Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2013
Volume
Vol. 33
Number
No. 4
Description
Journal Volume
Opuscula Mathematica
Vol. 33 (2013)
Projects
Pages
Articles
Concavity of solutions of a 2n-th order problem with symmetry
(2013) Al Twaty, Abdulmalik; Eloe, Paul W.
In this article we apply an extension of a Leggett-Williams type fixed point theorem to a two-point boundary value problem for a $2n$-th order ordinary differential equation. The fixed point theorem employs concave and convex functionals defined on a cone in a Banach space. Inequalities that extend the notion of concavity to $2n$-th order differential inequalities are derived and employed to provide the necessary estimates. Symmetry is employed in the construction of the appropriate Banach space.
Global existence and asymptotic behavior for a nonlinear degenerate SIS model
(2013) Ali Ziane, Tarik
In this paper we investigate the global existence and asymptotic behavior of a reaction diffusion system with degenerate diffusion arising in the modeling and the spatial spread of an epidemic disease.
On the longest path in a recursively partitionable graph
(2013) Bensmail, Julien
A connected graph $G$ with order $n \geq 1$ is said to be recursively arbitrarily partitionable (R-AP for short) if either it is isomorphic to $K_1$, or for every sequence $(n_1, \ldots , n_p)$ of positive integers summing up to n there exists a partition $(V_1, \ldots , V_p)$ of $V(G)$ such that each $V_i$ induces a connected R-AP subgraph of $G$ on $n_i$ vertices. Since previous investigations, it is believed that a R-AP graph should be »almost traceable« somehow. We first show that the longest path of a R-AP graph on $n$ vertices is not constantly lower than $n$ for every $n$. This is done by exhibiting a graph family $C$ such that, for every positive constant $c \geq 1$, there is a R-AP graph in $C$ that has arbitrary order $n$ and whose longest path has order $n-c$. We then investigate the largest positive constant $c' \lt 1$ such that every R-AP graph on n vertices has its longest path passing through $n \cdot c'$ vertices. In particular, we show that $c' \leq \frac{2}{3}.$ This result holds for R-AP graphs with arbitrary connectivity.
A note on k-Roman graphs
(2013) Bouchou, Ahmed; Blidia, Mostafa; Chellali, Mustapha
Let $G=(V,E)$ be a graph and let $k$ be a positive integer. A subset $D$ of $V(G)$ is a $k$-dominating set of $G$ if every vertex in $V\left( G\right) \backslash D$ has at least $k$ neighbours in $D$. The $k$-domination number $\gamma_{k}(G)$ is the minimum cardinality of a $k$-dominating set of $G$. A Roman $k$-dominating function on $G$ is a function $f\colon V(G)\longrightarrow\{0,1,2\}$ such that every vertex u for which $f(u)=0$ is adjacent to at least $k$ vertices $v_{1},v_{2},\ldots ,v_{k}$ with $f(v_{i})=2$ for $i=1,2,\ldots ,k.$ The weight of a Roman $k$-dominating function is the value $f(V(G))=\sum_{u\in V(G)}f(u)$ and the minimum weight of a Roman $k$-dominating function on $G$ is called the Roman $k$-domination number $\gamma_{kR}\left( G\right)$ of $G$. A graph $G$ is said to be a $k$-Roman graph if $\gamma_{kR}(G)=2\gamma_{k}(G).$ In this note we study $k$-Roman graphs.
Chaotic expansion in the G-expectation space
(2013) Boutabia, Hacѐne; Grabsia, Imen
In this paper, we are motivated by uncertainty problems in volatility. We prove the equivalent theorem of Wiener chaos with respect to $G$-Brownian motion in the framework of a sublinear expectation space. Moreover, we establish some relationship between Hermite polynomials and $G$-stochastic multiple integrals. An equivalent of the orthogonality of Wiener chaos was found.