Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2015
Volume
Vol. 35
Number
No. 3
Description
Journal Volume
Opuscula Mathematica
Vol. 35 (2015)
Projects
Pages
Articles
Hilderbrand's theorem for the essential spectrum
(2015) Bračič, Janko; Diogo, Cristina
We prove a variant of Hildebrandt’s theorem which asserts that the convex hull of the essential spectrum of an operator $A$ on a complex Hilbert space is equal to the intersection of the essential numerical ranges of operators which are similar to $A$. As a consequence, it is given a necessary and sufficient condition for zero not being in the convex hull of the essential spectrum of $A$.
A note on M2-edge colorings of graphs
(2015) Czap, Július
An edge coloring $\varphi$ of a graph $G$ is called an $M_2$-edge coloring if $|\varphi(v)|\le2$ for every vertex $v$ of $G$, where $\varphi(v)$ is the set of colors of edges incident with $v$. Let $K_{2}(G)$ denote the maximum number of colors used in an $M_2$-edge coloring of $G$. Let $G_1$, $G_2$ and $G_3$ be graphs such that $G_1\subseteq G_2\subseteq G_3$. In this paper we deal with the following question: Assuming that $K_2(G_1)=K_2(G_3)$, does it hold $K_2(G_1)=K_2(G_2)=K_2(G_3)$?
Frames and factorization of graph Laplacians
(2015) Jørgensen, Palle E.T.; Tian, Feng
Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $\mathscr{H}_{E}$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $\mathscr{H}_{E}$ we characterize the Friedrichs extension of the $\mathscr{H}_{E}$-graph Laplacian. We consider infinite connected network-graphs $G=(V,E)$, $V$ for vertices, and $E$ for edges. To every conductance function $c$ on the edges $E$ of $G$, there is an associated pair ($\mathscr{H}_{E}$, $\Delta$) where $\mathscr{H}_{E}$ in an energy Hilbert space, and $\Delta\left(=\Delta_{c}\right)$ is the $c$-graph Laplacian; both depending on the choice of conductance function $c$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $\mathscr{H}_{E}$ consisting of dipoles. Now $\Delta$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^{2}\left(V\right)$ and $\mathscr{H}_{E}$. It is known to automatically be essentially selfadjoint as an $l^{2}\left(V\right)$-operator, but generally not as an $\mathscr{H}_{E}$ operator. Hence as an $\mathscr{H}_{E}$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $l^{2}\left(V\right)$.
Ruin probability in a risk model with variable premium intensity and risky investments
(2015) Mišura, Ûliâ Stepanovna; Perestûk, Nikolaj Alekseevič; Ragulìna, Olena Ûrìïvna
We consider a generalization of the classical risk model when the premium intensity depends on the current surplus of an insurance company. All surplus is invested in the risky asset, the price of which follows a geometric Brownian motion. We get an exponential bound for the infinite-horizon ruin probability. To this end, we allow the surplus process to explode and investigate the question concerning the probability of explosion of the surplus process between claim arrivals.
Spectra of some selfadjoint Jacobi operators in the double root case
(2015) Motyka, Wojciech
In this paper we prove a mixed spectrum of Jacobi operators defined by $\lambda_n=s(n)(1+x(n))$ and $q_n=-2s(n)(1+y(n))$, where $(s(n))$ is a real unbounded sequence, $(x(n))$ and $(y(n))$ are some perturbations.