Artykuły (CN-OpMath)

Permanent URI for this collectionhttps://repo.agh.edu.pl/handle/AGH/102812

Artykuły czasopisma Opuscula Mathematica

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Access status: Open Access ,
Approximate solutions of a singular integral equation with Cauchy kernels in the quarter plane
(2008) Pylak, Dorota
In the paper, we present explicit formulae for the solution of the singular integral equation with Cauchy kernels in the quarter plane. Next, Jacobi and Chebyshev polynomials are used to derive approximate solutions of this equation.
Access status: Open Access ,
Fréchet differential of a power series in Banach algebras
(Wydawnictwa AGH, 2010) Silvestri, Benedetto
We present two new forms in which the Fréchet differential of a power series in a unitary Banach algebra can be expressed in terms of absolutely convergent series involving the commutant $C(T) : A \mapsto [A,T]$. Then we apply the results to study series of vector-valued functions on domains in Banach spaces and to the analytic functional calculus in a complex Banach space.
Access status: Open Access ,
A new composition theorem for Sp-weighted pseudo almost periodic functions and applications to semilinear differential equations
(2011) Zhao, Zhi-Han; Chang, Yong-Kui; N'Guérékata, G. M.
In this paper, we establish a new composition theorem for $S^p$-weighted pseudo almost periodic functions under weaker conditions than the Lipschitz ones currently encountered in the literatures. We apply this new composition theorem along with the Schauder's fixed point theorem to obtain new existence theorems for weighted pseudo almost periodic mild solutions to a semilinear differential equation in a Banach space.
Access status: Open Access ,
Towards theory of C-symmetries
(2017) Kužel', Sergìj Oleksandrovič; Sudìlovsʹka, Veronìka Igorìvna
The concept of $\mathcal{C}$-symmetry originally appeared in $\mathcal{PT}$-symmetric quantum mechanics is studied within the Krein spaces framework.
Access status: Open Access ,
Application of Chebyshev and trigonometric polynomials to the approximation of a solution of a singular integral equation with a multiplicative Cauchy kernel in the half-plane
(2008) Karczmarek, Paweł
In this article Chebyshev and trigonometric polynomials are used to constructan approximate solution of a singular integral equation with a multiplicative Cauchy kernelin the half-plane.
Access status: Open Access ,
Non-factorizable C-valued functions induced by finite connected graphs
(2017) Cho, Ilwoo
In this paper, we study factorizability of $C$-valued formal series at fixed vertices, called the graph zeta functions, induced by the reduced length on the graph groupoids of given finite connected directed graphs. The construction of such functions is motivated by that of Redei zeta functions. In particular, we are interested in (i) »non-factorizability« of such functions, and (ii) certain factorizable functions induced by non-factorizable functions. By constructing factorizable functions from our non-factorizable functions, we study relations between graph zeta functions and well-known number-theoretic objects, the Riemann zeta function and the Euler totient function.
Access status: Open Access ,
Bounds on the 2-domination number in cactus graphs
(2006) Chellali, Mustapha
A $2$-dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex not in $S$ is dominated at least twice. The minimum cardinality of a $2$-dominating set of $G$ is the $2$-domination number $\gamma_{2}(G)$. We show that if $G$ is a nontrivial connected cactus graph with $k(G)$ even cycles ($k(G)\geq 0$), then $\gamma_{2}(G)\geq\gamma_{t}(G)-k(G)$, and if $G$ is a graph of order n with at most one cycle, then $\gamma_{2}(G)\geqslant(n+\ell-s)/2$ improving Fink and Jacobson's lower bound for trees with $\ell>s$, where $\gamma_{t}(G)$, $\ell$ and $s$ are the total domination number, the number of leaves and support vertices of $G$, respectively. We also show that if $T$ is a tree of order $n\geqslant 3$, then $\gamma_{2}(T)\leqslant\beta(T)+s-1$, where $\beta(T)$ is the independence number of $T$.
Access status: Open Access ,
Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions
(2013) Ntouyas, Sotiris K.
This paper studies the boundary value problem of nonlinear fractional differential equations and inclusions of order $q \in (1,2]$ with nonlocal and integral boundary conditions. Some new existence and uniqueness results are obtained by using fixed point theorems.
Access status: Open Access ,
Criticality indices of 2-rainbow domination of paths and cycles
(2016) Bouchou, Ahmed; Blidia, Mostafa
A $2$-rainbow dominating function of a graph $G\left(V(G),E(G)\right)$ is a function $f$ that assigns to each vertex a set of colors chosen from the set ${1,2}$ so that for each vertex with $f(v)=\emptyset$ we have ${\textstyle\bigcup_{u\in N(v)}} f(u)=\{1,2\}$. The weight of a 2RDF $f$ is defined as $w\left( f\right)={\textstyle\sum\nolimits_{v\in V(G)}} |f(v)|$. The minimum weight of a $2$RDF is called the $2$-rainbow domination number of $G$, denoted by $\gamma_{2r}(G)$. The vertex criticality index of a $2$-rainbow domination of a graph $G$ is defined as $ci_{2r}^{v}(G)=(\sum\nolimits_{v\in V(G)}(\gamma_{2r}\left(G\right) -\gamma_{2r}\left( G-v\right)))/\left\vert V(G)\right\vert$, the edge removal criticality index of a $2$-rainbow domination of a graph $G$ is defined as $ci_{2r}^{-e}(G)=(\sum\nolimits_{e\in E(G)}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G-e\right)))/\left\vert E(G)\right\vert$ and the edge addition of a $2$-rainbow domination criticality index of $G$ is defined as $ci_{2r}^{+e}(G)=(\sum\nolimits_{e\in E(\overline{G})}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G+e\right)))/\left\vert E(\overline{G})\right\vert$, where $\overline{G}$ is the complement graph of $G$. In this paper, we determine the criticality indices of paths and cycles.
Access status: Open Access ,
On local antimagic total labeling of complete graphs amalgamation
(Wydawnictwa AGH, 2023) Lau, Gee-Choon; Shiu, Wai Chee
Let $G=(V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic (total) if $G$ admits a local antimagic (total) labeling. A bijection $g:E \to \{1,2,\dots,q\}$ is called a local antimagic labeling of $ if for any two adjacent vertices $u$ and $v$, we have $g^+(u) \ne g^+(v)$, where $g^+(u) = \sum_{e\in E(u)} g(e)$, and $E(u)$ is the set of edges incident to $u$. Similarly, a bijection $f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w_{f}(u) \ne w_{f}(v)$, where $w_f(u) = f(u) + \sum_{e\in E(u)} f(e)$. Thus, any local antimagic (total) labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $g^{+}(v)$ (respectively, $w_{f}(u)$). The local antimagic (total) chromatic number, denoted $\chi_{la}(G)$ (respectively $\chi_{lat}(G)$), is the minimum number of induced colors taken over local antimagic (total) labeling of $G$. In this paper, we determined $\chi_{lat}(G)$ where $G$ is the amalgamation of complete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of $K_1$ and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.
Access status: Open Access ,
Bifurcation in a nonlinear steady state system
(2010) Wang, Gen-Qiang; Cheng, Sui Sun
The steady state solutions of a nonlinear digital cellular neural network with $\omega$ neural units and a nonnegative variable parameter $\lambda$ are sought. We show that $\lambda = 1$ is a critical value such that the qualitative behavior of our network changes. More specifically, when $\omega$ is odd, then for $\lambda \in [0,1)$, there is one positive and one negative steady state, and for $\lambda \in [1,\infty)$, steady states cannot exist; while when $\omega$ is even, then for $\lambda \in [0,1)$, there is one positive and one negative steady state, and for $\lambda = 1$, there are no nontrivial steady states, and for $\lambda \in (1,\infty)$, there are two fully oscillatory steady states. Furthermore, the number of existing nontrivial solutions cannot be improved. It is hoped that our results are of interest to digital neural network designers.
Access status: Open Access ,
On some inverse problem for bi-parabolic equation with observed data in Lp spaces
(Wydawnictwa AGH, 2022) Nguyẽ̂n, Huy Tuá̂n
The bi-parabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for bi-parabolic equation when the observed data are obtained in $L^{p}$. We are interested in looking at three types of inverse problems. Regularization results in the $L^{2}$ space appears in many related papers, but the survey results are rare in $L^{p}$, $p \neq 2$. The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in $L^{p}$ spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in $L^{p}$, we obtain the approximated solution also in the space $L^{p}$. Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space $L^{p}$. This paper seems to generalize to previous results for bi-parabolic equation on this direction.
Access status: Open Access ,
From set-valued dynamical processes to fractals
(Wydawnictwa AGH, 2025) Guzik, Grzegorz; Kleszcz, Grzegorz
We present a general theory of topological semiattractors and attractors for set-valued semigroups. Our results extend and unify those previously obtained by Lasota and Myjak. In particular, we naturally generalize the concept of semifractals for the systems acting on Hausdorff topological spaces. The main tool in our analysis is the notion of topological (Kuratowski) limits. We especially focus on the forward asymptotic behavior of discrete set-valued processes generated by sequences of iterated function systems. In this context, we establish sufficient conditions for the existence of fractal-type limit sets.
Access status: Open Access ,
Perturbation series for Jacobi matrices and the quantum Rabi model
(Wydawnictwa AGH, 2021) Charif, Mirna; Zieliński, Lech
We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.
Access status: Open Access ,
On solvability of some difference-discrete equations
(2016) Vasil'ev, Aleksandr Vladimirović; Vasil'ev, Vladimir Borisović
Multidimensional difference equations in a discrete half-space are considered. Using the theory of periodic Riemann problems a general solution and solvability conditions in discrete Lebesgue spaces are obtained. Some statements of boundary value problems of discrete type are given.
Access status: Open Access ,
Influence of an Lp-perturbation on Hardy-Sobolev inequality with singularity a curve
(Wydawnictwa AGH, 2021) Ijaodoro, Idowu Esther; Thiam, El Hadji Abdoulaye
We consider a bounded domain $\Omega$ of $\mathbb{R}^{N}$, $N \geq 3$, $h$ and $b$ continuous functions on $\Omega.$ Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^{1}_{0}(\Omega)$ to the perturbed Hardy-Sobolev equation: $-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ in } \Omega,$ where $2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}$ is the critical Hardy-Sobolev exponent, $\sigma \in [0,2)$, $0\lt\delta\lt\frac{4}{N-2}$ and $\rho_{\Gamma}$ is the distance function to $\Gamma$. We show that the existence of minimizers does not depend on the local geometry of $\Gamma$ nor on the potential $h$. For $N=3$, the existence of ground-state solution may depends on the trace of the regular part of the Green function of $-\Delta+h$ and or on $b$. This is due to the perturbative term of order $1+\delta$.
Access status: Open Access ,
A note on attractivity for the intersection of two discontinuity manifolds
(Wydawnictwa AGH, 2020) Difonzo, Fabio V.
In piecewise smooth dynamical systems, a co-dimension 2 discontinuity manifold can be attractive either through partial sliding or by spiraling. In this work we prove that both attractivity regimes can be analyzed by means of the moments solution, a spiraling bifurcation parameter and a novel attractivity parameter, which changes sign when attractivity switches from sliding to spiraling attractivity or vice-versa. We also study what happens at what we call attractivity transition points, showing that the spiraling bifurcation parameter is always zero at those points.
Access status: Open Access ,
Asymptotic behavior of the solutions of operators that are sum of pseudo p-Laplace type
(Wydawnictwa AGH, 2025) Jana, Purbita
The article investigates a Poisson-type problem for operators that are finite sum of pseudo $p$-Laplace-type operators within long cylindrical domains. It establishes that the rate of convergence is exponential, which is considered optimal. In addition, the study analyzes the asymptotic behavior of the related energy functional. This research contributes to a deeper understanding of the mathematical properties and asymptotic analysis of solutions in this context.
Access status: Open Access ,
The asymptotic properties of the dynamic equation with a delayed argument
(2006) Čermák, Jan; Urbánek, Miroslav
In this paper, we present some asymptotic results related to the scalar dynamic equation with a delayed argument. Using the time scale calculus we generalize some results known in the differential and difference case to the more general dynamic case.
Access status: Open Access ,
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 $j\in\mathbb{Z}$.