings and tricolourings, for which, respectively, $s=2$ and $s=3$, distinguishing the cases $v=12k+1$ and $v=12k+9$. In particular, we settle completely the case of $s=2$, while for $s=3$ we determine upper and lower bounds for $c$.
Semicircular elements induced by p-adic number fields
(Wydawnictwa AGH, 2017) Cho, Ilwoo; Jørgensen, Palle E.T.
In this paper, we study semicircular-like elements, and semicircular elements induced by $p$-adic analysis, for each prime $p$. Starting from a $p$-adic number field $\mathbb{Q}_{p}$, we construct a Banach $∗$-algebra $\mathfrak{LS}_{p}$, for a fixed prime $p$, and show the generating elements $Q_{p,j}$ of $\mathfrak{LS}_{p}$ form weighted-semicircular elements, and the corresponding scalar-multiples $\Theta_{p,j}$ of $Q_{p,j}$ become semicircular elements, for all $j\in\mathbb{Z}$. The main result of this paper is the very construction of suitable linear functionals $\tau_{p,j}^{0}$ on $\mathfrak{LS}_{p}$, making $Q_{p,j}$ be weighted-semicircular, for all $j\in\mathbb{Z}$.
Existence of minimal and maximal solutions to RL fractional integro-differential initial value problems
(Wydawnictwa AGH, 2017) Denton Zachary; Ramírez, J.D.
In this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function. We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions. In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences. Finally, we illustrate our results with an example.
On 3-total edge product cordial connected graphs
(Wydawnictwa AGH, 2017) Ivančo, Jaroslav
A $k$-total edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize connected graphs of order at least 15 admitting a 3-total edge product cordial labeling.
On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model
(Wydawnictwa AGH, 2017) Pukach, Petro; Il'kiv, Volodymyr; Nytrebych, Zinovii; Vovk, Myroslava
The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable.