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Roczniki, numery i artykuły czasopisma Opuscula Mathematica
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- ArtykułOtwarty dostępUniform approximation by polynomials with integer coefficients(2016) Lipnicki, ArturLet r, n be positive integers with n≥6r. Let P be a polynomial of degree at most n on [0,1] with real coefficients, such that [formula] and [formula] are integers for k= 0,. . . , r − 1. It is proved that there is a polynomial Q of degree at most n with integer coefficients such that [formula] for x ∈ [0,1], where C1, C2 are some numerical constants. The result is the best possible up to the constants.
- ArtykułOtwarty dostępAsymptotic analysis for confluent hypergeometric function in two variables given by double integral(Wydawnictwa AGH, 2024) Haraoka, YoshishigeWe study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along $x=\infty$ and $y=\infty$. Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around $(\infty,\infty)$. Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new.We study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along $x=\infty$ and $y=\infty$. Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around $(\infty,\infty)$. Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new.
- ArtykułOtwarty dostępOn the solvability of some parabolic equations involving nonlinear boundary conditions with L1 data(Wydawnictwa AGH, 2024) Taourirte, Laila; Charkaoui, Abderrahim; Alaa, Nour EddineWe analyze the existence of solutions for a class of quasilinear parabolic equations with critical growth nonlinearities, nonlinear boundary conditions, and $L^1$ data. We formulate our problems in an abstract form, then using some techniques of functional analysis, such as Leray-Schauder's topological degree associated with the truncation method and very interesting compactness results, we establish the existence of weak solutions to the proposed models.
- ArtykułOtwarty dostępGeometric properties of the lattice of polynomials with integer coefficients(Wydawnictwa AGH, 2024) Lipnicki, Artur; Śmietański, Marek J.This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let $r$, $n$ be positive integers with $n \ge 6r$. Let $\boldsymbol{P}_n \cap \boldsymbol{M}_r$ be the space of polynomials of degree at most $n$ on $[0,1]$ with integer coefficients such that $P^{(k)}(0)/k!$ and $P^{(k)}(1)/k!$ are integers for $k=0,\dots,r-1$ and let $\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r$ be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of $\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r$ from $\boldsymbol{P}_n \cap \boldsymbol{M}_r$ in $L_2(0,1)$. We give rather precise quantitative estimations for successive minima of $\boldsymbol{P}_n^\mathbb{Z}$ in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in $\boldsymbol{P}_n \cap \boldsymbol{M}_r$.
- ArtykułOtwarty dostępStudy of fractional semipositone problems on RN(Wydawnictwa AGH, 2024) Biswas, NirjanLet $s\in (0,1)$ and $N\gt 2s$. In this paper, we consider the following class of nonlocal semipositone problems: $(-\Delta)^s u= g(x)f_a(u)\text{ in }\mathbb{R}^N,\quad u \gt 0\text{ in }\mathbb{R}^N,$ where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a\gt 0$ is a parameter, and $f_a \in \mathcal{C}(\mathbb{R})$ is strictly negative on $(-\infty,0]$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided a is near zero. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^r(\mathbb{R}^N)$ for every $r \in [\frac{2N}{N-2s}, \infty]$.
- ArtykułOtwarty dostępGraphs whose vertex set can be partitioned into a total dominating set and an independent dominating set(Wydawnictwa AGH, 2024) Haynes, Teresa W.; Henning, Michael A.A graph $G$ whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. We give constructions that yield infinite families of graphs that are TI-graphs, as well as constructions that yield infinite families of graphs that are not TI-graphs. We study regular graphs that are TI-graphs. Among other results, we prove that all toroidal graphs are TI-graphs.
- ArtykułOtwarty dostępPositive solutions for nonparametric anisotropic singular solutions(Wydawnictwa AGH, 2024) Papageorgiou, Nikolaos S.; Rǎdulescu, Vicenţiu; Sun, XueyingWe consider an elliptic equation driven by a nonlinear, nonhomogeneous differential operator with nonstandard growth. The reaction has the combined effects of a singular term and of a "superlinear" perturbation. There is no parameter in the problem. Using variational tools and truncation and comparison techniques, we show the existence of at least two positive smooth solutions.
- ArtykułOtwarty dostępReduction of positive self-adjoint extensions(Wydawnictwa AGH, 2024) Tarcsay, Zsigmond; Sebestyén, ZoltánWe revise Krein's extension theory of semi-bounded Hermitian operators by reducing the problem to finding all positive and contractive extensions of the "resolvent operator" $(I+T)^{-1}$ of $T$. Our treatment is somewhat simpler and more natural than Krein's original method which was based on the Krein transform $(I-T)(I+T)^{-1}$. Apart from being positive and symmetric, we do not impose any further constraints on the operator $T$: neither its closedness nor the density of its domain is assumed. Moreover, our arguments remain valid in both real or complex Hilbert spaces.
- ArtykułOtwarty dostępShifted model spaces and their orthogonal decompositions(Wydawnictwa AGH, 2024) Câmara, M. Cristina; Kliś-Garlicka, Kamila; Ptak, MarekWe generalize certain well known orthogonal decompositions of model spaces and obtain similar decompositions for the wider class of shifted model spaces, allowing us to establish conditions for near invariance of the latter with respect to certain operators which include, as a particular case, the backward shift $S^{*}$. In doing so, we illustrate the usefulness of obtaining appropriate decompositions and, in connection with this, we prove some results on model spaces which are of independent interest. We show moreover how the invariance properties of the kernel of an operator $T$, with respect to another operator, follow from certain commutation relations between the two operators involved.
- ArtykułOtwarty dostępA note on the general moment problem(Wydawnictwa AGH, 2024) El Azhar, Hamza; Hanine, Abdelouahab; Zerouali, El HassanIn this note we show that given an indeterminate Hamburger moment sequence, it is possible to perturb the first moment in such way that the obtained sequence remains an indeterminate Hamburger moment sequence. As a consequence we prove that every sequence of real numbers is a moment sequence for a signed discrete measure supported in $\mathbb{R}_{+}$.
- ArtykułOtwarty dostępFinitely additive functions in measure theory and applications(Wydawnictwa AGH, 2024) Alpay, Daniel; Jorgensen, PalleIn this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory, in particular, to the study of $\mu$-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures $\mu$ , and to adjoints of composition operators.
- ArtykułOtwarty dostępOn the Möbius invariant principal functions of Pincus(Wydawnictwa AGH, 2024) Ghosh, Sagar; Misra, GadadharIn this semi-expository short note, we prove that the only homogeneous pure hyponormal operator $T$ with $\operatorname{rank} (T^*T-TT^*) =1$, modulo unitary equivalence, is the unilateral shift.
- ArtykułOtwarty dostępJan Stochel, a stellar mathematician(Wydawnictwa AGH, 2024) Chavan, Sameer; Curto, Raúl; Jabłoński, Zenon Jan; Jung, Il Bong; Putinar, MihaiThe occasion for this survey article was the 70th birthday of Jan Stochel, professor at Jagiellonian University, former head of the Chair of Functional Analysis and a prominent member of the Kraków school of operator theory. In the course of his mathematical career, he has dealt, among other things, with various aspects of functional analysis, single and multivariable operator theory, the theory of moments, the theory of orthogonal polynomials, the theory of reproducing kernel Hilbert spaces, and mathematical aspects of quantum mechanics.
- ArtykułOtwarty dostępCesàro summability of Taylor series in higher order weighted Dirichlet-type spaces(Wydawnictwa AGH, 2024) Ghara, Soumitra; Gupta, Rajeev; Reza, Md. RamizFor a positive integer $m$ and a finite non-negative Borel measure $\mu$ on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces $\mathcal H_{\mu, m}$. We show that if $\alpha\gt\frac{1}{2}$, then for any $f$ in $\mathcal H_{\mu, m}$ the sequence of generalized Ces?ro sums $\{\sigma_n^{\alpha}[f]\}$ converges to $f$. We further show that if $\alpha=\frac{1}{2}$ then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer $m$.
- ArtykułOtwarty dostępDegenerate singular parabolic problems with natural growth(Wydawnictwa AGH, 2024) El Ouardy, Mounim; El Hadfi, Youssef; Sbai, AbdelaazizIn this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term $\$ \begin{cases}\frac{\partial u}{\partial t}-\operatorname{div}((a(x,t)+u^{q})|\nabla u|^{p-2}\nabla u)+d(x,t)\frac{|\nabla u|^{p}}{u^{\gamma}}=f & \text{ in } Q,\\ u(x,t)=0 & \text{ on } \Gamma, \\ u(x,t=0)=u_{0}(x) & \text{ in } \Omega, \end{cases} \$$ where $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$, $N\gt 2$, $Q$ is the cylinder $\Omega \times (0,T)$, $T\gt 0$, $\Gamma$ the lateral surface $\partial \Omega \times (0,T)$, $2\leq p\lt N$, $a(x,t)$ and $b(x,t)$ are positive measurable bounded functions, $q\geq 0$, $0\leq\gamma\lt 1$, and $f$ non-negative function belongs to the Lebesgue space $L^{m}(Q)$ with $m\gt 1$, and $u_{0}\in L^{\infty}(\Omega)$ such that $\$\forall\omega\subset\subset\Omega\, \exists D_{\omega}\gt 0:\, u_{0}\geq D_{\omega}\text{ in }\omega.\$$ More precisely, we study the interaction between the term $u^{q}$ $(q>0)$ and the singular lower order term $d(x,t)|\nabla u|^{p}u^{-\gamma}$ $(0\lt\gamma\lt 1)$ in order to get a solution to the above problem. The regularizing effect of the term $u^q$ on the regularity of the solution and its gradient is also analyzed.
- CzasopismoOpuscula MathematicaAGH University Press (2005-)
ISSN: 1232-9274 e-ISSN: 2300-6919 - Rocznik czasopismaOpuscula MathematicaVol. 29 (2009)
- Numer czasopismaOpuscula Mathematica2009 - Vol. 29 - No. 1
- Numer czasopismaOpuscula Mathematica2009 - Vol. 29 - No. 3
- Numer czasopismaOpuscula Mathematica2009 - Vol. 29 - No. 4