Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2019
Volume
Vol. 39
Number
No. 5
Description
Journal Volume
Opuscula Mathematica
Vol. 39 (2019)
Projects
Pages
Articles
On properties of minimizers of a control problem with time-distributed functional related to parabolic equations
(Wydawnictwa AGH, 2019) Astašova, Irina Viktorovna; Filinovskij, Aleksej Vladislavovič
We consider a control problem given by a mathematical model of the temperature control in industrial hothouses. The model is based on one-dimensional parabolic equations with variable coefficients. The optimal control is defined as a minimizer of a quadratic cost functional. We describe qualitative properties of this minimizer, study the structure of the set of accessible temperature functions, and prove the dense controllability for some set of control functions.
On the imaginary part of coupling resonance points
(Wydawnictwa AGH, 2019) Azamov, Nurulla A.; Daniels, Tom
We prove for rank one perturbations that the imaginary part of a coupling resonance point is inversely proportional by a factor of $-2$ to the rate of change of the scattering phase, as a function of the coupling variable, evaluated at the real part of the resonance point. This equality is analogous to the Breit-Wigner formula from quantum scattering theory. For more general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of coupling resonance points, non-real and real.
On 1-rotational decompositions of complete graphs into tripartite graphs
(Wydawnictwa AGH, 2019) Bunge, Ryan C.
Consider a tripartite graph to be any simple graph that admits a proper vertex coloring in at most 3 colors. Let $G$ be a tripartite graph with $n$ edges, one of which is a pendent edge. This paper introduces a labeling on such a graph $G$ used to achieve 1-rotational $G$-decompositions of $K_{2nt}$ for any positive integer $t$. It is also shown that if $G$ with a pendent edge is the result of adding an edge to a path on $n$ vertices, then $G$ admits such a labeling.
Direct and inverse spectral problems for Dirac systems with nonlocal potentials
(Wydawnictwa AGH, 2019) Dębowska, Kamila; Nižnik, Leonid Pavlovič
The main purposes of this paper are to study the direct and inverse spectral problems of the one-dimensional Dirac operators with nonlocal potentials. Based on informations about the spectrum of the operator, we find the potential and recover the form of the Dirac system. The methods used allow us to reduce the situation to the one-dimensional case. In accordance with the given assumptions and conditions we consider problems in a specific way. We describe the spectrum, the resolvent, the characteristic function etc. Illustrative examples are also given.
Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions
(Wydawnictwa AGH, 2019) Hai, D. D.; Wang, X.
We prove the existence of positive solutions for the $p$-Laplacian problem $\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}$
where $\phi (s)=|s|^{p-2}s$, $p \gt 1$, $H_{i}:\mathbb{R}\rightarrow\mathbb{R}$ can be nonlinear, $i=1,2$, $f:(0,\infty)\rightarrow \mathbb{R}$ is $p$-superlinear or $p$-sublinear at $\infty$ and is allowed be singular $(\pm\infty)$ at $0$, and $\lambda$ is a positive parameter.