Kernel Of Operator Research Articles

A noncommutative weak type (1,1) estimate for a square function from ergodic theory

In this paper, we investigate the boundedness of a square function operator from ergodic theory acting on noncommutative Lp-spaces. The main result is a weak type (1,1) estimate of this operator. We also show the (L∞,BMO) estimate, and thus all the strong type (Lp,Lp) estimates by interpolation. The main new difficulty lies in the fact that the kernel of this square function operator does not enjoy any regularity, while the Lipschitz regularity assumption is crucial in showing such endpoint estimates for the noncommutative Calderón-Zygmund singular integrals.

Finite Element Method for Fractional Parabolic Integro-Differential Equations with Smooth and Nonsmooth Initial Data

We study the space-time finite element discretizations for time fractional parabolic integro-differential equations in a bounded convex polygonal domain in $${\mathbb {R}}^d (d=1,2,3)$$ . Both spatially semidiscrete and fully discrete finite element approximations are considered and analyzed. We use piecewise linear and continuous finite elements to approximate the space variable whereas the time discretization uses two fully discrete schemes based on the convolution quadrature, namely the backward Euler and the second-order backward difference. For the spatially discrete scheme, optimal order a priori error estimates are derived for smooth initial data, i.e., when $$u_0\in H_0^1(\varOmega )\cap H^2(\varOmega )$$ . Moreover, for the homogeneous problem, almost optimal error estimates for positive time are established for nonsmooth initial data, i.e., when the initial function $$u_0$$ is only in $$ L^2(\varOmega )$$ . The error estimates for the fully discrete methods are shown to be optimal in time for both smooth and nonsmooth initial data under the specific choice of the kernel operator in the integral. Finally, we provide some numerical illustrations to verify our theoretical analysis.

Approximation Theorems for Localized Baskakov Operators

We investigate the relation between the convergence of the localized Baskakov operators and the number of their terms N + 1. We obtain the limit functions and the approximation orders by the localized Baskakov operators under some conditions for N. A new estimate about the kernel of the Baskakov operators is achieved by using one of the central limit theorems in probability theory as well.

Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by hyperplanes at the two ends. On each domain a family of orthogonal polynomials, related to the Gegebauer polynomials, is studied and shown to share two characteristic properties of spherical harmonics: they are eigenfunctions of a second order linear differential operator with eigenvalues depending only on the polynomial degree, and they satisfy an addition formula that provides a closed form formula for the reproducing kernel of the orthogonal projection operator. The addition formula leads to a convolution structure, which provides a powerful tool for studying the Fourier orthogonal series on these domains. Furthermore, another family of orthogonal polynomials, related to the Hermite polynomials, is defined and shown to be the limit of the first family, and their properties are derived accordingly.

Algorithm for Construction of Quadrature Formulas with Exponential Convergence for Linear Operators Acting on Periodic Functions

We propose an algorithm for construction of quadrature formulas for linear operators acting on periodic functions. For analytic functions, the order of accuracy of quadrature formulas grows unboundedly with the growth of the number of the grid nodes. Within rather general restrictions on the kernels of linear operators, an exponential estimate of a quadrature formula is proved. To exemplify, we find quadrature formulas for the calculation of integral operators with logarithmic singularities which are used in the boundary element method to obtain overconvergent numerical schemes for solution of boundary value problems for harmonic and biharmonic equations on the plane.

Transmutation operators and a new representation for solutions of perturbed Bessel equations

New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form are obtained. The integral kernel is represented as a Fourier‐Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to ω. For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones previously obtained by the authors for large values of ω and ℓ and for noninteger values of ℓ. The results are based on application of several ideas from the classical transmutation (transformation) operator theory, asymptotic formulas for the solution, results connecting the decay rate of the Fourier transform with the smoothness of a function, the Paley‐Wiener theorem, and some results from constructive approximation theory. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.

Establishment of PM10 and PM2.5 emission inventories from wind erosion source and simulation of its environmental impact based on WEPS-Models3 in southern Xinjiang, China

Southern Xinjiang located in the far northwest of China is experiencing serious particulate matter (PM) pollution. The wind erosion has been recognized as a great contributor of PM10 and PM2.5 concentrations in southern Xinjiang. In this study we developed a method that used the Wind Erosion Prediction System (WEPS) to establish the PM10 and PM2.5 emission inventories from the wind erosion source in southern Xinjiang with a 10 km × 10 km spatial resolution and a temporal resolution of one month. The PM10 and PM2.5 emission inventories from wind erosion sources were provided as input to the Models3/SMOKE (Sparse Matrix Operator Kernel Emission). The Community Multi-scale Air Quality (CMAQ) model was employed to simulate PM10 and PM2.5 concentrations from wind erosion source and the results were compared with the monitoring data to verify the emission inventories. The total PM10 and PM2.5 emissions were 12 × 106 t and 4 × 106 t, respectively, and the emission per unit area were 14.6 t/km2 and 4.9 t/km2, respectively, in southern Xinjiang in 2016. The PM10 and PM2.5 emissions per unit area were highest in Kashi being 19.1 t/km2 and 9.1 t/km2, respectively. The total emissions peaked at 5.4 × 106 t/km2 in PM10 and 1.4 × 106 t/km2 in PM2.5 in Bazhou. The PM10 and PM2.5 emissions were highest in spring, followed by summer, winter and autumn. The Normalized Mean Error (NME) between the predicted monthly average PM10 and PM2.5 concentrations from wind erosion source and the results from multiplying the monitoring data by the source apportionment results of PM10 and PM2.5 were 25.3% and 26.1%, respectively, which indicated that the accuracies of the PM10 and PM2.5 emission inventories from wind erosion source in southern Xinjiang were satisfactory.

On continuous endomorphisms of entire functions

The paper is concerned with continuous linear operators on the space of entire functions. The properties of such operators that are related to the definition of convolution-type operators in spaces of analytic functions are investigated. Corollaries refining both the approximation theorem for the kernel of a symmetric convolution operator and the dual definition of a differential operator in a complex domain are stated. Bibliography: 20 titles.

Метод регуляризации для нелинейных интегродифференциальных уравнений с нулевым оператором дифференциальной части и несколькими быстро изменяющимися ядрами

A nonlinear integro-differential equation with a zero operator of the differential part and several rapidly changing kernels is considered. The work is a continuation of the research conducted earlier for a single rapidly changing core. The main ideas of this generalization and the subtleties that arise in the development of the corresponding algorithm for the regularization method are fully visible in the case of two rapidly changing kernels, so for the sake of reducing the calculations, this particular case is taken. A similar problem with a single spectral value of the kernel of an integral operator is analyzed in one of the authors ' papers. In this case, the singularities in the solution of the problem are described only by the spectral value of the kernel. However, the influence of the zero operator of the differential part affects the fact that in the first approximation, the asymptotics of the solution of the problem under consideration will not contain the functions of the boundary layer, and the limit operator itself will become degenerate (but not zero). The conditions for the solvability of the corresponding iterative problems, as in the linear case, will not be in the form of differential equations (as was the case in problems with a non-zero operator of the differential part), but integro-differential equations, and the formation of these equations is significantly influenced by nonlinearity. Note that, in contrast to the linear case, there is no inhomogeneity of the corresponding linear problem in the right part of the problem under study. As it was shown earlier, its presence in the problem would lead to the appearance in the asymptotic solution of terms with negative powers of a small parameter, and in the nonlinear case there would be innumerable such powers, and the corresponding formal asymptotic solution would have the form of a Laurent series. This would make the creation of an algorithm for asymptotic solutions problematic, so in this paper, wanting to remain within the framework of asymptotic solutions of the type of Taylor series, inhomogeneity is excluded. In addition, in the nonlinear case, so-called resonances may occur, which significantly complicate the development of the corresponding algorithm for the regularization method. This publication deals with the non-resonant case. It is assumed that the study of an alternative variant (a more complex resonant problem) will be carried out in the future.

The Krein-von Neumann extension of a regular even order quasi-differential operator

We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to Möller and Zettl.

Existence results for a resonant third-order problem with two dimensional kernel on the half-line

The purpose of this research is to establish conditions that guarantee the solvability of a resonant third-order boundary value problem (BVP) on the half-line. The resonant problem is subject to integral and m-point boundary conditions while, the kernel of the differential operator is equal to two. The coincidence degree theory will be applied and an illustrative example will be considered to demonstrate the result.

A Comprehensive Price Prediction System Based on Inverse Multiquadrics Radial Basis Function for Portfolio Selection

Price prediction plays a crucial role in portfolio selection (PS). However, most price prediction strategies only make a single prediction and do not have efficient mechanisms to make a comprehensive price prediction. Here, we propose a comprehensive price prediction (CPP) system based on inverse multiquadrics (IMQ) radial basis function. First, the novel radial basis function (RBF) system based on IMQ function rather than traditional Gaussian (GA) function is proposed and centers on multiple price prediction strategies, aiming at improving the efficiency and robustness of price prediction. Under the novel RBF system, we then create a portfolio update strategy based on kernel and trace operator. To assess the system performance, extensive experiments are performed based on 4 data sets from different real-world financial markets. Interestingly, the experimental results reveal that the novel RBF system effectively realizes the integration of different strategies and CPP system outperforms other systems in investing performance and risk control, even considering a certain degree of transaction costs. Besides, CPP can calculate quickly, making it applicable for large-scale and time-limited financial market.

Supermixing and hypermixing of strongly continuous semigroups and their direct sum

Supermixing and hypermixing strongly continuous semigroups are introduced in this paper. It is proved that supermixing preserves under quasiconjugacy. Moreover, it is established that if a strongly continuous semigroup is supermixing(hypermixing), then any discretization of it, is supermixing(hypermixing). Also, it is proved that supermixing(hypermixing) of (Tt ) t ≥ 0 implies that any of Tt 's is a supermixing(hypermixing). Some sufficient conditions for supermixing and hypermixing are stated that based on dense sets and kernels of operators of semigroup. It is proved that the supermixing(hypermixing) of two strongly continuous semigroups imply that their direct sum is supermixing(hypermixing). Also, supermixing(hypermixing) of the direct sum of finite strongly continuous semigroups indicates that any of these semigroups is supermixing(hypermixing). Outcomes of this study have shown that extension of the concepts of supermixing and hypermixing to strongly continuous semigroups leads to get access to a proper subset of the set of hypercyclic semigroups and reach valuable results about them.

Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear

<abstract> In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, $ 1 &lt; q &lt; 2 $, in the equation on the behavior of numerical solutions of fractal-fractional reaction diffusion equations (FFRDE). Iterative approximations to the solutions of these equations are constructed by applying the theory of fractional calculus with the help of Lagrange polynomial functions. In key parameter regimes, all these iterative solutions based on a power kernel, an exponential kernel and a generalized Mittag-Leffler kernel are very close. Hence, iterative solutions obtained using one of these kernels are compared with full numerical solutions of the FFRDE and excellent agreement is found. All numerical solutions in this paper were obtained using Mathematica. </abstract>

Kernels of Unbounded Toeplitz Operators and Factorization of Symbols

We consider kernels of unbounded Toeplitz operators in H^p({mathbb {C}}^{+}) in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in H^p({mathbb {C}}^{+}), we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of z. We apply the results to describe the kernels of Toeplitz operators with non-vanishing piecewise continuous symbols.

Short-time existence of the α-Dirac-harmonic map flow and applications

In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for α-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator has nontrivial minimal kernel, we first prove the short time existence of the heat flow for α-Dirac-harmonic maps. The obstacle to the global existence is the singular time when the kernel of the Dirac operator no longer stays minimal along the flow. In this case, the kernel may not be continuous even if the map is smooth with respect to time. To overcome this issue, we use the analyticity of the target manifold to obtain the density of the maps along which the Dirac operator has minimal kernel in the homotopy class of the given initial map. Then, when we arrive at the singular time, this density allows us to pick another map which has lower energy to restart the flow. Thus, we get a flow which may not be continuous at a set of isolated points. Furthermore, with the help of small energy regularity and blow-up analysis, we finally get the existence of nontrivial α-Dirac-harmonic maps () from closed surfaces. Moreover, if the target manifold does not admit any nontrivial harmonic sphere, then the map part stays in the same homotopy class as the given initial map.

The Beurling-Lax-Halmos theorem for infinite multiplicity

In this paper we consider several questions emerging from the Beurling-Lax-Halmos Theorem, which characterizes the shift-invariant subspaces of vector-valued Hardy spaces. The Beurling-Lax-Halmos Theorem states that a backward shift-invariant subspace is a model space H(Δ)≡HE2⊖ΔHE2, for some inner function Δ. Our first question calls for a description of the set F in HE2 such that H(Δ)=EF⁎, where EF⁎ denotes the smallest backward shift-invariant subspace containing the set F. In our pursuit of a general solution to this question, we are naturally led to take into account a canonical decomposition of operator-valued strong L2-functions. Next, we ask: Is every shift-invariant subspace the kernel of a (possibly unbounded) Hankel operator? Consideration of the question on the structure of shift-invariant subspaces leads us to study and coin a new notion of “Beurling degree” for an inner function. We then establish a deep connection between the spectral multiplicity of the model operator (the truncated backward shift) and the Beurling degree of the corresponding characteristic function. At the same time, we consider the notion of meromorphic pseudo-continuations of bounded type for operator-valued functions, and then use this notion to study the spectral multiplicity of model operators between separable complex Hilbert spaces. In particular, we consider the case of multiplicity-free: more precisely, for which characteristic function Δ of the model operator T does it follow that T is multiplicity-free, i.e., T has multiplicity 1? We show that if Δ has a meromorphic pseudo-continuation of bounded type in the complement of the closed unit disk and the adjoint of the flip of Δ is an outer function, then T is multiplicity-free.

Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes III. Code optimization via tensor contraction

The present study deals with the code optimization and its implementation of the direct reconstruction method (DRM) using the complete-search tensor contraction (CsTC) framework to extract the best performance of high-order methods on modern computing architectures. DRM was originally proposed to overcome severe computational costs of the physical domain-based discontinuous Galerkin (DG) method on mixed-curved meshes. In this work, the performance of DRM is further enhanced through the code optimization via the CsTC technique. Required kernels for tensor operations in the DRM solution algorithm are analyzed and optimized by completely searching all candidates of GEMM (General Matrix Multiplication) subroutines. The computational performance is thoroughly examined by simulating a turbulent flow over a circular cylinder at ReD=3900 by DG-P3 and -P5 approximations. Compared to a quadrature-based approach with the full integration, the optimized DRM significantly reduces the memory requirements and the number of floating-point operations to compute the DG residual on a linear mesh as well as high-order curved meshes. On a P3-mesh, the optimized DRM provides 13.74× and 23.03× speed-ups in DG-P3 and -P5, respectively, while the amount of memory required is reduced to 1/16.6 and 1/19.9. On a linear mesh, it even yields 1.25× and 1.12× speed-ups in DG-P3 and -P5, respectively. The memory requirement is reduced to 1/1.27 and 1/1.15, respectively. In particular, it is observed that the optimized DRM on a P3-mesh performs better than the optimized quadrature-based method on a P1-mesh.

Mathematical modeling parametric vibrations of the pipeline with pulsating fluid flow

A mathematical model of the dynamics of a straight viscoelastic pipe conveying pulsating fluid has been developed in the paper. Using the Bubnov-Galerkin method, a mathematical model of the problem is reduced to solving a system of ordinary integro-differential equations, where time is an independent variable. Solution of integro-differential equations is determined by a numerical method based on the elimination of a singularity in the relaxation kernel of an integral operator. A system of algebraic equations is obtained according to the numerical method for unknowns. To solve the system of algebraic equations, the Gauss method is used. A computational algorithm is developed for solving the problems of the dynamics of viscoelastic pipelines conveying fluid.

A transmutation operator method for solving the inverse quantum scattering problem * *Research was supported by CONACYT, Mexico via the projects 222478 and 284470. Research of Vladislav Kravchenko was supported by the Regional mathematical center of the Southern Federal University, Russia.

The inverse quantum scattering problem for the perturbed Bessel equation is considered. A direct and practical method for solving the problem is proposed. It allows one to reduce the inverse problem to a system of linear algebraic equations, and the potential is recovered from the first component of the solution vector of the system. The approach is based on a special form Fourier–Jacobi series representation for the transmutation operator kernel and the Gelfand–Levitan equation which serves for obtaining the system of linear algebraic equations. The convergence and stability of the method are proved as well as the existence and uniqueness of the solution of the truncated system. Numerical realization of the method is discussed. Results of numerical tests are provided revealing a remarkable accuracy and stability of the method.