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

Editorial

Author: Starý Novotný Vít
Publication venue: Československé sdružení uživatelů TeXu
Publication date: 01/01/2025
Field of study

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

OBITUARY

Author
Publication venue: Institute of Information Theory and Automation AS CR
Publication date: 01/01/2025
Field of study

Optimality conditions for interval-valued vector equilibrium problems

Author: Prasad Ashish Kumar
Khatri Julie
Ahmad Izhar
Publication venue: Institute of Information Theory and Automation AS CR
Publication date: 01/01/2025
Field of study

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

Author
Publication venue: Union of Czech Mathematicians and Physicists
Publication date: 01/01/2025
Field of study

Multilinear fractional maximal and integral operators with homogeneous kernels, Hardy-Littlewood-Sobolev and Olsen-type inequalities

Author: Chen Cong
Yang Kaikai
Wang Hua
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2025
Field of study

summary:Let

m\in \mathbb {N}

and

0<\alpha <mn

. Let

\mathcal {T}_{\Omega ,\alpha ;m}

be the multilinear fractional integral operator with homogeneous kernels, and let

\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

\mathcal {T}_{\Omega ,\alpha ;m}

and

\mathcal {M}_{\Omega ,\alpha ;m}

are bounded from

L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n) \times \nobreak \cdots \times L^{p_m}(\mathbb R^n)

into

L^q(\mathbb R^n)

provided that

\vec {\Omega }=(\Omega _1,\Omega _2,\dots ,\Omega _m)\in [L^s({\bf S}^{n-1})]^{m}

s'<p_1,p_2,\dots ,p_m<\infty

s'/m<p<n/{\alpha }

\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

\vec {\Omega }=(\Omega _1,\Omega _2,\dots ,\Omega _m) \in [L^s({\bf S}^{n-1})]^{m}

s'\leq p_1,p_2,\dots ,p_m<\infty

s'/m\leq p<n/{\alpha }

and

(*)

, the multilinear operators

\mathcal {T}_{\Omega ,\alpha ;m}

and

\mathcal {M}_{\Omega ,\alpha ;m}

are bounded from

L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n) \times \cdots \times L^{p_m}(\mathbb R^n)

into

L^{q,\infty }(\mathbb R^n)

, which are completely new. Moreover, we will use the idea of Adams to show that

\mathcal {T}_{\Omega ,\alpha ;m}

and

\mathcal {M}_{\Omega ,\alpha ;m}

are bounded from

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

L^{q,\kappa }(\mathbb R^n)

whenever

s'<p_1,p_2,\dots ,p_m<\infty

0<\kappa <1

s'/m<p<{n(1-\kappa )}/{\alpha }

\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

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

WL^{q,\kappa }(\mathbb R^n)

whenever

s'\leq p_1,p_2,\dots ,p_m<\infty

0<\kappa <1

s'/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

Author: Florchinger Patrick
Publication venue: Institute of Information Theory and Automation AS CR
Publication date: 01/01/2025
Field of study

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

Author: Dudáš Adam
Peregrín Tomáš
Publication venue: Institute of Information Theory and Automation AS CR
Publication date: 01/01/2025
Field of study

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]

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

Author: Dlab Vlastimil
Publication venue: The Union of Czech Mathematicians and Physicists
Publication date: 01/01/2025
Field of study

Capacity solutions for a degenerate $p_{i}(x)$ -Laplacian thermistor system with electrical conductivities

Author: Khelifi Hichem
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2025
Field of study

summary:We establish the existence of a capacity solution for a degenerate anisotropic stationary system with variable exponents and electrical conductivity. The system is a generalization of the thermistor problem, addressing the interaction between temperature and electric potential within semiconductor material

$H_{\infty }$ analysis of cooperative multi-agent systems by adaptive interpolation

Author: Tomljanović Zoran
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2025
Field of study

summary:We consider a projection-based model reduction approach to computing the maximal impact, one agent or a group of agents has on the cooperative system. As a criterion for measuring the agent-team impact on multi-agent systems, we use the

H_{\infty }

norm, and output synchronization is taken as the underlying cooperative control scheme. We investigate a projection-based model reduction approach that allows efficient

H_{\infty }

norm calculation. The convergence of this approach depends on initial interpolation points, so we present approaches to their determination. Since the analysis of multi-agent systems is important from different perspectives, several comparisons are presented in the section on numerical experiments. A graph Laplacian matrix of an inter-agent interaction graph is a foundational element in modeling and analyzing multi-agent systems. We consider various graph topology matrices, system parameters, and excitations of different agents. Different strategies for selecting initial interpolation points are also compared with baseline approaches for calculating the

H_{\infty }

norm

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

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