Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator

In this paper, the boundedness of the Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of the p-adic fractional Hausdorff operator on weighted p-adic weak Lebesgue spaces. We also obtain the sufficient condition of commutators of the p-adic fractional Hausdorff operator by taking symbol function from Lipschitz spaces. Moreover, strong type estimates for fractional Hausdorff operator and its commutator on weighted p-adic Lorentz spaces are also acquired.

Analysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators

Given a linear self-adjoint differential operator $$\mathscr {L}$$ L along with a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), in many numerical applications it is crucial to understand how good the (relative) approximation of the whole spectrum of the discretized operator $$\mathscr {L}\,^{(n)}$$ L ( n ) is, compared to the spectrum of the continuous operator $$\mathscr {L}$$ L . The theory of Generalized Locally Toeplitz sequences allows to compute the spectral symbol function $$\omega $$ ω associated to the discrete matrix $$\mathscr {L}\,^{(n)}$$ L ( n ) . Inspired by a recent work by T. J. R. Hughes and coauthors, we prove that the symbol $$\omega $$ ω can measure, asymptotically, the maximum spectral relative error $$\mathscr {E}\ge 0$$ E ≥ 0 . It measures how the scheme is far from a good relative approximation of the whole spectrum of $$\mathscr {L}$$ L , and it suggests a suitable (possibly non-uniform) grid such that, if coupled to an increasing refinement of the order of accuracy of the scheme, guarantees $$\mathscr {E}=0$$ E = 0 .

Weighted Estimates for Commutator of Rough p -Adic Fractional Hardy Operator on Weighted p -Adic Herz–Morrey Spaces

The current article investigates the boundedness criteria for the commutator of rough p -adic fractional Hardy operator on weighted p -adic Lebesgue and Herz-type spaces with the symbol function from weighted p -adic bounded mean oscillations and weighted p -adic Lipschitz spaces.

Estimates for Commutators of Bilinear Fractional p -Adic Hardy Operator on Herz-Type Spaces

In the current article, we investigate the boundedness of commutators of the bilinear fractional p -adic Hardy operator on p -adic Herz spaces and p -adic Morrey-Herz spaces by considering the symbol function from central bounded mean oscillations and Lipschitz spaces.

Two-Weight, Weak-Type Norm Inequalities for Fractional Integral Operators and Commutators on Weighted Morrey and Amalgam Spaces

Let 0<γ<n and Iγ be the fractional integral operator of order γ, Iγfx=∫ℝnx−yγ−nfydy and let b,Iγ be the linear commutator generated by a symbol function b and Iγ, b,Iγfx=bx⋅Iγfx−Iγbfx. This paper is concerned with two-weight, weak-type norm estimates for such operators on the weighted Morrey and amalgam spaces. Based on weak-type norm inequalities on weighted Lebesgue spaces and certain Ap-type conditions on pairs of weights, we can establish the weak-type norm inequalities for fractional integral operator Iγ as well as the corresponding commutator in the framework of weighted Morrey and amalgam spaces. Furthermore, some estimates for the extreme case are also obtained on these weighted spaces.

A Characterization of Central BMO Space via the Commutator of Fractional Hardy Operator

This paper is devoted in characterizing the central BMO ℝn space via the commutator of the fractional Hardy operator with rough kernel. Precisely, by a more explicit decomposition of the operator and the kernel function, we will show that if the symbol function belongs to the central BMO ℝn space, then the commutator are bounded on Lebesgue space. Conversely, the boundedness of the commutator implies that the symbol function belongs to the central BMO ℝn space by exploiting the center symmetry of the Hardy operator deeply.

Text and its Problem in the Culture Philosophy of the XX Century

This article discusses the text and its problem in the culture of philosophy of the XX century. It is shown that the peculiarity of this approach is the disclosure of the essence of dialogical thinking in culture through the study of the text as the “primary reality” of any Humanities. The article analyzes the scientific research that the text as a universal form of communication can not be reduced only to the semiotic or only hermeneutic understanding. The paper presents an attempt to chronologically consider the problem of the text in different periods of society. The concept of “test epoch” is revealed in great detail . In conclusion, the article reveals the cultural and philosophical traditions of the XX century in the Russian intellectual field, where of particular interest is the direction of philosophical thought, which seeks to justify the specifics of the methodological approach to social and humanitarian knowledge, which differs from the natural science and mathematical methodology. Central to this area of research are the concepts of symbol, function, communication, game, and text. The main advantage of this study is that such a view will be interesting to specialists in the field of philosophy and Philology.

Standard pairs and existence of symmetric multiscaling functions

Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed.

CONTACT PROBLEM FOR A TWO-LAYERED CYLINDER

Introduction. The investigation of the contact problems for cylindrical bodies is urgent due to the engineering contact strength analysis on shafts, cores and pipe-lines. In the present paper, a new contact problem of elastostatics on the interaction between a rigid band and an infinite two-layered cylinder, which consists of an internal continuous cylinder and an outer hollow one, with a frictionless contact between the cylinders, is studied. The outer cylindrical band of finite length is press fitted. By using a Fourier integral transformation, the problem is reduced to an integral equation with respect to the unknown contact pressure.Materials and Methods. Different combinations of linearly elastic materials of the composite cylinder are considered. Asymptotics of the symbol function of the integral equation kernel at zero and infinity is analyzed. This plays an important role for the application of the analytical solution methods. A key dimensionless geometric parameter is introduced, and a singular asymptotic technique is employed to solve the integral equation.Research Results. On the basis of the symbol function properties, a special easily factorable approximation being applicable in a wide variation range of the problem parameters is suggested. The Monte-Carlo method is used to determine the approximation parameters. The asymptotic formulas are derived both for the contact pressure, and for its integral characteristic. Calculations are made for different materials and for various relative thickness of the cylindrical layer including thin-walled layers.Discussion and Conclusions. The asymptotic solutions are effective for relatively wide bands when the contact zone length is bigger than the diameter of the composite cylinder. It is significant that the method is applicable also for those cases when the outer cylindrical layer is treated as a cylindrical shell. The asymptotic solutions can be recommended to engineers for the contact strength analysis of the elastic barrels with a flexible coating of another material.