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Journal articles on the topic 'Quasi-linear operator'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Gjonbalaj, Qefsere Doko, and Valdete Rexhëbeqaj Hamiti. "On M−quasi Paranormal Operators." European Journal of Pure and Applied Mathematics 15, no. 3 (July 31, 2022): 830–40. http://dx.doi.org/10.29020/nybg.ejpam.v15i3.4392.

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In this paper we introduce a new class of operators called M−quasi paranormal operators. A bounded linear operator T in a complex Hilbert space H is said to be a M−quasi paranormal operator if it satisfies ∥T 2x∥ 2 ≤ M∥T 3x∥ · ∥T x∥, ∀x ∈ H, where M is a real positive number. We prove basic properties, the structural and spectral properties of this class of operators.

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2

Wang, Han, and Jianrong Wu. "The norm of continuous linear operator between two fuzzy quasi-normed spaces." AIMS Mathematics 7, no. 7 (2022): 11759–71. http://dx.doi.org/10.3934/math.2022655.

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<abstract> <p>In this paper, firstly, we introduce the concepts of continuity and boundedness of linear operators between two fuzzy quasi-normed spaces with general continuous <italic>t</italic>-norms, prove the equivalence of them, and point out that the set of all continuous linear operators forms a convex cone. Secondly, we establish the family of star quasi-seminorms on the cone of continuous linear operators, and construct a fuzzy quasi-norm of a continuous linear operator.</p> </abstract>

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Abanin, Alexander V., and Julia V. Korablina. "Compactness of Linear Operators on Quasi-Banach Spaces of Holomorphic Functions." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 4-1 (216-1) (December 28, 2022): 83–89. http://dx.doi.org/10.18522/1026-2237-2022-4-1-83-89.

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We state conditions under which some classical operators acting from abstract quasi-Banach spaces of functions holomorphic in a plain domain into a weighted space of the same functions with sup-norm are compact. It is obtained abstract criteria for the compactness of a linear operator on an arbitrary quasi-Banach space which are stated in terms of delta-functions and formulate their realizations for both classical and generalized Fock spaces. The above results are applied to the weighted composition operator. It is established some conditions for the compactness of this operator which are given in terms of norms of delta-functions in the corresponding dual spaces. These results are essential generalizations of the known Zorboska’s ones. Namely, we significantly extended the class of weighted spaces of holomorphic functions with uniform norms for which one can state some conditions for the compactness of an arbitrary linear operator or the weighted composition operator.

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4

Bakery, Awad A., and Mustafa M. Mohammed. "Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means." Open Mathematics 17, no. 1 (December 31, 2019): 1703–15. http://dx.doi.org/10.1515/math-2019-0135.

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Abstract Let E be a generalized Cesáro sequence space defined by weighted means and by using s-numbers of operators from a Banach space X into a Banach space Y. We give the sufficient (not necessary) conditions on E such that the components $$\begin{array}{} \displaystyle S_{E}(X, Y):=\Big\{T\in L(X, Y):((s_{n}(T))_{n=0}^{\infty}\in E\Big\}, \end{array}$$ of the class SE form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal SE, the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small and the pre-quasi Banach operator ideal constructed by s-numbers is simple Banach space. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and this sequence space is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to this sequence space.

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5

Mohsen, Salim Dawood, and Hanan Khalid Mousa. "Another Results Related of Fuzzy Soft Quasi Normal Operator in Fuzzy Soft Hilbert Space." Journal of Physics: Conference Series 2322, no. 1 (August 1, 2022): 012050. http://dx.doi.org/10.1088/1742-6596/2322/1/012050.

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Abstract The goal of this paper, is to introduce another classes of the fuzzy soft bounded linear operator in the fuzzy soft Hilbert space which is a fuzzy soft quasi normal operator, as well as, give some properties about this concept with investigating the relationship among this types of the fuzzy soft bounded linear operator on fuzzy soft Hilbert space with other kinds of fuzzy soft bounded linear operators.

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6

Lohaj, Shqipe. "Structural and Spectral Properties of k-Quasi Class Q(N) and k-Quasi Class Q*(N) Operators." European Journal of Pure and Applied Mathematics 15, no. 4 (October 31, 2022): 1836–53. http://dx.doi.org/10.29020/nybg.ejpam.v15i4.4580.

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Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce two new classes of operators: k−quasi class Q(N ) and k−quasi class Q*(N ). An operator T ∈ L(H) is of k−quasi class Q(N ) for a fixed real number N ≥ 1 and k a natural number, if T satisfies N ∥T^k+1(x)∥^2 ≤ ∥T^k+2(x)∥^2 + ∥T^k(x)∥^2, for all x ∈ H. An operator T ∈ L(H) is of k−quasi class Q*(N ) for a fixed real number N ≥ 1 and k a natural number, if T satisfiesN ∥T*T^k(x)∥^2 ≤ ∥T^k+2(x)∥^2 + ∥T^k(x)∥^2, for all x ∈ H. We study structural and spectral properties of these classes of operators. Also we compare this new classes of operators with other known classes of operators

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7

Çavuş, Abdullah, Djavvat Khadjiev, and Seda Öztürk. "On periodic solutions to nonlinear differential equations in Banach spaces." Filomat 30, no. 4 (2016): 1069–76. http://dx.doi.org/10.2298/fil1604069c.

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Let A denote the generator of a strongly continuous periodic one-parameter group of bounded linear operators in a complex Banach space H. In this work, an analog of the resolvent operator which is called quasi-resolvent operator and denoted by R? is defined for points of the spectrum, some equivalent conditions for compactness of the quasi-resolvent operators R? are given. Then using these, some theorems on existence of periodic solutions to the non-linear equations ?(A)x = f (x) are given, where ?(A) is a polynomial of A with complex coefficients and f is a continuous mapping of H into itself.

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8

Malik, Saroj, and Néstor Thome. "On a revisited Moore-Penrose inverse of a linear operator on Hilbert spaces." Filomat 31, no. 7 (2017): 1927–31. http://dx.doi.org/10.2298/fil1707927m.

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For two given Hilbert spaces H and K and a given bounded linear operator A ? L(H,K) having closed range, it is well known that the Moore-Penrose inverse of A is a reflexive g-inverse G ? L(K,H) of A which is both minimum norm and least squares. In this paper, weaker equivalent conditions for an operator G to be the Moore-Penrose inverse of A are investigated in terms of normal, EP, bi-normal, bi-EP, l-quasi-normal and r-quasi-normal and l-quasi-EP and r-quasi-EP operators.

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9

Bakery, Awad A., and Mustafa M. Mohammed. "Small Pre-Quasi Banach Operator Ideals of Type Orlicz-Cesáro Mean Sequence Spaces." Journal of Function Spaces 2019 (May 2, 2019): 1–9. http://dx.doi.org/10.1155/2019/7265010.

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In this paper, we give the sufficient conditions on Orlicz-Cesáro mean sequence spaces cesφ, where φ is an Orlicz function such that the class Scesφ of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers which belong to cesφ forms an operator ideal. The completeness and denseness of its ideal components are specified and Scesφ constructs a pre-quasi Banach operator ideal. Some inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are explained. Moreover, we have presented the sufficient conditions on cesφ such that the pre-quasi Banach operator ideal generated by approximation number is small. The above results coincide with that known for cesp (1<p<∞).

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10

Micic, Jadranka, and Kemal Hot. "Inequalities among quasi-arithmetic means for continuous field of operators." Filomat 26, no. 5 (2012): 977–91. http://dx.doi.org/10.2298/fil1205977m.

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In this paper we study inequalities among quasi-arithmetic means for a continuous field of self-adjoint operators, a field of positive linear mappings and continuous strictly monotone functions which induce means. We present inequalities with operator convexity and without operator convexity of appropriate functions. Also, we present a general formulation of converse inequalities in each of these cases. Furthermore, we obtain refined inequalities without operator convexity. As applications, we obtain inequalities among power means.

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11

Dikhaminjia, N., J. Rogava, and M. Tsiklauri. "Operator Splitting for Quasi-Linear Abstract Hyperbolic Equation." Journal of Mathematical Sciences 218, no. 6 (September 28, 2016): 737–41. http://dx.doi.org/10.1007/s10958-016-3058-9.

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12

Mecheri, Salah. "On Quasi-Class A Operators." Annals of the Alexandru Ioan Cuza University - Mathematics 59, no. 1 (January 1, 2013): 163–72. http://dx.doi.org/10.2478/v10157-012-0020-0.

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Abstract Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let A;B be operators in B(H). In this paper we prove that if A is quasi-class A and B* is invertible quasi-class A and AX = XB, for some X ∈ C2 (the class of Hilbert-Schmidt operators on H), then A*X = XB*. We also prove that if A is a quasi-class A operator and f is an analytic function on a neighborhood of the spectrum of A, then f(A) satisfies generalized Weyl's theorem. Other related results are also given.

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13

Harju, Markus, Jaakko Kultima, Valery Serov, and Teemu Tyni. "Two-dimensional inverse scattering for quasi-linear biharmonic operator." Inverse Problems & Imaging 15, no. 5 (2021): 1015. http://dx.doi.org/10.3934/ipi.2021026.

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<p style='text-indent:20px;'>The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space <inline-formula><tex-math id="M1">\begin{document}$ W^1_{\infty}( \mathbb{{R}}^2) $\end{document}</tex-math></inline-formula>. Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator. Moreover, we present a numerical method for the reconstruction of unknown coefficients, which from the practical point of view can be thought of as recovery of the coefficients from fixed energy measurements.</p>

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14

Dikhaminjia, Nana, Jemal Rogava, and Mikheil Tsiklauri. "Construction and numerical resolution of high-order accuracy decomposition scheme for a quasi-linear evolution equation." Georgian Mathematical Journal 25, no. 3 (September 1, 2018): 337–48. http://dx.doi.org/10.1515/gmj-2018-0004.

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AbstractIn the present work the Cauchy problem for an abstract evolution equation with a Lipschitz-continuous operator is considered, where the main operator represents the sum of positive definite self-adjoint operators. The fourth-order accuracy decomposition scheme is constructed for an approximate solution of the problem. The theorem on the error estimate of an approximate solution is proved. Numerical calculations for different model problems are carried out using the constructed scheme. The obtained numerical results confirm the theoretical conclusions.

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15

Drnovšek, Roman, Nika Novak, and Vladimir Müller. "An operator is a product of two quasi-nilpotent operators if and only if it is not semi-Fredholm." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 5 (October 2006): 935–44. http://dx.doi.org/10.1017/s0308210500004819.

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We prove that a (bounded, linear) operator acting on an infinite-dimensional, separable, complex Hilbert space can be written as a product of two quasi-nilpotent operators if and only if it is not a semi-Fredholm operator. This solves the problem posed by Fong and Sourour in 1984. We also consider some closely related questions. In particular, we show that an operator can be expressed as a product of two nilpotent operators if and only if its kernel and co-kernel are both infinite dimensional. This answers the question implicitly posed by Wu in 1989.

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16

Cao, Jianbing, and Yifeng Xue. "The Quasi-Linear Operator Outer Generalized Inverse with Prescribed Range and Kernel in Banach Spaces." Journal of Operators 2013 (August 4, 2013): 1–7. http://dx.doi.org/10.1155/2013/204587.

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Let and be Banach spaces, and let be a bounded linear operator. In this paper, we first define and characterize the quasi-linear operator (resp., out) generalized inverse (resp., ) for the operator , where and are homogeneous subsets. Then, we further investigate the perturbation problems of the generalized inverses and . The results obtained in this paper extend some well-known results for linear operator generalized inverses with prescribed range and kernel.

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17

Duggal, B. P. "A remark on the essential spectra of quasi-similar dominant contractions." Glasgow Mathematical Journal 31, no. 2 (May 1989): 165–68. http://dx.doi.org/10.1017/s0017089500007680.

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We consider operators, i.e. bounded linear transformations, on an infinite dimensional separable complex Hilbert space H into itself. The operator A is said to be dominant if for each complex number λ there exists a number Mλ(≥l) such that ∥(A – λ)*x∥ ≤ Mλ∥A – λ)x∥ for each x∈H. If there exists a number M≥Mλ for all λ, then the dominant operator A is said to be M-hyponormal. The class of dominant (and JW-hyponormal) operators was introduced by J. G. Stampfli during the seventies, and has since been considered in a number of papers, amongst then [7], [11]. It is clear that a 1-hyponormal is hyponormal. The operator A*A is said to be quasi-normal if Acommutes with A*A, and we say that A is subnormal if A has a normal extension. It is known that the classes consisting of these operators satisfy the following strict inclusion relation:

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18

Zhao, Jing, and Hang Zhang. "Solving Split Common Fixed-Point Problem of Firmly Quasi-Nonexpansive Mappings without Prior Knowledge of Operators Norms." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/389689.

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Very recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms for the split common fixed-point problem concerned two bounded linear operators. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. It is the purpose of this paper to introduce a viscosity iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. We prove the strong convergence of the proposed algorithms for split common fixed-point problem governed by the firmly quasi-nonexpansive operators. As a consequence, we obtain strong convergence theorems for split feasibility problem and split common null point problems of maximal monotone operators. Our results improve and extend the corresponding results announced by many others.

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19

Zhdanov, Michael S., Sheng Fang, and Gábor Hursán. "Electromagnetic inversion using quasi‐linear approximation." GEOPHYSICS 65, no. 5 (September 2000): 1501–13. http://dx.doi.org/10.1190/1.1444839.

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Three‐dimensional electromagnetic inversion continues to be a challenging problem in electrical exploration. We have recently developed a new approach to the solution of this problem based on quasi‐linear approximation of a forward modeling operator. It generates a linear equation with respect to the modified conductivity tensor, which is proportional to the reflectivity tensor and the complex anomalous conductivity. We solved this linear equation by using the regularized conjugate gradient method. After determining a modified conductivity tensor, we used the electrical reflectivity tensor to evaluate the anomalous conductivity. Thus, the developed inversion scheme reduces the original nonlinear inverse problem to a set of linear inverse problems. The developed algorithm has been realized in computer code and tested on synthetic 3-D EM data. The case histories include interpretation of a 3-D magnetotelluric survey conducted in Hokkaido, Japan, and the 3-D inversion of the tensor controlled‐source audio magnetotelluric data over the Sulphur Springs thermal area, Valles Caldera, New Mexico, U.S.A.

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20

Malyshev, Igor. "On some perturbation techniques for quasi-linear parabolic equations." Journal of Applied Mathematics and Stochastic Analysis 3, no. 3 (January 1, 1990): 169–75. http://dx.doi.org/10.1155/s1048953390000168.

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We study a nonhomogeneous quasi-linear parabolic equation and introduce a method that allows us to find the solution of a nonlinear boundary value problem in explicit form. This task is accomplished by perturbing the original equation with a source function, which is then found as a solution of some nonlinear operator equation.

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Jafarizad, S., and A. Ranjbari. "Some applications of the open mapping theorem in locally convex cones." Ukrains’kyi Matematychnyi Zhurnal 73, no. 3 (March 19, 2021): 425–30. http://dx.doi.org/10.37863/umzh.v73i3.222.

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UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones. Specially, we prove some relations between an open linear operator and its adjoint in uc-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).

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Imanbaev, N. S., and Ye Kurmysh. "On zeros of an entire function coinciding with exponential typequasi-polynomials, associated with a regular third-order differential operator on an interval." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 103, no. 3 (September 30, 2021): 44–53. http://dx.doi.org/10.31489/2021m3/44-53.

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In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W^3_2(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed.

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23

Rashid, M. H. M., and M. S. M. Noorani. "On relaxation normality in the Fuglede-Putnam theorem for a quasi-class $A$ operators." Tamkang Journal of Mathematics 40, no. 3 (September 30, 2009): 307–12. http://dx.doi.org/10.5556/j.tkjm.40.2009.508.

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Let $T$ be a bounded linear operator acting on a complex Hilbert space $ \mathcal{H} $. In this paper, we show that if $A$ is quasi-class $A$, $ B^* $ is invertible quasi-class $A$, $X$ is a Hilbert-Schmidt operator, $AX=XB$ and $ \left\Vert |A^*| \right\Vert \left\Vert |B|^{-1} \right\Vert \leq 1 $, then $ A^* X = X B^* $.

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24

Mukhamedov, Farrukh, and Abduaziz Abduganiev. "On Pure Quasi-Quantum Quadratic Operators of 𝕄2(ℂ)." Open Systems & Information Dynamics 20, no. 04 (November 25, 2013): 1350018. http://dx.doi.org/10.1142/s1230161213500182.

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In this paper we study quasi-quantum quadratic operators (quasi-QQO) acting on the algebra of 2 × 2 matrices 𝕄2(ℂ). It is known that a channel is called pure if it sends pure states to pure ones. In this paper, we introduce a weaker condition for the channel called q-purity. To study q-pure channels, we concentrate on quasi-QQO acting on 𝕄2(ℂ). We describe all trace-preserving quasi-QQO on 𝕄2(ℂ), which allows us to prove that if a trace-preserving symmetric quasi-QQO is such that the corresponding quadratic operator is linear, then its q-purity implies its positivity. If a symmetric quasi-QQO has a Haar state τ, then its corresponding quadratic operator is nonlinear, and it is proved that such q-pure symmetric quasi-QQO cannot be positive. We think that such a result will allow one to check whether a given mapping from 𝕄2(ℂ) to 𝕄2(ℂ) ⊗ 𝕄2(ℂ) is pure or not. On the other hand, our study is related to the construction of pure quantum nonlinear channels. Moreover, we also indicate that nonlinear dynamics associated with pure quasi-QQO may have different kind of dynamics, i.e. it may behave chaotically or trivially.

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Pedregal, Pablo. "Weak continuity and weak lower semicontinuity for some compensation operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 113, no. 3-4 (1989): 267–79. http://dx.doi.org/10.1017/s0308210500024136.

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SynopsisWe study a special class of linear differential operators well-behaved with respect to weakconvergence. Questions related to weak lower semicontinuity, associated Young measures, weak continuity and quasi-convexity are addressed. Specifically, it is shown that the well-known necessary conditions for weak lower semicontinuity are also sufficient in this case. Some examples are given, including a discussion on how well the operator curl fits inthis context.

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Saranya, K., V. Piramanantham, and E. Thandapani. "Oscillation Results for Third-Order Semi-Canonical Quasi-Linear Delay Differential Equations." Nonautonomous Dynamical Systems 8, no. 1 (January 1, 2021): 228–38. http://dx.doi.org/10.1515/msds-2020-0135.

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Abstract The main purpose of this paper is to study the oscillatory properties of solutions of the third-order quasi-linear delay differential equation ℒ y ( t ) + f ( t ) y β ( σ ( t ) ) = 0 {\cal L}y(t) + f(t){y^\beta }(\sigma (t)) = 0 where ℒy(t) = (b(t)(a(t)(y 0(t)) )0)0 is a semi-canonical differential operator. The main idea is to transform the semi-canonical operator into canonical form and then obtain new oscillation results for the studied equation. Examples are provided to illustrate the importance of the main results.

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Morchid Alaoui, Moulay Driss, Abdelouahed El Khalil, and Abdelfattah Touzani. "Multiplicity results for quasi linear problems involving p(x)-Laplace operator." Complex Variables and Elliptic Equations 63, no. 11 (November 24, 2017): 1664–74. http://dx.doi.org/10.1080/17476933.2017.1400540.

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Boussetila, Nadjib, and Faouzia Rebbani. "A Modified Quasi-Reversibility Method for a Class of Ill-Posed Cauchy Problems." gmj 14, no. 4 (December 2007): 627–42. http://dx.doi.org/10.1515/gmj.2007.627.

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Abstract The goal of this paper is to present some extensions of the method of quasi-reversibility applied to an ill-posed Cauchy problem associated with an unbounded linear operator in a Hilbert space. The key point to our proof is the use of a new perturbation to construct a family of regularizing operators for the considered problem. We show the convergence of this method, and we estimate the convergence rate under a priori regularity assumptions on the problem data.

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Dmytryshyn, M. I., and O. V. Lopushansky. "Spectral approximations of strongly degenerate elliptic differential operators." Carpathian Mathematical Publications 11, no. 1 (June 30, 2019): 48–53. http://dx.doi.org/10.15330/cmp.11.1.48-53.

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We establish analytical estimates of spectral approximations errors for strongly degenerate elliptic differential operators in the Lebesgue space $L_q(\Omega)$ on a bounded domain $\Omega$. Elliptic operators have coefficients with strong degeneration near boundary. Their spectrum consists of isolated eigenvalues of finite multiplicity and the linear span of the associated eigenvectors is dense in $L_q(\Omega)$. The received results are based on an appropriate generalization of Bernstein-Jackson inequalities with explicitly calculated constants for quasi-normalized Besov-type approximation spaces which are associated with the given elliptic operator. The approximation spaces are determined by the functional $E\left(t,u\right)$, which characterizes the shortest distance from an arbitrary function ${u\in L_q(\Omega)}$ to the closed linear span of spectral subspaces of the given operator, corresponding to the eigenvalues such that not larger than fixed ${t>0}$. Such linear span of spectral subspaces coincides with the subspace of entire analytic functions of exponential type not larger than ${t>0}$. The approximation functional $E\left(t,u\right)$ in our cases plays a similar role as the modulus of smoothness in the functions theory.

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Penenko, Alexey, Vladimir Penenko, Elena Tsvetova, Alexander Gochakov, Elza Pyanova, and Viktoriia Konopleva. "Sensitivity Operator Framework for Analyzing Heterogeneous Air Quality Monitoring Systems." Atmosphere 12, no. 12 (December 18, 2021): 1697. http://dx.doi.org/10.3390/atmos12121697.

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Air quality monitoring systems differ in composition and accuracy of observations and their temporal and spatial coverage. A monitoring system’s performance can be assessed by evaluating the accuracy of the emission sources identified by its data. In the considered inverse modeling approach, a source identification problem is transformed to a quasi-linear operator equation with the sensitivity operator. The sensitivity operator is composed of the sensitivity functions evaluated on the adjoint ensemble members. The members correspond to the measurement data element aggregates. Such ensemble construction allows working in a unified way with heterogeneous measurement data in a single-operator equation. The quasi-linear structure of the resulting operator equation allows both solving and predicting solutions of the inverse problem. Numerical experiments for the Baikal region scenario were carried out to compare different types of inverse problem solution accuracy estimates. In the considered scenario, the projection to the orthogonal complement of the sensitivity operator’s kernel allowed predicting the source identification results with the best accuracy compared to the other estimate types. Our contribution is the development and testing of a sensitivity-operator-based set of tools for analyzing heterogeneous air quality monitoring systems. We propose them for assessing and optimizing observational systems and experiments.

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CALVO, T., G. MAYOR, J. TORRENS, J. SUÑER, M. MAS, and M. CARBONELL. "GENERATION OF WEIGHTING TRIANGLES ASSOCIATED WITH AGGREGATION FUNCTIONS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 08, no. 04 (August 2000): 417–51. http://dx.doi.org/10.1142/s0218488500000290.

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In this work, we present several ways to obtain different types of weighting triangles, due to these types characterize some interesting properties of Extended Ordered Weighted Averaging operators, EOWA, and Extended Quasi-linear Weighted Mean, EQLWM, as well as of their reverse functions. We show that any quantifier determines an EOWA operator which is also an Extended Aggregation Function, EAF, i.e., the weighting triangle generated by a quantifier is always regular. Moreover, we present different results about generation of weighting triangles by means of sequences and fractal structures. Finally, we introduce a degree of orness of a weighting triangle associated with an EOWA operator. After that, we mention some results on each class of triangle, considering each one of these triangles as triangles associated with their corresponding EOWA operator, and we calculate the ornessof some interesting examples.

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Francomano, E., C. Lodato, S. Lopes, and A. Tortorici. "An Algorithm for Optical Flow Computation Based on a Quasi-Interpolant Operator." Computing Letters 2, no. 1-2 (March 6, 2006): 93–106. http://dx.doi.org/10.1163/157404006777491954.

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A fundamental problem in the processing of image sequences is the computation of the velocity field of the apparent motion of brightness patterns usually referred to optical flow. In this paper a novel optical flow estimator based on a bivariate quasi-interpolant operator is presented. Namely, a non linear minimizing technique has been employed to compute the velocity vectors by modeling the flow field with a 2D quasi-interpolant operator based on centered cardinal B-spline functions. In this way an efficient computational scheme for optical flow estimate is provided. In addition the large solving linear systems involved in the process are sparse. Experiments on several image sequences have been carried out in order to investigate the performance of the optical flow estimator.

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Tahir, Jawad Kadhim. "Numerical Computations for One Class of Dynamical Mathematical Models in Quasi-Sobolev Space." Mathematical Modelling of Engineering Problems 8, no. 2 (April 28, 2021): 267–72. http://dx.doi.org/10.18280/mmep.080214.

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The article studies some mathematical models that represent one class of dynamical equations in quasi-Sobolev space. The analytical investigation of solvability of the Cauchy problem in the quasi-Sobolev space and theoretical results used to enhance and develop an algorithm structure of the numerical procedures to find approximate solutions for models, the steps of algorithm based on the theoretical investigation of models, new algorithm of numerical method allowing to find approximate solutions of mathematical models under study in quasi-Sobolev space. Construction a program implements an algorithm of numerical method that allow finding approximate solutions for models. To construct the theory of degenerate holomorphic semigroups of operators in quasi-Banach spaces of sequences, we used the classical methods of functional analysis, theory of linear bounded operators, spectral theory. To construct the operators of resolving semigroups we used the Laplace transform of operator-valued functions in quasi-Banach spaces of sequences. The numerical investigation for models generate some approximate solutions which are normally based on the modified projection method. The convergence of the approximate solution to the exact one theoretically is justified by the convergence of the corresponding series, the agreement of approximate computations with the theoretical solution is established.

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El-Houari, H., L. S. Chadli, and H. Moussa. "On a Class of Schrödinger System Problem in Orlicz–Sobolev Spaces." Journal of Function Spaces 2022 (March 30, 2022): 1–13. http://dx.doi.org/10.1155/2022/2486542.

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Using the mountain pass theorem, we obtain the existence of a nontrivial and nonnegative weak solution of a quasi-linear Schrödinger system driven by the ω ⋅ -Laplacian operator in Orlicz–Sobolev spaces.

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35

Esposito, Giampiero. "A parametrix for quantum gravity?" International Journal of Geometric Methods in Modern Physics 13, no. 05 (April 21, 2016): 1650060. http://dx.doi.org/10.1142/s0219887816500602.

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In the 60s, DeWitt discovered that the advanced and retarded Green functions of the wave operator on metric perturbations in the de Donder gauge make it possible to define classical Poisson brackets on the space of functionals that are invariant under the action of the full diffeomorphism group of spacetime. He therefore tried to exploit this property to define invariant commutators for the quantized gravitational field, but the operator counterpart of such classical Poisson brackets turned out to be a hard task. On the other hand, in the mathematical literature, it is by now clear that, rather than inverting exactly an hyperbolic (or elliptic) operator, it is more convenient to build a quasi-inverse, i.e. an inverse operator up to an operator of lower order which plays the role of regularizing operator. This approximate inverse, the parametrix, which is, strictly, a distribution, makes it possible to solve inhomogeneous hyperbolic (or elliptic) equations. We here suggest that such a construction might be exploited in canonical quantum gravity provided one understands what is the counterpart of classical smoothing operators in the quantization procedure. We begin with the simplest case, i.e. fundamental solution and parametrix for the linear, scalar wave operator; the next step are tensor wave equations, again for linear theory, e.g. Maxwell theory in curved spacetime. Last, the nonlinear Einstein equations are studied, relying upon the well-established Choquet-Bruhat construction, according to which the fifth derivatives of solutions of a nonlinear hyperbolic system solve a linear hyperbolic system. The latter is solved by means of Kirchhoff-type formulas, while the former fifth-order equations can be solved by means of well-established parametrix techniques for elliptic operators. But then the metric components that solve the vacuum Einstein equations can be obtained by convolution of such a parametrix with Kirchhoff-type formulas. Some basic functional equations for the parametrix are also obtained, that help in studying classical and quantum version of the Jacobi identity.

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36

Feldman, Richard M., Bryan L. Deuermeyer, and Ciriaco Valdez-Flores. "Utilization of the method of linear matrix equations to solve a quasi-birth-death problem." Journal of Applied Probability 30, no. 3 (September 1993): 639–49. http://dx.doi.org/10.2307/3214772.

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The steady-state analysis of a quasi-birth-death process is possible by matrix geometric procedures in which the root to a quadratic matrix equation is found. A recent method that can be used for analyzing quasi-birth–death processes involves expanding the state space and using a linear matrix equation instead of the quadratic form. One of the difficulties of using the linear matrix equation approach regards the boundary conditions and obtaining the norming equation. In this paper, we present a method for calculating the boundary values and use the operator-machine interference problem as a vehicle to compare the two approaches for solving quasi-birth-death processes.

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37

Feldman, Richard M., Bryan L. Deuermeyer, and Ciriaco Valdez-Flores. "Utilization of the method of linear matrix equations to solve a quasi-birth-death problem." Journal of Applied Probability 30, no. 03 (September 1993): 639–49. http://dx.doi.org/10.1017/s0021900200044375.

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The steady-state analysis of a quasi-birth-death process is possible by matrix geometric procedures in which the root to a quadratic matrix equation is found. A recent method that can be used for analyzing quasi-birth–death processes involves expanding the state space and using a linear matrix equation instead of the quadratic form. One of the difficulties of using the linear matrix equation approach regards the boundary conditions and obtaining the norming equation. In this paper, we present a method for calculating the boundary values and use the operator-machine interference problem as a vehicle to compare the two approaches for solving quasi-birth-death processes.

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38

BRIHAYE, YVES, JEAN NDIMUBANDI, and BHABANI PRASAD MANDAL. "QES SYSTEMS, INVARIANT SPACES AND POLYNOMIALS RECURSIONS." Modern Physics Letters A 22, no. 19 (June 21, 2007): 1423–38. http://dx.doi.org/10.1142/s0217732307023912.

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Let us denote [Formula: see text], the finite-dimensional vector spaces of functions of the form ψ(x) = pn(x)+f(x)pm(x), where pn(x) and pm(x) are arbitrary polynomials of degree at most n and m in the variable x while f(x) represents a fixed function of x. Conditions on m, n and f(x) are found such that families of linear differential operators exist which preserve [Formula: see text]. A special emphasis is accorded to the cases where the set of differential operators represents the enveloping algebra of some abstract algebra. These operators can be transformed into linear matrix valued differential operators. In the second part, such types of operators are considered and a connection is established between their solutions and series of polynomials-valued vectors obeying three terms recurrence relations. When the operator is quasi-exactly solvable, it possesses a finite-dimensional invariant vector space. We study how this property leads to the truncation of the polynomials series.

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39

Clason, Christian, and Andrej Klassen. "Quasi-solution of linear inverse problems in non-reflexive Banach spaces." Journal of Inverse and Ill-posed Problems 26, no. 5 (October 1, 2018): 689–702. http://dx.doi.org/10.1515/jiip-2018-0026.

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Abstract We consider the method of quasi-solutions (also referred to as Ivanov regularization) for the regularization of linear ill-posed problems in non-reflexive Banach spaces. Using the equivalence to a metric projection onto the image of the forward operator, it is possible to show regularization properties and to characterize parameter choice rules that lead to a convergent regularization method, which includes the Morozov discrepancy principle. Convergence rates in a suitably chosen Bregman distance can be obtained as well. We also address the numerical computation of quasi-solutions to inverse source problems for partial differential equations in {L^{\infty}(\Omega)} using a semi-smooth Newton method and a backtracking line search for the parameter choice according to the discrepancy principle. Numerical examples illustrate the behavior of quasi-solutions in this setting.

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40

Jovanović, Boško, Sergey Lemeshevsky, and Peter Matus. "On the Stability of Differential-operator Equations and Operator-difference Schemes as t → ∞." Computational Methods in Applied Mathematics 2, no. 2 (2002): 153–70. http://dx.doi.org/10.2478/cmam-2002-0010.

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AbstractFor the abstract Cauchy problem for a parabolic equation a priori estimates of the global and asymptotic stability in various energy norms have been obtained. Similar problems are also considered for the second-order equation. In the latter case, a priori estimates of the asymptotic stability by the initial data have been obtained. The corresponding estimates of the global stability for three-level operator difference schemes have been proved. Estimates of the asymptotic behavior of the solution for quasi-linear multidimensional equations with unbounded nonlinearity have been obtained. The corresponding mathematical apparatus permitting one to prove unconditional monotonicity of the difference schemes approximating nonlinear problems is presented.

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41

Popkov, Yuri S. "Controlled Positive Dynamic Systems with an Entropy Operator: Fundamentals of the Theory and Applications." Mathematics 9, no. 20 (October 14, 2021): 2585. http://dx.doi.org/10.3390/math9202585.

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Controlled dynamic systems with an entropy operator (DSEO) are considered. Mathematical models of such systems were used to study the dynamic properties in demo-economic systems, the spatiotemporal evolution of traffic flows, recurrent procedures for restoring images from projections, etc. Three problems of the study of DSEO are considered: the existence and uniqueness of singular points and the influence of control on them; stability in “large” of the singular points; and optimization of program control with linear feedback. The theorems of existence, uniqueness, and localization of singular points are proved using the properties of equations with monotone operators and the method of linear majorants of the entropy operator. The theorem on asymptotic stability of the DSEO in “large” is proven using differential inequalities. Methods for the synthesis of quasi-optimal program control and linear feedback control with integral quadratic quality functional, and ensuring the existence of a nonzero equilibrium, were developed. A recursive method for solving the integral equations of the DSEO using the multidimensional functional power series and the multidimensional Laplace transform was developed. The problem of managing regional foreign direct investment is considered, the distribution of flows is modeled by the corresponding DSEO. It is shown that linear feedback control is a more effective tool than program control.

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42

Itou, Hiromichi, Victor A. Kovtunenko, and Kumbakonam R. Rajagopal. "Crack problem within the context of implicitly constituted quasi-linear viscoelasticity." Mathematical Models and Methods in Applied Sciences 29, no. 02 (February 2019): 355–72. http://dx.doi.org/10.1142/s0218202519500118.

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A quasi-linear viscoelastic relation that stems from an implicit viscoelastic constitutive body containing a crack is considered. The abstract form of the response function is given first in [Formula: see text], [Formula: see text], due to power-law hardening; second in [Formula: see text] due to limiting small strain. In both the cases, sufficient conditions on admissible response functions are formulated, and corresponding existence theorems are proved rigorously based on the variational theory and using monotonicity methods. Due to the presence of a Volterra convolution operator, an auxiliary-independent variable of velocity type is employed. In the case of limiting small strain, the generalized solution of the problem is provided within the context of bounded measures and expressed by a variational inequality.

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43

Lapin, Alexander V., Vladimir V. Shaydurov, and Ruslan M. Yanbarisov. "Finite difference scheme for a non-linear subdiffusion problem with a fractional derivative along the trajectory of motion." Russian Journal of Numerical Analysis and Mathematical Modelling 38, no. 1 (February 1, 2023): 23–35. http://dx.doi.org/10.1515/rnam-2023-0003.

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Abstract The article is devoted to the construction and study of a finite-difference scheme for a one-dimensional diffusion–convection equation with a fractional derivative with respect to the characteristic of the convection operator. It develops the previous results of the authors from [5, 6] in the following ways: the differential equation contains a fractional derivative of variable order along the characteristics of the convection operator and a quasi-linear diffusion operator; a new accuracy estimate is proved, which singles out the dependence of the accuracy of mesh scheme on the curvature of the characteristics.

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Lapin, A. "Mixed Hybrid Finite Element Method for a Variational Inequality with a Quasi-linear Operator." Computational Methods in Applied Mathematics 9, no. 4 (2009): 354–67. http://dx.doi.org/10.2478/cmam-2009-0023.

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Abstract A mixed hybrid finite element method has been applied to a variational inequality with a potential second-order quasi-linear differential operator. The Lagrange multiplier method for a dual problem has been used to construct this finite element scheme. The existence and uniqueness of a solution for the resulting finite- dimensional problem has been proved, the solution iterative methods are discussed. The non-overlapping domain decomposition method combined with the mixed hybrid finite element approximation is analyzed.

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45

Zhdanov, Michael S., Vladimir I. Dmitriev, Sheng Fang, and Gábor Hursán. "Quasi‐analytical approximations and series in electromagnetic modeling." GEOPHYSICS 65, no. 6 (November 2000): 1746–57. http://dx.doi.org/10.1190/1.1444859.

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The quasi‐linear approximation for electromagnetic forward modeling is based on the assumption that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background (normal) field through an electrical reflectivity tensor λ⁁. In the original formulation of the quasi‐linear approximation, λ⁁ was determined by solving a minimization problem based on an integral equation for the scattering currents. This approach is much less time‐consuming than the full integral equation method; however, it still requires solution of the corresponding system of linear equations. In this paper, we present a new approach to the approximate solution of the integral equation using λ⁁ through construction of quasi‐analytical expressions for the anomalous electromagnetic field for 3-D and 2-D models. Quasi‐analytical solutions reduce dramatically the computational effort related to forward electromagnetic modeling of inhomogeneous geoelectrical structures. In the last sections of this paper, we extend the quasi‐analytical method using iterations and develop higher order approximations resulting in quasi‐analytical series which provide improved accuracy. Computation of these series is based on repetitive application of the given integral contraction operator, which insures rapid convergence to the correct result. Numerical studies demonstrate that quasi‐analytical series can be treated as a new powerful method of fast but rigorous forward modeling solution.

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46

Sumin, Vladimir I. "Volterra funktional equations in the stability problem for the existence of global solutions of distributed controlled systems." Russian Universities Reports. Mathematics, no. 132 (2020): 422–40. http://dx.doi.org/10.20310/2686-9667-2020-25-132-422-440.

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Earlier the author proposed a rather general form of describing controlled initial–boundary value problems (CIBVPs) by means of Volterra functional equations (VFE) z(t)=f(t,A[z](t),v(t) ), t≡{t^1,⋯,t^n }∈Π⊂R^n, z∈L_p^m≡(L_p (Π) )^m, where f(.,.,.):Π×R^l×R^s→R^m; v(.)∈D⊂L_k^s – control function; A:L_p^m (Π)→L_q^l (Π)- linear operator; the operator A is a Volterra operator for some system T of subsets of the set Π in the following sense: for any H∈T, the restriction A├ [z]┤|_H does not depend on the values of ├ z┤|_(Π\H); (this definition of the Volterra operator is a direct multidimensional generalization of the well-known Tikhonov definition of a functional Volterra type operator). Various CIBVP (for nonlinear hyperbolic and parabolic equations, integro-differential equations, equations with delay, etc.) are reduced by the method of conversion the main part to such functional equations. The transition to equivalent VFE-description of CIBVP is adequate to many problems of distributed optimization. In particular, the author proposed (using such description) a scheme for obtaining sufficient stability conditions (under perturbations of control) of the existence of global solutions for CIBVP. The scheme uses continuation local solutions of functional equation (that is, solutions on the sets H∈T). This continuation is realized with the help of the chain {H_1⊂H_2⊂⋯⊂H_(k-1)⊂H_k≡Π}, where H_i∈T, i=¯(1,k.) A special local existence theorem is applied. This theorem is based on the principle of contraction mappings. In the case p=q=k=∞ under natural assumptions, the possibility of applying this principle is provided by the following: the right-hand side operator F_v [z(.) ](t)≡f(t,A[z](t),v(t)) satisfies the Lipschitz condition in the operator form with the quasi-nilpotent «Lipschitz operator». This allows (using well-known results of functional analysis) to introduce in the space L_∞^m (H) such an equivalent norm in which the operator of the right-hand side will be contractive. In the general case 1≤p,q,k ≤∞, (this case covers a much wider class of CIBVP), the operator F_v; as a rule, does not satisfy such Lipschitz condition. From the results obtained by the author earlier, it follows that in this case there also exists an equivalent norm of the space L_p^m (H), for which the operator F_v is a contraction operator. The corresponding basic theorem (equivalent norm theorem) is based on the notion of equipotential quasi-nilpotency of a family of linear operators, acting in a Banach space. This article shows how this theorem can be applied to obtain sufficient stability conditions (under perturbations of control) of the existence of global solutions of VFE.

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BRIHAYE, Y., and A. NININAHAZWE. "DIRAC OSCILLATORS AND QUASI-EXACTLY SOLVABLE OPERATORS." Modern Physics Letters A 20, no. 25 (August 20, 2005): 1875–85. http://dx.doi.org/10.1142/s0217732305018128.

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The Dirac equation is formulated in the background of three types of physically relevant potentials: scalar, vector and "Dirac-oscillator" potentials. Assuming these potentials to be spherically-symmetric and with generic polynomial forms in the radial variable, we construct the corresponding radial Dirac equation. Cases where this linear spectral equation is exactly solvable or quasi-exactly solvable are worked out in details. When available, relations between the radial Dirac operator and some super-algebra are pointed out.

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Barratt, Dylan, Ton Stefan van den Bremer, and Thomas Alan Adcock Adcock. "MNLS simulations of surface wave groups with directional spreading in deep and finite depth waters." Journal of Ocean Engineering and Marine Energy 7, no. 3 (June 10, 2021): 261–75. http://dx.doi.org/10.1007/s40722-021-00201-2.

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AbstractWe simulate focusing surface gravity wave groups with directional spreading using the modified nonlinear Schrödinger (MNLS) equation and compare the results with a fully-nonlinear potential flow code, OceanWave3D. We alter the direction and characteristic wavenumber of the MNLS carrier wave, to assess the impact on the simulation results. Both a truncated (fifth-order) and exact version of the linear dispersion operator are used for the MNLS equation. The wave groups are based on the theory of quasi-determinism and a narrow-banded Gaussian spectrum. We find that the truncated and exact dispersion operators both perform well if: (1) the direction of the carrier wave aligns with the direction of wave group propagation; (2) the characteristic wavenumber of the carrier wave coincides with the initial spectral peak. However, the MNLS simulations based on the exact linear dispersion operator perform significantly better if the direction of the carrier wave does not align with the wave group direction or if the characteristic wavenumber does not coincide with the initial spectral peak. We also perform finite-depth simulations with the MNLS equation for dimensionless depths ($$k_{\text {p}}d$$ k p d ) between 1.36 and 5.59, incorporating depth into the boundary conditions as well as the dispersion operator, and compare the results with those of fully-nonlinear potential flow code to assess the finite-depth limitations of the MNLS.

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49

Rudakov, I. A. "Oscillation Problem for an I-Beam with Fixed and Hinged End Supports." Herald of the Bauman Moscow State Technical University. Series Natural Sciences, no. 84 (June 2019): 4–21. http://dx.doi.org/10.18698/1812-3368-2019-3-4-21.

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The paper investigates the problem concerning time-periodic solutions to a quasi-linear equation describing forced oscillations of an I-beam with fixed and hinged end supports. The non-linear term and the right side of the equation are time-periodic functions. We seek a Fourier series solution to the equation. In order to construct an orthonormal system, we studied the eigenvalue problem for a differential operator representing the original equation. We estimated the roots of the respective transcendental equation while investigating eigenvalue asymptotic of this problem. We derived conditions under which the differential operator kernel is finite-dimensional and the inverse operator is completely continuous over the complement to the kernel. We prove a lemma on existence and regularity of solutions to the respective linear problem. The regularity proof involved studying the sums of Fourier series. We prove a theorem on existence and regularity of a periodic solution when the non-linear term satisfies a non-resonance condition at infinity. The proof included prior estimation of solutions to the respective operator equation and made use of the Leray --- Schauder fixed point theorem. We determine additional conditions under which the periodic solution found via the main theorem is a singular solution.

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50

Kalton, N. J. "Banach Envelopes of Non-Locally Convex Spaces." Canadian Journal of Mathematics 38, no. 1 (February 1, 1986): 65–86. http://dx.doi.org/10.4153/cjm-1986-004-2.

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Let X be a quasi-Banach space whose dual X* separates the points of X. Then X* is a Banach space under the normFrom X we can construct the Banach envelope Xc of X by defining for x ∊ X, the normThen Xc is the completion of (X, ‖ ‖c). Alternatively ‖ ‖c is the Minkowski functional of the convex hull of the unit ball. Xc has the property that any bounded linear operator L:X → Z into a Banach space extends with preservation of norm to an operator .

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