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  1. 16 Demonstratio Mathematica
  2. 14 Opuscula Mathematica
  3. 5 Bulletin of the Polish Academy of Sciences. Mathematics
  4. 4 Journal of Applied Analysis
  5. 4 Probability and Mathematical Statistics
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  1. 3 Syroka Z.
  2. 2 Babiarz A.
  3. 2 Gil M.
  4. 2 Houjun W.
  5. 2 Klamka J.
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  1. 2 2024
  2. 1 2023
  3. 7 2022
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1
Content available Finitely additive functions in measure theory and applications
Alpay Daniel, Jorgensen Palle
Opuscula Mathematica
|
2024
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Vol. 44, no. 3
323--339
EN
In 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 μ-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 μ, and to adjoints of composition operators.
2
Content available An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components
Gil’ Michael
Opuscula Mathematica
|
2024
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Vol. 44, no. 2
241--248
EN
Let A be a bounded linear operator in a complex separable Hilbert space, A∗ be its adjoint one and AI := (A − A∗)/(2i). Assuming that AI is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of A. Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality [formula], where λk(A) (k = 1, 2, . . .) are the eigenvalues of A and N2(·) is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators.
3
Direct proofs of intrinsic properties of prox-regular sets in Hilbert spaces
Konstantinov Matey, Zlateva Nadia
Journal of Applied Analysis
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2023
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Vol. 29, nr 1
143--149
EN
We provide new proofs of two intrinsic properties of prox-regular sets in Hilbert spaces.
4
Content available Artificial Neural Network based on mathematical models used in quantum computing
Ruciński Dariusz
Studia Informatica : systems and information technology
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2022
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Vol. 2(27)
27--48
EN
The article is a proposition of a new approach to building a neural model based on the system of Day-Ahead Market operating at TGE S.A. The reason for the proposed method is an attempt to find a better model for the DAM system. The proposed methodology is based on using mathematical models used in quantum computing. All calculations performed on learning the Artificial Neuron Network are based on operations described in Hilbert space. The main idea of calculations is to replace the data from the decimal system into the quantum state in Hilbert space and perform learning operations for a neural model of the DAM system in a special manner which relay on the teaching model for each position of the quantum register for all data. The obtained results were compared to the “classical” neural model with the use of a comparative model.
5
Content available remote Sharp sufficient condition for the convergence of greedy expansions with errors in coefficient computation
Valiullin Artur R., Valiullin Albert R., Solodov Alexei P.
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
254--264
EN
Generalized approximate weak greedy algorithms (gAWGAs) were introduced by Galatenko and Livshits as a generalization of approximate weak greedy algorithms, which, in turn, generalize weak greedy algorithm and thus pure greedy algorithm. We consider a narrower case of gAWGA in which only a sequence of absolute errors {ξn}∞n=1 is nonzero. In this case sufficient condition for a convergence of a gAWGA expansion to an expanded element obtained by Galatenko and Livshits can be written as ∑∞n=1ξ2n<∞ . In the present article, we relax this condition and show that the convergence is guaranteed for ξn=o(1√n) . This result is sharp because the convergence may fail to hold for ξn≍1√n .
6
Content available remote Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces
Kondo Atsumasa
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
752--759
EN
In this article, we prove a common fixed point theorem for commutative nonlinear mappings that jointly satisfy a certain condition. From the main theorem, a common fixed point theorem for commutative generalized hybrid mappings is derived as a special case. Our novel approach significantly expands the applicable range of mappings for well-known fixed point theorems to be effective. Examples are presented to explicitly illustrate this contribution.
7
Content available remote Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces
Kim Kyung Soo
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
81--89
EN
The aim of this paper is to study existence and uniqueness of coupled fixed point for a family of self-mappings satisfying a new coupled implicit relation in a Hilbert space. We also prove well-posedness of a coupled fixed point problem.
8
Content available remote An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems
Okeke Chibueze C., Ugwunnadi Godwin C., Jolaoso Lateef O.
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
506--527
EN
The purpose of this article is to study and analyse a new extragradient-type algorithm with an inertial extrapolation step for solving split fixed-point problems for demicontractive mapping, equilibrium problem, and pseudomonotone variational inequality problem in real Hilbert spaces. One of the advantages of the proposed algorithm is that a strong convergence result is achieved without a prior estimate of the Lipschitz constant of the cost operator, which is very difficult to find. In addition, the stepsize is generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the Lipschitz constant of the cost operator. Some numerical experiments are reported to show the performance and behaviour of the sequence generated by our algorithm. The obtained results in this article extend and improve many related recent results in this direction in the literature.
9
Content available remote Recognizing the topologies of spaces of metrics with the topology of uniform convergence
Koshino Katsuhisa
Bulletin of the Polish Academy of Sciences. Mathematics
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2022
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Vol. 70, no. 2
165--171
EN
Given a metrizable space X of density κ, we study the topological structure of the space PM(X) of continuous bounded pseudometrics on X, which is endowed with the topology of uniform convergence. We prove that PM(X) is homeomorphic to [0,1)κ(κ−1)/2 if X is finite, to ℓ2(2<κ) if X is infinite and generalized compact, and to ℓ2(2κ) if X is not generalized compact. We also show that for an infinite σ-compact metrizable space X, the space M(X)⊂PM(X) of continuous bounded metrics on X and the space AM(X)⊂M(X) of bounded admissible metrics on X are homeomorphic to ℓ2 if X is compact, and to ℓ∞ if X is not compact.
10
Content available remote Most Cantor sets in RN are in general position with respect to all projections
Frolkina Olga
Bulletin of the Polish Academy of Sciences. Mathematics
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2022
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Vol. 70, no. 1
53--62
EN
We prove the theorem stated in the title. This answers a question of John Cobb (1994). We also consider the case of the Hilbert space ℓ2.
11
Content available On some extensions of the a-model
Jursenas Rytis
Opuscula Mathematica
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2020
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Vol. 40, no. 5
569--597
EN
The A-model for finite rank singular perturbations of class [formula], is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces [formula] admit an orthogonal decomposition [formula], with the corresponding projections satisfying [formula], nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.
12
Content available remote Structure of n-quasi left m-invertible and related classes of operators
Duggal Bhagwati Prashad, Kim In Hyun
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
249--268
EN
Given Hilbert space operators T, S ∈ B(H), let Δ and δ ∈ B(B (H)) denote the elementary operators ΔT,S(X) = (LT RS − I) (X) = TXS - X and δT,S(X) = (LT – RS)(X) = TX - XS. Let d = Δ or δ. Assuming T commutes with S∗, and choosing X to be the positive operator S∗nSn for some positive integer n, this paper exploits properties of elementary operators to study the structure of n-quasi [m, d]-operators dm T,S (X) = 0 to bring together, and improve upon, extant results for a number of classes of operators, such as n-quasi left m-invertible operators, n-quasi m-isometric operators, n-quasi m-self-adjoint operators and n-quasi (m, C) symmetric operators (for some conjugation C of H). It is proved that Sn is the perturbation by a nilpotent of the direct sum of an operator Sn1 = (…)n satisfying dmT1S1(I1) = 0 , T1 = (…) , with the 0 operator; if S is also left invertible, then Sn is similar to an operator B such that dmB∗,B(I) = 0. For power bounded S and T such that ST∗ - T∗S = 0 and ΔTS(S∗nSn) = 0, S is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators T, S satisfying dmT,S (I) = 0, given certain commutativity properties, transfers to operators satisfying S∗ndmT,S (I)Sn = 0.
13
Content available Geometry Interpretation of Differentiability of Rock Types in Hilbert‘s Space
Feriancikova Katarina, Lazarova Edita, Krulakova Maria, Ivanicova Lucia, Lesso Igor
Inżynieria Mineralna
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2019
|
R. 21, nr 1/2
49--54
EN
Signal of vibrations accompanying the rotary drilling of three rock types (andesite, limestone and granite) by diamond core-drill bits was processed and evaluated in order to track the signal characteristics of tested rock types. Mathematical procedures of Hilbert’s abstract space were applied to express the differences between the rock types based on vibration signal. Experiments were performed using the laboratory drilling rig designed and constructed at the Institute of Geotechnics SAS providing automated continuous monitoring of key process parameters (thrust force, rotation speed, torque, advance rate, etc.). Nominal regime of thrust force 5000 N and rotation speed 1000 rpm was used in the experiments along with monitoring with sampling frequency 17 kHz. The vibration signal was recorded by accelerometers in three orthogonal directions: axial in the drilling directions and two radial directions in horizontal and vertical planes. For the purposes of evaluation, only the vibrations in axial direction were assessed as their signal exhibits the highest entropy. A method providing the expression of mutual differences between the vibrations formed during the drilling of different rock types was developed, which enables to set the differences in abstract space to the planar visualization.
PL
Sygnały drgań pochodzących z wierceniu obrotowego trzech rodzajów skał (andezyt, wapień i granit) za pomocą diamentowych wierteł rdzeniowych został przetworzony i oceniony w celu śledzenia charakterystyk sygnałowych badanych rodzajów skał. Zastosowano matematyczne procedury przestrzeni Hilberta, aby wyrazić różnice między rodzajami skał w oparciu o sygnał wibracyjny. Eksperymenty przeprowadzono na laboratoryjnej platformie wiertniczej zaprojektowanej i skonstruowanej w Instytucie Geotechniki SAS, zapewniającej zautomatyzowane ciągłe monitorowanie kluczowych parametrów procesu (siły ciągu, prędkości obrotowej, momentu obrotowego, prędkości posuwu itp.). W doświadczeniach zastosowano nominalną wartość siły nacisku 5000 N i prędkości obrotowej 1000 rpm wraz z monitorowaniem częstotliwości 17 kHz. Sygnał drgań został zarejestrowany przez akcelerometry w trzech kierunkach ortogonalnych: osiowym w kierunkach wiercenia i dwóch promieniowych w płaszczyznach poziomej i pionowej. Do celów oceny oceniono jedynie drgania w kierunku osiowym, ponieważ ich sygnał wykazuje najwyższą entropię. Opracowano metodę wyrażania wzajemnych różnic między drganiami powstającymi podczas wiercenia różnych rodzajów skał, która umożliwia przeniesienie różnic z przestrzeni Hilberta na wizualizację dwuwymiarową.
14
Content available remote Reproducing Kernel Hilbert Space and Coalescence Hidden-variable Fractal Interpolation Functions
Prasad Srijanani Anurag
Demonstratio Mathematica
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2019
|
Vol. 52, nr 1
467--474
EN
Reproducing Kernel Hilbert Spaces (RKHS) and their kernel are important tools which have been found to be incredibly useful in many areas like machine learning, complex analysis, probability theory, group representation theory and the theory of integral operator. In the present paper, the space of Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is demonstrated to be an RKHS and its associated kernel is derived. This extends the possibility of using this new kernel function, which is partly self-affine and partly non-self-affine, in diverse fields wherein the structure is not always self-affine.
15
Content available remote A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces
Jolaoso Lateef Olakunle, Taiwo Adeolu, Alakoya Timilehin Opeyemi, Mewomo Oluwatosin Temitope
Demonstratio Mathematica
|
2019
|
Vol. 52, nr 1
183--203
EN
We consider a new subgradient extragradient iterative algorithm with inertial extrapolation for approximating a common solution of variational inequality problems and fixed point problems of a multivalued demicontractive mapping in a real Hilbert space. We established a strong convergence theorem for our proposed algorithm under some suitable conditions and without prior knowledge of the Lipschitz constant of the underlying operator. We present numerical examples to show that our proposed algorithm performs better than some recent existing algorithms in the literature.
16
Content available remote Ergodic and fixed point theorems for sequences and nonlinear mappings in a Hilbert space
Rouhani B. D.
Demonstratio Mathematica
|
2018
|
Vol. 51, nr 1
27--36
EN
In this paper, we introduce the notion of 2-generalized hybrid sequences, extending the notion of nonexpansive and hybrid sequences introduced and studied in our previous work [Djafari Rouhani B., Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. thesis, Yale University, 1981; and other published in J. Math. Anal. Appl., 1990, 2002, and 2014; Nonlinear Anal., 1997, 2002, and 2004], and prove ergodic and convergence theorems for such sequences in a Hilbert space H. Subsequently, we apply our results to prove new fixed point theorems for 2-generalized hybrid mappings, first introduced in [Maruyama T., Takahashi W., Yao M., Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal., 2011, 12, 185-197] and further studied in [Lin L.-J., Takahashi W., Attractive point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math., 2012, 16, 1763-1779], defined on arbitrary nonempty subsets of H.
17
Content available remote A unified algorithm for solving split generalized mixed equilibrium problem, and for finding fixed point of nonspreading mapping in Hilbert spaces
Jolaoso L. O., Oyewole K. O., Okeke C. C., Mewomo O. T.
Demonstratio Mathematica
|
2018
|
Vol. 51, nr 1
211--232
EN
The purpose of this paper is to study a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. We introduce a new iterative algorithm and prove its strong convergence for approximating a common solution of a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. Our algorithm is developed by combining a modified accelerated Mann algorithm and a viscosity approximation method to obtain a new faster iterative algorithm for finding a common solution of these problems in real Hilbert spaces. Also, our algorithm does not require any prior knowledge of the bounded linear operator norm. We further give a numerical example to show the efficiency and consistency of our algorithm. Our result improves and compliments many recent results previously obtained in this direction in the literature.
18
Content available The basis property of eigenfunctions in the problem of a nonhomogeneous damped string
Rzepnicki Ł.
Opuscula Mathematica
|
2017
|
Vol. 37, no. 1
141--165
EN
The equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator iA. We prove that the set of root vectors of the operator A forms a basis of subspaces in a certain Hilbert space H. Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator A is a Riesz basis for H.
19
Content available remote Synthesis of Transition Systems from Quantum Logics
Bernardinello L., Ferigato C., Pomello L., Puerto Aubel A.
Fundamenta Informaticae
|
2017
|
Vol. 154, nr 1/4
25--36
EN
The set of elementary regions of a transition system, ordered by set inclusion, forms an orthomodular poset, also referred to as quantum logic, which is regular and rich. Starting from an abstract regular and rich quantum logic, one can construct an elementary transition system such that the orginal logic embeds into its set of regions, and which is saturated of transitions. We study the problem of selecting subsets of transitions on the same set of states, which generate the same set of regions.
20
Content available Schauder’s fixed-point theorem in approximate controllability problems
Babiarz A., Klamka J., Niezabitowski M.
International Journal of Applied Mathematics and Computer Science
|
2016
|
Vol. 26, no. 2
263--275
EN
The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder’s fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.
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