SUR LE MÉTHODE DE L'INTÉGRALE GÉNÉRALISÉE DE LAPLACE APPLIQUÉE AUX ÉQUATIONS ARÉOLAIRES LINÉAIRES
L'équation aréolaire linéaire de second ordre (1) aux coefficients polynomiaux est résolue par l'emploi d'une méthodologie qui présente une combinaison des méthodes classiques de Riemann et Frobenius sur les équations différentielles ordinaires complexes (11), et l'intégrale aréolaire de Laplace (3), avec les contours de Heinkel.
Copyright (c) 1991 Matematichki Bilten https://creativecommons.org/licenses/by/4.0
EXTENSION OF AN INTERPOLATION OF JENSEN'S INEQUALITY
Extension of interpolation of Jensen's inequality for convex functiоns given by S. S. Dragomir is obtained.
A refinement of the well-known inequality due to Jensen and some natural applications are given.
Copyright (c) 1991 https://creativecommons.org/licenses/by/4.0
LINEARIZATION OF A PRODUCT OF Q-APPELL POLYNOMIALS
За полиномите (1) се определува продуктот (2), чии коефициенти се дадени со (3).
A new coherent category of coherent inverse systеms will be defined which from the earlier known category COH defined in [6] and [7] differs in the definition of coherent homotopy. The obtained category of coherent inverse systеms \(X=(X_a,p_{a_0\ldots a_n},A)\) with this new definition of coherent homotopy satisfies the requirements of the theory announced in [1] and [2] without explicitely given formulas. For inverse systеms \((X_a,p_{a_0a_1},A)\), which are special case of coherent inverse systеms such a theory is developed in [4] (also [3]).
ON THE STRICTLY PARALLEL SUBBUNDLES OF VECTOR BUNDLES
In this paper are considered four propositions about strictly parallel subbundles of vector bundles. In propositions 1 and 2 are generalized some results of Walker [1] and Wong [2] in case of an arbitrary vector bundle, instead of tangent bundle. Proposition 3 generalizes one Wong’s result [2] in global case. In proposition 4 are given necessary and sufficient conditions for existence of a connection in a vector bundle \(\xi\) such that given \(r\) linearly independent vector fields and \(s\) linearly independent 1-forms in \(\xi\) are parallel. That condition is given by (3).
ON SHARP RESULTS ON COEFFICIENT MULTIPLIERS OF HARMONIC FUNCTION SPACES OF SEVERAL VARIABLES AND RELATED PROBLEMS
The first sharp results concerning coefficient multipliers in various spaces of harmonic functions in the unit disk were obtained by A. Shields and D. Williams several decades ago. Since then, extending these theorems to higher-dimensional harmonic function spaces remained an open problem for a long time. In 1998, R. F. Shamoyan provided the first such sharp results in higher dimensions using spherical harmonics. These results were subsequently extended in various directions to harmonic function spaces on the unit ball by R. F. Shamoyan and M. Arsenović, as well as by E. Malinnikova and A. Eikrem. The main goal of this note is to collect and survey these sharp results within a single unified framework. In the second part of this note, we also briefly review recent advances on this topic for analytic function spaces of several variables.
Copyright (c) 2025 Matematichki Bilten https://creativecommons.org/licenses/by/4.0
NORMAL FILTER IN PSEUDO QUASI-ORDERED RESIDUATED SYSTEMS
The concept of quasi-ordered residuated systems is a generalization of both commutative residual lattices and hoop-algebras. On the other hand, it is a BE-algebra with respect to the residue operation. Pseudo quasi-ordered residuated systems is a noncommutative generalization of quasi-ordered residuated systems. In the present paper, we introduce the notion of normal filters in a pseudo quasi-ordered residuated system and give some characterizations of it. We state and prove some theorems which determine the relationship between this concept and the other types of filters in a pseudo quasi-ordered residuated system such as implicative and comparative filters.
In this paper, we introduce the concepts of fuzzy SSP-irresolute continuous mappings, fuzzy SSP-irresolute open (closed) mappings, and SSP homeomorphisms. We also investigate several of their properties and examine their relationships with other forms of fuzzy continuity.
A NOTE ON THE INVERSE THEOREM FOR A GENERALISED MODULUS OF SMOOTHNESS
We prove the theorem inverse to Jackson’s theorem for a modulus of smoothness of the first order generalised by means of asymmetric operator of generalised translation.
Copyright (c) 2001 Matematichki Bilten https://creativecommons.org/licenses/by/4.0
SEVERAL INEQUALITIES FOR OPERATOR MONOTONE FUNCTIONS ON FINITE INTERVALS
In this paper we show that, if the function \(f:(-1,1)\to \mathbb{R}\) is operator monotone in \((-1,1)\), then there exists a positive measure \(\mu\) on \([-1,1]\) such that \[ [f(B)-f(A)](B-A) = f'(0)\int_{-1}^{1}\left(\int_{0}^{1}\left[(1-\lambda((1-t)A+tB))^{-1}(B-A)\right]^2\,dt\right)d\mu(\lambda) \] for all \(A,B\) with \(\operatorname{Sp}(A), \operatorname{Sp}(B)\subset(-1,1)\). Some necessary and sufficient conditions for the operators \(A,B\) with \(\operatorname{Sp}(A), \operatorname{Sp}(B)\subset(-1,1)\) such that the inequality \[ f(B)B+f(A)A\geq f(A)B+f(B)A \] holds for any operator monotone function \(f\) on \((-1,1)\) are also given.
Copyright (c) 2021 Matematichki Bilten https://creativecommons.org/licenses/by/4.0
CHAIN CONNECTEDNESS HOMOTOPY AND SHAPE INVARIANT
The characterization of connectedness by coverings i.e. chain connectedness led to some new concepts. For example chain components coincide with quasicomponents. We present some important results on chain connectedness and on chain components.
Copyright (c) 2023 Matematichki Bilten https://creativecommons.org/licenses/by/4.0
DELAYED-FEEDBACK STABILIZATION OF EQUILIBRIA IN THE FRACTIONAL-ORDER RÖSSLER AND LORENZ SYSTEMS: A COMPARATIVE ANALYSIS
This study investigates the stabilisation of equilibria in Rössler and Lorenz systems incorporating time-delayed feedback control (TDFC) with delay parameter \(\tau\) and feedback gain \(K\). The research identifies combinations of feedback gain and delay that stabilize the system’s equilibria through two complementary approaches: a local spectral test based on the characteristic equation and Matignon’s criterion, and direct simulation using a fractional forward Euler of Grünwald-Letnikov type. Stabilization in simulations is certified by requiring all trajectories to satisfy a finite-time tolerance \(10^{-2}\), \(\forall\,t\ge200\). The results indicate that TDFC can suppress chaotic dynamics where present and/or enforce convergence to the target equilibrium: The Rössler system exhibits two disjoint stability islands and \(19.5\%\) contraction between theoretical and numerical stability sets, whereas the Lorenz system shows full-domain stability with \(100\%\) agreement; this contrast is traced to a tenfold difference in the oscillation frequency of the unstable eigenvalues. The framework developed here is intended as a template for subsequent studies of other Lorenz-type systems.
In this paper, we find out the cosine operator functions \begin{equation*} C(t)= \begin{bmatrix} a(t) & b(t) & c(t) \\ 0 & d(t) & e(t) \\ 0 & 0 & f(t) \end{bmatrix} \end{equation*} in a real vector space \(\mathbb{R}^3 \, ( t\in \mathbb{R})\), as the solutions of the second order Cauchy problem \begin{equation*} C''(t)=\mathcal{A}\cdot C(t),\, C(0)=I, \, C'(0)=0. \end{equation*} We find the solutions for the various cases of a given real matrix \begin{equation*} \mathcal{A}= \begin{bmatrix} A & B & C \\ 0 & D & E \\ 0 & 0 & F \end{bmatrix}, \end{equation*} which is a generator of these cosine operator functions \(C(t), \, t\in \mathbb{R}\).
Copyright (c) 2024 Matematichki Bilten https://creativecommons.org/licenses/by/4.0
ON A TYPE OF IDEAL GENERALIZED SEMIOPEN SETS
In this paper, we obtain two weakly ideal generalized semiopen sets via (\mu\)-semiopen sets in ideal \(\tau_{\mu}\)-topological spaces and study some of the ideal \(\tau_{\mu}\)-topological characterizations through these sets.
QUASI-VALUATION MAPS ON JU-ALGEBRAS INDUCED BY FILTERS
The subject of this report is the observation of the concept of JU-filters in JU-algebras, which are a generalization of KU-algebras. In such algebras, the concept of quasi-valuation maps based on the ideal in them has already been developed. Additionally, the design of a quasi-metric space on a JU-algebra induced by that quasi-valuation map is considered. In this paper, the concept of quasi-valuation maps on JU-algebras based on a filter was designed. Then, some of its fundamental properties are shown. In addition, the concepts of quasi-valuation maps on JU-algebra based on implicative and comparative filters were constructed and analyzed. The interconnections of these three types of quasi-valuation maps on JU-algebras are also registered.
PROXIMATE APPROXIMATIVE RETRACTS IN COMPACT METRIC SPACES
Retracts are fundamental concepts in topology that help in understanding the structure and properties of spaces. A retract of a topological space \(X\) is a subspace \(A\) with a continuous map (retraction) from \(X\) onto \(A\) that leaves \(A\) fixed. This notion is crucial as it provides insights into simplifying spaces while retaining essential topological features. This paper introduces the concept of proximate approximative retracts (proxAR) in compact metric spaces. It discusses definitions and properties of various retraction types, including proximate retracts (PR), approximative proximate retracts (APR), weak retracts (WR), and approximative weak retracts (AWR). The relationships between these retraction types and proxAR are explored.
ON (NON-STANDARD) PSEUDO-VALUATIONS ON JU-ALGEBRAS
JU-algebra, as a generalization of KU-algebra, is the subject of study by several researchers in the second decade of this century. In addition to the previous one, the concept of pseudo-valuations on such algebras, generated by the ideal was introduced in 2019. This paper, as a continuation of previous research, refers to the design of the congruence relation on such an algebra and the analysis of its properties induced by a pseudo-valuation on it. Also, three important properties of the JU-algebra quotient structure created in this way are proved.
2-HOP ROMAN DOMINATION OF SOME PRODUCT GRAPHS
In this paper, we introduce the study of 2-hop Roman dominating functions of some product graphs. A 2-hop Roman dominating (2HRD) function on a graph \(G=(V,E)\)is a function \(f:V(G)\rightarrow\{0,1,2\}\) such that for every vertex \(v\) with \(f(v)=0\), there are two vertices \(x,y\) with \(f(x)=f(y)=2\) and \(d(v,x)=d(v,y)=2\). The weight of 2HRD function is the sum of its function values over all the vertices. The 2-hop Roman domination number of \(G\) denoted by \(\gamma_{2hR}(G)\) is the minimum weight of a 2HRD function in \(G\). We present the upper bond of 2-hop Roman domination number of Cartesian product graph. Also, we determine the 2-hop Roman domination number of tensor product graphs.
NOTES ON RECENT ADVANCES ON BERGMAN-TYPE PROJECTIONS IN FUNCTION SPACES OF SEVERAL VARIABLES
The intention of this survey is to collect in a single paper many recent results and advances related to Bergman-type projections acting on various spaces of analytic functions of several complex variables in the unit ball, tubular domains over symmetric cones, and bounded strongly pseudoconvex domains. Various new and interesting extensions of classical results on Bergman projections are presented in our survey. All these results were previously obtained by the first author in various papers. Bergman-type projections have many important applications in the complex function theory of several complex variables in tubular domains over symmetric cones and in bounded strongly pseudoconvex domains. Our results can be seen as direct extensions of previously known results of E. Stein, D. Bekolle, D. Debertol, B. F. Sehba, W. S. Cohn, C. Nana, L. Chen, Sh. Zhang and others, who related function spaces of the same dimension. In this paper, various new and interesting problems in this research area will also be formulated and posed by the authors. Theorems on Bergman-type projections between analytic function spaces of different dimensions over domains in $\mathbb{C}^n$ may be viewed as a new research area and may have important applications in complex function theory.
ON THE CONVERGENCE OF THE PROXIMAL GRADIENT METHOD WITH VARIABLE STEP SIZES
Composite optimization problems arise frequently in modeling, since the objective function might contain components that do not possess some "nice" properties like differentiability; the case of $\ell_1$ (LASSO) regularization is one such example. The proximal gradient methods are designed to handle this kind of optimization problems, and can solve them efficiently when the proximal mapping has a closed-form solution. Theoretical analyses of the convergence properties of the proximal gradient method with constant step size have showed sublinear and linear convergence for convex and strongly convex objective functions respectively. In this paper we show that under standard assumptions the same kind of convergence result can be established for the proximal gradient method with variable step sizes in the general setting of bounded step sizes. Further, a recently proposed step size selection for the proximal gradient method with variable step sizes is considered, and the above convergence analysis is implemented for this method.
A UNIFORMLY CONVERGENT METHOD FOR A LINEAR SINGULARLY-PERTURBED REACTION-DIFFUSION PROBLEM ON A LISEIKIN TYPE OF MESH
In this paper, we consider the numerical solution of a linear singularly perturbed reaction-diffusion boundary value problem, where a small perturbation parameter multiplies the second-order derivative. To achieve this, we approximate the differential equation using a central difference scheme on a Liseikin-type mesh. Furthermore, it is shown that the proposed method is uniformly convergent with respect to the perturbation parameter with an order of convergence of 2. Finally, we provide two numerical examples that illustrate the theoretical results regarding the uniform convergence of the discrete problem, as well as the robustness of the method.
ON SEQUENCE CONVERGENCE IN (3, j)-METRIC SPACES, j ∈ {1, 2}
In this article, we show that a convergent sequence in \((3, 2)\)-metric spaces has a unique limit. We give several examples in \((3, 1, \rho)\)-metric spaces and \((3, 1)\)-metric spaces in which a convergent sequence has more than one limit. We obtain sufficient conditions for a sequence in \((3, 1)\)-metric space to have a unique limit.
COMPARISON OF NUMERICAL AND EXACT SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH CONSTANT DELAY
Since it is well known that the analytical solution of fractional differential equations with delays is challenging, if not impossible, we have compared the exact solutions of two such equations using four numerical methods: Fractional Euler Method (FEM), Backward Euler Method (BEM), Weighted Difference Method (WDM), and Fractional Adams Method (FAM) in this study. The efficiency of each numerical approach is evaluated based on the relative difference between the exact and approximate solutions at predetermined time points, presented graphically and in tabular form. To provide a comprehensive comparison, the execution time is calculated by taking the average of over 10 executions for each method. From the results, we conclude that FAM has the lowest relative errors, is more accurate, and has a lower computational cost. FAM is the best alternative for accurate solutions and short execution times over extended timescales, FEM offers satisfactory accuracy and efficiency at intermediate timescales, WDM shows progress over time but has higher computational costs, and BEM consistently exhibits significant error rates and the variability of the execution time in the two examples.
NUMERICALLY VERIFYING THE ASSOUAD DIMENSION: THE SET {0} ∪ {1/n : n ∈ N}
In this case study, we aim to rigorously verify established theoretical properties concerning the Assouad dimension of the set {0} ∪ {1/n : n ∈ N}through detailed calculations. This work serves as a foundational step towards identifying general principles governing the convergence behavior of Assouad dimension calculations for arbitrary fractals. Specifically, we investigate the necessary precision for the generation and measurement of such sets to achieve a target accuracy in the estimation of their Assouad dimensions.
DESINGULARIZATION FORMULAS FOR THE STOCKWELL AND DIRECTIONAL STOCKWELL TRANSFORMS
Abstract. We establish a connection between the Stockwell transform, the Radon transform, and the directional Stockwell transform. We also provide desingularization formulas for the Stockwell transform and the directional Stockwell transform.
CHAIN RELATION IN A FAMILY OF SETS. CHAIN FUNCTION
In this paper we define connectedness of a family of sets, by defining the terms finite chain in the family and chain relation between two sets. Furthermore, we define a chain function between two families of sets. Some properties of connected families and chain functions are obtained.
COMPARATIVE STRONG FILTERS IN HYPER QUASI-ORDERED RESIDUATED SYSTEMS
The concept of quasi-ordered residuated systems as a generalization of both quasi-ordered commutative residuated lattices and hoop-algebras was developed in 2018 by Bonzio and Chajda. The ideas of the theory of hyper structures were applied to this system by this author and designed the concept of hyper quasi-ordered residuated system. In this paper, the concept of comparative strong filters in a hyper quasi-ordered residuated system is constructed and analyzed. This concept is compared to a (strong) filter in this algebraic structure, also.
SCALE INVARIANT STOCHASTIC GRADIENT METHOD WITH MOMENTUM
Optimization in noisy environments arises frequently in applications. Solving this problem quickly, efficiently, and accurately is therefore of great importance. The stochastic gradient descent (SGD) method has proven to be a fundamental and an effective tool which is flexible enough to allow modifications for improving its convergence properties. In this paper we propose a new algorithm for solving an unconstrained optimization problems in noisy environments which combines the SGD with a modified momentum term using a twopoint step size estimation in the Barzilai-Borwein (BB) framework. We perform a high probability analysis for the proposed algorithm and we establish its convergence under the standard assumptions. Numerical experiments demonstrate a promising behavior of the proposed method compared to the "vanilla" SGD with momentum in noise-free and in noisy environment when the objective function is scaled.
FANTASTIC FILTERS IN QUASI-ORDERED RESIDUATED SYSTEMS
The notion of quasi-ordered residuated systems was introduced by Bonzio and Chajda in 2018 as a generalization of both commutative residual lattices and hoop-algebras. With respect to the residuum, this system is a BE-algebra. After that, the concepts of ideals and filters in such systems as well as some types of filters in them were introduced by this author a little differently than it was done in the mentioned structures. In this paper, the concept of fantastic filters in a quasi-ordered residuated system is introduced. In addition, the relationships between this newly designed filter in the quasi-ordered residated system and other types of filters in it are analyzed.
Copyright (c) 2022 Matematichki Bilten https://creativecommons.org/licenses/by/4.0