Abstract: The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical physics to which (in the rapidly decreasing case) the method of the inverse scattering problem is applicable. These solutions are such that the spectrum of their associated linear differential operators has a finite-zone structure. The set of linear operators with a given finite-zone spectrum is the Jacobian variety of a Riemann surface, which is determined by the structure of the spectrum. We give an explicit solution of the corresponding non-linear equations in the language of the theory of Abelian functions.

Abstract: This paper explores the application of Koopman operator theory to the control of robotic systems. The operator is introduced as a method to generate data-driven models that have utility for model-based control methods. We then motivate the use of the Koopman operator towards augmenting model-based control. Specifically, we illustrate how the operator can be used to obtain a linearizable data-driven model for an unknown dynamical process that is useful for model-based control synthesis. Simulated results show that with increasing complexity in the choice of the basis functions, a closed-loop controller is able to invert and stabilize a cart- and VTOL-pendulum systems. Furthermore, the specification of the basis function are shown to be of importance when generating a Koopman operator for specific robotic systems. Experimental results with the Sphero SPRK robot explore the utility of the Koopman operator in a reduced state representation setting where increased complexity in the basis function improve open- and closed-loop controller performance in various terrains, including sand.

Abstract: There are many aggregation operators have been defined up to date, but in this work, we define the interval valued Pythagorean fuzzy weighted geometric (IPFWG) operator, the interval-valued Pythago...

Abstract: A modification of Szasz–Mirakyan operators is presented that reproduces the functions 1 and \(e^{2ax}\), \(a>0\) fixed. We prove uniform convergence, order of approximation via a certain weighted modulus of continuity, and a quantitative Voronovskaya-type theorem. A comparison with the classical Szasz–Mirakyan operators is given. Some shape preservation properties of the new operators are discussed as well. Using a natural transformation, we also present a uniform error estimate for the operators in terms of the first- and second-order moduli of smoothness.

Abstract: Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman operator itself is linear but infinite-dimensional (evolves on a function space). This linear operator captures the full information of the dynamics described by the original nonlinear system. In particular, spectral properties of the Koopman operator play a crucial role in analyzing the original system. In the first part of this paper, we review the so-called Koopman operator theory for nonlinear dynamical systems, with emphasis on modal decomposition and computation that are direct to wide applications. Then, in the second part, we present a series of applications of the Koopman operator theory to power systems technology. The applications are established as data-centric methods, namely, how to use massive quantities of data obtained numerically and experimentally, through spectral analysis of the Koopman operator: coherency identification of swings in coupled synchronous generators, precursor diagnostic of instabilities in the coupled swing dynamics, and stability assessment of power systems without any use of mathematical models. Future problems of this research direction are identified in the last concluding part of this paper.

Abstract: We use the method of similar operators to perform the asymptotic analysis of the spectrum of a differential operator with an involution. We show that such operators have compact resolvent, and that their large eigenvalues are determined by the values of (the Fourier coefficients) of their potential up to a summable sequence.

Abstract: Using the approach based on conformal symmetry we calculate the three-loop (NNLO) contribution to the evolution equation for flavor-nonsinglet leading twist operators in the MS scheme. The explicit expression for the three-loop kernel is derived for the corresponding light-ray operator in coordinate space. The expansion in local operators is performed and explicit results are given for the matrix of the anomalous dimensions for the operators up to seven covariant derivatives. The results are directly applicable to the renormalization of the pion light-cone distribution amplitude and flavor-nonsinglet generalized parton distributions.

Abstract: We construct and estimate the fundamental solution of highly anisotropic space-inhomogeneous integro-differential operators. We use the Levi method. We give applications to the Cauchy problem for such operators.

Abstract: We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles . For , the flat vector bundle is the direct image of , where is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kahler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

Abstract: We shall say that a bounded linear operator T acting on a Banach space X admits a generalized Kato–Riesz decomposition if there exists a pair of T-invariant closed subspaces (M, N) such that , the reduction is Kato and is Riesz. In this paper, we define and investigate the generalized Kato–Riesz spectrum of an operator. For T is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator S acting on X such that , , is Riesz. We investigate generalized Drazin-Riesz invertible operators and also characterize bounded linear operators which can be expressed as a direct sum of a Riesz operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator. In particular, we characterize the single-valued extension property at a point in the case that admits a generalized Kato–Riesz decomposition.

Abstract: In this paper, we introduce a Kantorovich type generalization of q-Bernstein-Stancu operators. We study the convergence of the introduced operators and also obtain the rate of convergence by these operators in terms of the modulus of continuity. Further, we study local approximation property and Voronovskaja type theorem for the said operators. We show comparisons and some illustrative graphics for the convergence of operators to a certain function.

Abstract: A new binary relation associated with the core–EP inverse is presented and studied on the corresponding subset of all generalized Drazin invertible bounded linear Hilbert space operators. Using the...

Abstract: In this paper, we study semicircular-like elements, semicircular elements and operators generated by these elements in a free product Banach \(*\) -algebra generated by measurable functions over p-adic number fields \(\mathbb { Q}_{p},\) over primes p.

Abstract: In this paper we introduce fractional powers of quaternionic operators. Their definition is based on the theory of slice-hyperholomorphic functions and on the $S$-resolvent operators of the quaternionic functional calculus. The integral representation formulas of the fractional powers and the quaternionic version of Kato's formula are based on the notion of $S$-spectrum of a quaternionic operator. The proofs of several properties of the fractional powers of quaternionic operators rely on the $S$-resolvent equation. This equation, which is very important and of independent interest, has already been introduced in the case of bounded quaternionic operators, but for the case of unbounded operators some additional considerations have to be taken into account. Moreover, we introduce a new series expansion for the pseudo-resolvent, which is of independent interest and allows to investigate the behavior of the $S$-resolvents close to the $S$-spectrum. The paper is addressed to researchers working in operator theory and in complex analysis.

Abstract: In the present paper, we study the problem of approximation to a function by means of positive linear operators in modular spaces in the sense of power series method. Indeed, in order to get stronger results than the classical cases, we use the power series method which also includes both Abel and Borel methods. An application that satisfies our theorem is also provided.

Abstract: This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.

Abstract: The “Up-and-down” theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result in operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result is used to prove a theorem of domination for order narrow positive abstract Uryson operators from a vector lattice E to a Banach lattice F with an order continuous norm.

Abstract: We present a novel construction of recursion operators for scalar second-order integrable multidimensional PDEs with isospectral Lax pairs written in terms of first-order scalar differential operators. Our approach is quite straightforward and can be readily applied using modern computer algebra software. It is illustrated by examples, two of which are new.

Abstract: We extend some basic results known for finite range operators to long range operators with off-diagonal decay. Namely, we prove an analogy of Sch'nol's theorem. We also establish the connection between the almost sure spectrum of long range random operators and the spectra of deterministic periodic operators.

Abstract: We firstly give a modification of the known Hermite-Hadamard type inequalities for the generalized k-fractional integral operators of a function with respect to another function. We secondly establish several Hermite-Hadamard type inequalities for the generalized k-fractional integral operators of a function with respect to another function. The results presented here, being very general, are pointed out to be specialized to yield some known results. Relevant connections of the various results presented here with those involving relatively simple fractional integral operators are also indicated.

Abstract: We consider Hormander type symbols on a family of spaces associated with non-negative self-adjoint operators, and we prove boundedness of the corresponding pseudodifferential operators on both classical and non-classical Besov and Triebel–Lizorkin spaces. Consequently, this also covers the case of Sobolev spaces. As an application, we obtain boundedness of spectral multipliers on the mentioned spaces.

Abstract: In the present paper, we investigate stability of trajectories of Lotka–Volterra (LV) type operators defined in finite dimensional simplex. We prove that any LV type operator is a surjection of the simplex. It is introduced a new class of LV-type operators, called MLV type ones, and we show that trajectories of the introduced operators converge. Moreover, we show that such kind of operators have totally different behavior than \({\mathbf {f}}\)-monotone LV type operators.

Abstract: We give necessary and sufficient conditions on the symbols to guarantee that the corresponding pseudo-differential operators are nuclear from \(L^{p_1}({\mathbb {S}}^1)\) into \(L^{p_2}({\mathbb {S}}^1)\) for \(1\le p_1,p_2<\infty \). Applications are given to adjoints of nuclear pseudo-differential operators from \(L^{p_2'}({\mathbb {S}}^1)\) into \(L^{p_1'}({\mathbb {S}}^1)\) for \(1\le p_1,p_2<\infty \) and products of nuclear pseudo-differential operators on \(L^p({\mathbb {S}}^1),\,1\le p<\infty \).