Institute of Mathematics AS CR, v. v. i.
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    Editorial

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    Československé sdružení uživatelů TeXu
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    summary:Úvodní slovo shrnuje obsah čísla.summary:The editorial presents an overview of the articles from this issue

    OBITUARY

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      Institute of Information Theory and Automation AS CR
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      Optimality conditions for interval-valued vector equilibrium problems

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      Institute of Information Theory and Automation AS CR
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      summary:In the article, one formulates Fritz John type and Karush-Kuhn-Tucker type necessary conditions for an interval-valued vector equilibrium problem having a locally LU-efficient solution, where convexificators demonstrate the solutions that are regular. Sufficient conditions for a locally weak LU-efficient solution have been entrenched by imposing appropriate assumptions along with generalized convexity. Some applications are presented for a constrained interval-valued vector variational inequality and a constrained interval-valued vector optimization problem

      Instruction for authors

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        Union of Czech Mathematicians and Physicists
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        Multilinear fractional maximal and integral operators with homogeneous kernels, Hardy-Littlewood-Sobolev and Olsen-type inequalities

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        Matematický ústav AV ČR
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        summary:Let m∈Nm\in \mathbb {N} and 0<α<mn0<\alpha <mn. Let TΩ,α;m\mathcal {T}_{\Omega ,\alpha ;m} be the multilinear fractional integral operator with homogeneous kernels, and let MΩ,α;m\mathcal {M}_{\Omega ,\alpha ;m} be the multilinear fractional maximal operator with homogeneous kernels. We will use the idea of Hedberg to reprove that the multilinear operators TΩ,α;m\mathcal {T}_{\Omega ,\alpha ;m} and MΩ,α;m\mathcal {M}_{\Omega ,\alpha ;m} are bounded from Lp1(Rn)×Lp2(Rn)×⋯×Lpm(Rn)L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n) \times \nobreak \cdots \times L^{p_m}(\mathbb R^n) into Lq(Rn)L^q(\mathbb R^n) provided that Ω⃗=(Ω1,Ω2,…,Ωm)∈[Ls(Sn−1)]m\vec {\Omega }=(\Omega _1,\Omega _2,\dots ,\Omega _m)\in [L^s({\bf S}^{n-1})]^{m}, s′<p1,p2,…,pm<∞s'<p_1,p_2,\dots ,p_m<\infty , s′/m<p<n/αs'/m<p<n/{\alpha }, 1p=1p1+1p2+⋯+1pmand1q=1p−αn. \frac {1}{p}=\frac {1}{p_1}+\frac {1}{p_2}+\cdots +\frac {1}{p_m} \quad \text {and} \quad \frac {1}{q}=\frac {1}{p}-\frac {\alpha }{n}. This result was first obtained by Chen and Xue. We also prove that under the assumptions that Ω⃗=(Ω1,Ω2,…,Ωm)∈[Ls(Sn−1)]m\vec {\Omega }=(\Omega _1,\Omega _2,\dots ,\Omega _m) \in [L^s({\bf S}^{n-1})]^{m}, s′≤p1,p2,…,pm<∞s'\leq p_1,p_2,\dots ,p_m<\infty , s′/m≤p<n/αs'/m\leq p<n/{\alpha } and (∗)(*), the multilinear operators TΩ,α;m\mathcal {T}_{\Omega ,\alpha ;m} and MΩ,α;m\mathcal {M}_{\Omega ,\alpha ;m} are bounded from Lp1(Rn)×Lp2(Rn)×⋯×Lpm(Rn)L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n) \times \cdots \times L^{p_m}(\mathbb R^n) into Lq,∞(Rn)L^{q,\infty }(\mathbb R^n), which are completely new. Moreover, we will use the idea of Adams to show that TΩ,α;m\mathcal {T}_{\Omega ,\alpha ;m} and MΩ,α;m\mathcal {M}_{\Omega ,\alpha ;m} are bounded from Lp1,κ(Rn)×Lp2,κ(Rn)×⋯×Lpm,κ(Rn)L^{p_1,\kappa }(\mathbb R^n)\times L^{p_2,\kappa }(\mathbb R^n) \times \cdots \times L^{p_m,\kappa }(\mathbb R^n) into Lq,κ(Rn)L^{q,\kappa }(\mathbb R^n) whenever s′<p1,p2,…,pm<∞s'<p_1,p_2,\dots ,p_m<\infty , 0<κ<10<\kappa <1, s′/m<p<n(1−κ)/αs'/m<p<{n(1-\kappa )}/{\alpha }, 1p=1p1+1p2+⋯+1pmand1q=1p−αn(1−κ), \frac {1}{p}=\frac {1}{p_1}+\frac {1}{p_2}+\cdots +\frac {1}{p_m} \quad \text {and} \quad \frac {1}{q}=\frac {1}{p}-\frac {\alpha }{n(1-\kappa )}, and also bounded from Lp1,κ(Rn)×Lp2,κ(Rn)×⋯×Lpm,κ(Rn)L^{p_1,\kappa }(\mathbb R^n)\times L^{p_2,\kappa }(\mathbb R^n) \times \cdots \times L^{p_m,\kappa }(\mathbb R^n) into WLq,κ(Rn)WL^{q,\kappa }(\mathbb R^n) whenever s′≤p1,p2,…,pm<∞s'\leq p_1,p_2,\dots ,p_m<\infty , 0<κ<10<\kappa <1, s′/m≤p<n(1−κ)/αs'/m\leq p<{n(1-\kappa )}/{\alpha } and (∗∗)(**). These results mentioned above are also completely new. In addition, some new estimates in the limiting cases are also established. Applications to the Hardy-Littlewood-Sobolev and Olsen-type inequalities are discussed as well

        A stochastic version of Vidyasagar theorem on the stabilization of interconnected systems

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        Institute of Information Theory and Automation AS CR
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        summary:The purpose of this paper is to provide sufficient conditions for the feedback asymptotic stabilization in probability for a class of affine in the control nonlinear stochastic differential systems. In fact, under the assumptions stated in this paper we prove the existence of a control Lyapunov function that according to the stochastic version of Artstein's theorem guarantees the asymptotic stability in probability by means of a state feedback law that is smooth except eventually at the equilibrium. This result generalizes the well-known theorem of Vidyasagar concerning the feedback stabilization problem for interconnected control systems

        Honeycomb graphs for parametric identification of correlation classes in multidimensional datasets

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        Institute of Information Theory and Automation AS CR
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        summary:In the process of gaining knowledge from large sets of data, one of the most significant methods from the area of descriptive statistics −- correlation analysis −- is applied to determine direct functional relationships between pairs of attributes. Even though the results of correlation analysis are measured through a crisp correlation coefficient, whose values belong to the [−1,1][-1,1] interval, human interpretation of these values is conventionally vague and uses linguistic classes of correlation to describe the strength of relationships between attribute pairs. However, this interpretative vagueness −- and the correlation classes themselves −- are not commonly employed in the decision-making processes. Therefore, this work focuses on the design and implementation of so-called Honeycomb Graphs −- a visualization method for parametric identification of correlation classes in multidimensional datasets based on graphical models. After implementing the proposed visualization technique, two case studies on benchmark datasets are conducted, and the model is evaluated from both qualitative and quantitative points of view. The results of these studies highlight interactive exploration of correlation analysis while adhering to qualitative and quantitative standards of scientific visualizations and high utilization potential of the method in feature selection tasks, making it a valuable tool for predictive analysis and data exploration

        Číselná charakteristika podobných trojúhelníků

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        The Union of Czech Mathematicians and Physicists
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        Sakaguchi type functions defined by balancing polynomials

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        Matematický ústav AV ČR
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        summary:The class of Sakaguchi type functions defined by balancing polynomials has been introduced as a novel subclass of bi-univalent functions. The bounds for the Fekete-Szegö inequality and the initial coefficients ∣a2∣\vert a_{2}\vert and ∣a3∣\vert a_{3}\vert have also been estimated

        Dynamic behavior of vector solutions of a class of 2-D neutral differential systems

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        Matematický ústav AV ČR
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        summary:This work deals with the analysis pertaining some dynamic behavior of vector solutions of first order two-dimensional neutral delay differential systems of the form ddt[u(t)+pu(t−τ)v(t)+pv(t−τ)]=[abcd][u(t−α)v(t−β)]. \frac {{\rm d}}{{\rm d}t} \begin {bmatrix} u(t)+pu(t-\tau )\\ v(t)+pv(t-\tau )\\ \end {bmatrix} = \begin {bmatrix} a & b \\ c & d \\ \end {bmatrix} \begin {bmatrix} u(t-\alpha )\\ v(t-\beta )\\ \end {bmatrix}. The effort has been made to study ddt[x(t)−p(t)h1(x(t−τ))y(t)−p(t)h2(y(t−τ))]+[a(t)b(t)c(t)d(t)][f1(x(t−α))f2(y(t−β))]=0, \frac {{\rm d}}{{\rm d}t} \begin {bmatrix} x(t)-p(t)h_{1}(x(t-\tau ))\\ y(t)-p(t)h_{2}(y(t-\tau )) \end {bmatrix} + \begin {bmatrix} a(t) & b(t)\\ c(t) & d(t) \end {bmatrix} \begin {bmatrix} f_{1}(x(t-\alpha ))\\ f_{2}(y(t-\beta )) \end {bmatrix} =0, where p,a,b,c,d,h1,h2,f1,f2∈C(R,R)p,a,b,c,d,h_1,h_2,f_1,f_2 \in C(\mathbb {R},\mathbb {R}); α,β,τ∈R+\alpha ,\beta ,\tau \in \mathbb {R}^+. We verify our results with the examples

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