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 ,
(1,2)-PDS in graphs with the small number of vertices of large degrees
(Wydawnictwa AGH, 2025) Bednarz, Urszula; Pirga, Mateusz
We define and study a perfect $(1,2)$-dominating set which is a special case of a $(1,2)$-dominating set. We discuss the existence of a perfect $(1,2)$-dominating set in graphs with at most two vertices of maximum degree. In particular, we present a complete solution if the maximum degree equals $n-1$ or $n-2$.
Access status: Open Access ,
2-biplacement without fixed points of (p,q)-bipartite graphs
(2005) Orchel, Beata
In this paper we consider 2-biplacement without fixed points of paths and $(p, q)$-bipartite graphs of small size. We give all $(p, q)$-bipartite graphs $G$ of size q for which the set $\mathcal{S}^{*}(G)$ of all 2-biplacements of $G$ without fixed points is empty.
Access status: Open Access ,
2-hyperreflexivity and hyporeflexivity of power partial isometries
(2016) Piwowarczyk, Kamila; Ptak, Marek
Power partial isometries are not always hyperreflexive neither reflexive. In the present paper it will be shown that power partial isometries are always hyporeflexive and $2$-hyperreflexive.
Access status: Open Access ,
2-splittable and cordial graphs
(2010) Cichacz-Przeniosło, Sylwia
E. Miller and G. E. Stevens proved in [E. Miller, G. E. Stevens, <i>Some graphs for which even size is sufficient for splittability</i>, Congressus Numerantium 173 (2005), 137–147] the existence of certain families of $2$-splittable caterpillars. In this paper we characterize other families of $2$-splittable caterpillars. Moreover, we show that for some of them there exists a friendly labeling inducing two isomorphic subgraphs.
Access status: Open Access ,
3-biplacement of bipartite graphs
(2008) Adamus, Lech; Leśniak, Edyta; Orchel, Beata
Let $G=(L,R;E)$ be a bipartite graph with color classes $L$ and $R$ and edge set $E$. A set of two bijections $\{\varphi_1 , \varphi_2\}$, $\varphi_1 , \varphi_2 :L \cup R \to L \cup R$, is said to be a $3$-biplacement of $G$ if $\varphi_1(L)= \varphi_2(L) = L$ and $E \cap \varphi_1^*(E)=\emptyset$, $E \cap \varphi_2^*(E)=\emptyset$, $\varphi_1^*(E) \cap \varphi_2^*(E)=\emptyset$, where$\varphi_1^*$, $\varphi_2^*$ are the maps defined on $E$, induced by $\varphi_1$, $\varphi_2$, respectively. We prove that if $|L|=p$, $|R|=q$, $3 \leq p \leq q$, then every graph $G=(L,R;E)$ of size at most $p$ has a $3$-biplacement.
Access status: Open Access ,
A characterization of convex φ-functions
(2012) Micherda, Bartosz
The properties of four elements $(LPFE)$ and $(UPFE)$, introduced by Isac and Persson, have been recently examined in Hilbert spaces, $L^p$-spaces and modular spaces. In this paper we prove a new theorem showing that a modular of form $\rho_{\Phi}(f)=\int_{\Omega}\Phi(t,|f(t)|)d\mu(t)$ satisfies both $(LPFE)$ and $(UPFE)$ if and only if $\Phi$ is convex with respect to its second variable. A connection of this result with the study of projections and antiprojections onto latticially closed subsets of the modular space $L^{\Phi}$ is also discussed.
Access status: Open Access ,
A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions
(2011) Wang, JinRong; Yan, X.; Zhang, X.-H.; Wang, T.-M.; Li, X.-Z.
In this paper, we discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type $\begin{aligned}D_{t}^{q}x(t)=Ax(t)+\int\limits_{0}^{t}B(t-s)x(s)ds+t^{n}f\left(t,x(t)\right),&\;t\in [0,T],\;n\in Z^{+},\\&q\in(0,1],\;x(0)=g(x)+x_{0}.\end{aligned}$ Some sufficient conditions for the existence of mild solutions for the above system are given. The main tools are the resolvent operators and fixed point theorems due to Banach's fixed point theorem, Krasnoselskii's fixed point theorem and Schaefer's fixed point theorem. At last, an example is given for demonstration.
Access status: Open Access ,
A comprehensive review on the existence of normalized solutions for four classes of nonlinear elliptic equations
(Wydawnictwa AGH, 2025) Chen, Sitong; Tang, Xianhua
This paper provides a comprehensive review of recent results concerning the existence of normalized solutions for four classes of nonlinear elliptic equations: Schrödinger equations, Schrödinger-Poisson equations, Kirchhoff equations, and Choquard equations
Access status: Open Access ,
A direct approach to linear-quadratic stochastic control
(Wydawnictwa AGH, 2017) Duncan, Tyrone E.; Pasik-Duncan, Bozenna
A direct approach is used to solve some linear-quadratic stochastic control problems for Brownian motion and other noise processes. This direct method does not require solving Hamilton-Jacobi-Bellman partial differential equations or backward stochastic differential equations with a stochastic maximum principle or the use of a dynamic programming principle. The appropriate Riccati equation is obtained as part of the optimization problem. The noise processes can be fairly general including the family of fractional Brownian motions.
Access status: Open Access ,
A distribution associated with the Kontorovich-Lebedev transform
(2006) Yakubovich, Semyon B.
We show that in a sense of distributions $\lim_{\varepsilon\to 0+} {1\over \pi^2} \tau\sinh\pi\tau \int_{\varepsilon}^{\infty} K_{i\tau}(y)K_{ix}(y){dy\over y} =\delta(\tau-x),$ where $\delta$ is the Dirac distribution, $\tau#, $x\in\mathbb{R}$ and $K_{\nu}(x)$ is the modified Bessel function. The convergence is in $\mathcal{E}^{\prime}(\mathbb{R})$ for any even $\varphi(x)\in\mathcal{E}(\mathbb{R})$ being a restriction to $\mathbb{R}$ of a function $\varphi(z)$ analytic in a horizontal open strip $G_a=\{z\in\mathbb{C}\colon\,|\text{Im}\,z|\lt a, \ a\gt 0\}$ and continuous in the strip closure. Moreover, it satisfies the condition $\varphi(z)=O\bigl(|z|^{-\text{Im}\,z-\alpha}e^{-\pi|\text{Re}\,z|/2}\bigr)$, $|\text{Re}\,z|\to\infty$ uniformly in $\overline{G_a}$. The result is applied to prove the representation theorem for the inverse Kontorovich-Lebedev transformation on distributions.
Access status: Open Access ,
A double index transform with a product of Macdonald's functions revisited
(2009) Âkuboviĉ, Semën B.
We prove an inversion theorem for a double index transform, which is associated with the product of Macdonald's functions $K_{i \tau}(\sqrt{x^2+y^2}-y) K_{i \tau}(\sqrt{x^2+y^2}+y)$, where $(x, y) \in \mathbb{R}_+ \times \mathbb{R}_+$ and $i \tau, \tau \in \mathbb{R}_+$ is a pure imaginary index. The results obtained in the sequel are applied to find particular solutions of integral equations involving the square and the cube of the Macdonald function $K_{i \tau}(t)$ as a kernel.
Access status: Open Access ,
A dynamical inverse problem for a parabolic equations
(2006) Maksimov, Vâčeslav
A problem of dynamical reconstruction of unknown distributed or boundary disturbances acting upon nonlinear parabolic equations is discussed. A regularized algorithm which allows us to reconstruct disturbances synchro with the process under consideration is designed. This algorithm is stable with respect to informational noises and computational errors.
Access status: Open Access ,
A finite difference method for nonlinear parabolic-elliptic systems of second-order partial differential equations
(2007) Malec, Marian; Sapa, Lucjan
This paper deals with a finite difference method for a wide class of weakly coupled nonlinear second-order partial differential systems with initial condition and weakly coupled nonlinear implicit boundary conditions. One part of each system is of the parabolic type (degenerated parabolic equations) and the other of the elliptic type (equations with a parameter) in a cube in $\mathbf{R}^{1+n}$. A suitable finite difference scheme is constructed. It is proved that the scheme has a unique solution, and the numerical method is consistent, convergent and stable. The error estimate is given. Moreover, by the method, the differential problem has at most one classical solution. The proof is based on the Banach fixed-point theorem, the maximum principle for difference functional systems of the parabolic type and some new difference inequalities. It is a new technique of studying the mixed-type systems. Examples of physical applications and numerical experiments are presented.
Access status: Open Access ,
A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices
(2008) Pchelintseva, Irina
We consider self-adjoint unbounded Jacobi matrices with diagonal $q_n = b_{n}n$ and off-diagonal entries $\lambda_n = n$, where bn is a $2$-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum of the operator is either purely absolutely continuous or discrete. We study the situation where the spectral phase transition occurs, namely the case of $b_{1}b_{2} = 4$. The main motive of the paper is the investigation of asymptotics of generalized eigenvectors of the Jacobi matrix. The pure point part of the spectrum is analyzed in detail.
Access status: Open Access ,
A general 2-part Erdȍs-Ko-Rado theorem
(Wydawnictwa AGH, 2017) Katona, Gyula O.H.
A two-part extension of the famous Erdȍs-Ko-Rado Theorem is proved. The underlying set is partitioned into $X_1$ and $X_2$. Some positive integers $k_i$, $\ell_i$ ($1\leq i\leq m$) are given. We prove that if $\mathcal{F}$) is an intersecting family containing members $F$ such that $|F\cap X_1|=k_i$, $|F\cap X_2|=\ell_i$ holds for one of the values $i$ ($1\leq i\leq m$) then $|\mathcal{F}|$ cannot exceed the size of the largest subfamily containing one element.
Access status: Open Access ,
A general boundary value problem and its Weyl function
(2007) Ryzhov, Vladimir
We study the abstract boundary value problem defined in terms of the Green identity and introduce the concept of Weyl operator function $M(\cdot)$ that agrees with other definitions found in the current literature. In typical cases of problems arising from the multidimensional partial equations of mathematical physics the function $M(\cdot)$ takes values in the set of unbounded densely defined operators acting on the auxiliary boundary space. Exact formulae are obtained and essential properties of $M(\cdot)$ are studied. In particular, we consider boundary problems defined by various boundary conditions and justify the well known procedure that reduces such problems to the 'equation on the boundary' involving the Weyl function, prove an analogue of the Borg-Levinson theorem, and link our results to the classical theory of extensions of symmetric operators.
Access status: Open Access ,
A general elliptic equation with intrinsic operator
(Wydawnictwa AGH, 2025) Motreanu, Dumitru
Existence and bound of a solution is established for a general elliptic equation with intrinsic operator subject to Dirichlet boundary condition. This provides a sufficient condition to the fundamental question if there is a solution belonging to a prescribed ball in the function space. An application deals with an equation involving a convolution product.
Access status: Open Access ,
A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes
(2013) Alpay, Daniel; Kipnis, Alon
Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.
Access status: Open Access ,
A hierarchy of maximal intersecting triple systems
(Wydawnictwa AGH, 2017) Polcyn, Joanna; Ruciński, Andrzej
We reach beyond the celebrated theorems of Erdȍs-Ko-Rado and Hilton-Milner, and a recent theorem of Han-Kohayakawa, and determine all maximal intersecting triples systems. It turns out that for each $n\geq 7$ there are exactly 15 pairwise non-isomorphic such systems (and 13 for n=6). We present our result in terms of a hierarchy of Turán numbers $\operatorname{ex}^{(s)}(n; M_2^{3})$, $s\geq 1$, where $M_2^{3}$ is a pair of disjoint triples. Moreover, owing to our unified approach, we provide short proofs of the above mentioned results (for triple systems only). The triangle $C_3$ is defined as $C_3=\{\{x_1,y_3,x_2\},\{x_1,y_2,x_3\},\{x_2,y_1,x_3\}\}$. Along the way we show that the largest intersecting triple system $H$ on $n\geq 6$ vertices, which is not a star and is triangle-free, consists of $\max\{10,n\}$ triples. This facilitates our main proof's philosophy which is to assume that $H$ contains a copy of the triangle and analyze how the remaining edges of $H$ intersect that copy.
Access status: Open Access ,
A linear time algorithm to compute vertices that belong to all, some and no minimum dominating sets in a tree and its consequences
(Wydawnictwa AGH, 2025) Ziemann, Radosław; Żyliński, Paweł
We provide a linear time algorithm for determining the sets of vertices that belong to all, some and no minimum dominating sets of a tree, respectively, thus improving the quadratic time algorithm of Benecke and Mynhardt in 2008 [S. Benecke, C.M. Mynhardt, Trees with domination subdivision number one, Australas. J. Comb. 42 (2008), 201-209]. Some algorithmic consequences are also discussed.