Norm Estimates Research Articles

Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay. By using new conformable delayed matrix functions and the method of variation, we obtain a representation of their solutions. As an application, we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayed matrix functions. The obtained results are new, and they extend and improve some existing ones. Finally, an example is presented to illustrate the validity of our theoretical results.

A fractal uncertainty principle for the short-time Fourier transform and Gabor multipliers

We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire functions. Specifically for signals in L 2 ( R d ) , we obtain norm estimates of Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, which we also use to derive explicit upper bound asymptotes for the multidimensional Cantor iterates, in particular. Finally, we translate the fractal uncertainty principle to discrete Gaussian Gabor multipliers.

Norm estimates for operators in norm-attainable C*-algebras

Norm estimates for operators in norm-attainable C*-algebras

On the convergence of difference schemes of high accuracy for the equation of ion-acoustic waves in a magnetized plasma

Multiparametric difference schemes of the finite element method of a high order of accuracy for the Sobolevtype equation of the fourth-order in time are studied. In particular, the first boundary value problem for the equation of ion-acoustic waves in a magnetized plasma is considered. A high-order accuracy of the scheme is achieved due to the special discretization of time and space variables. The presence of parameters in the scheme makes it possible to regularize the accuracy of the schemes and optimize the implementation algorithm. An a priori estimate in a weak norm is obtained by the method of energy inequality. Based on this estimate and the Bramble-Hilbert lemma, the convergence of the constructed algorithms in classes of generalized solutions is proved. An algorithm for implementing the difference scheme is proposed.

THE LAW OF UKRAINE "ON COPYRIGHT AND RELATED RIGHTS": ANALYSIS OF THE AMENDED PROVISIONS

ANNOTATION. Introduction The European integration vector of Ukraine's legislative activity creates a powerful platform for improving domestic legislation. Ukraine, as a party to the Association Agreement between Ukraine and the European Union, undertook to harmonize a large number of domestic legal acts with the legislation of the European Union. The implementation of the set tasks in the outlined areas is achieved by adopting a number of laws that update the existing ones or add novelty to the provisions of the current legislation on intellectual property rights. The most recent "step" towards the approximation of domestic norms to the standards of the European Union was the adoption of a new edition of the Law of Ukraine "On Copyright and Related Rights".Studying the provisions of the latter in general gives a positive impression, taking into account the completeness of the wording of the procedures for acquiring exclusive property rights to individual copyright objects and their compliance with the Directives of the European Union. However, clarification of some provisions of the legislation, in terms of inaccuracies of the categorical apparatus, etc., requires a detailed doctrinal study. Summary of the main results: The article provides the author's vision of the concept of the category "originality of the coposition", which is interpreted through the prism of the author's personal creative work and the presence of the undisputed presumption of legality of the work, which is proven in judicial or extrajudicial dispute resolution procedures. The statement regarding the emergence and existence of a new object of copyright - an NFT work - is proved. The conditions for providing legal protection of an NFT- work are defined, among which a specific one stands out - the need to destroy (destruction) a physical object of art in cases of tokenization of objects of fine art. The author expresses an opinion regarding the nearest prospect of securing such an object of copyright at the legislative level, which is determined by the requirements of practice. An opinion is expressed regarding the possibility of the existence of an NFT work, both as an original and as a derivative composition. Conclusion. The conducted doctrinal study of the main provisions of the Copyright Law "On Copyright and Related Rights" made it possible to review the revised provisions from the "angle" of criticism and identify existing gaps with a list of recommendations for overcoming and eliminating the latter. However, the researched list of outlined issues is not exhaustive and contributes to further development of research in the specified field.

A Computational Framework for 2D Crack Growth Based on the Adaptive Finite Element Method

As a part of a damage tolerance assessment, the goal of this research is to estimate the two-dimensional crack propagation trajectory and its accompanying stress intensity factors (SIFs) using the adaptive finite element method. The adaptive finite element code was developed using the Visual Fortran language. The advancing-front method is used to construct an adaptive mesh structure, whereas the singularity is represented through construction of quarter-point single elements around the crack tip. To generate an optimal mesh, an adaptive mesh refinement procedure based on the posteriori norm stress error estimator is used. The splitting node strategy is used to model the fracture, and the trajectory follows the successive linear extensions for every crack increment. The stress intensity factors (SIFs) for each crack extension increment are calculated using the displacement extrapolation technique. The direction of crack propagation is determined using the theory of maximum circumferential stress. The present study is carried out for two geometries, namely a rectangular structure with two holes and one central crack, and a cracked plate with four holes. The results demonstrate that, depending on the position of the hole, the crack propagates in the direction of the hole due to the unequal stresses at the crack tip, which are caused by the hole’s influence. The results are consistent with other numerical investigations for predicting crack propagation trajectories and SIFs.

Hausdorff measure of noncompactness of matrix mappings on certain difference sequence spaces

The well-known difference sequence spaces were introduced by Kızmaz [17] in 1981 and have been generalized by many authors uptill now. These spaces were fi rst time extended by Sarıgöl [31] to the sequence spaces$l_{\infty }(\Delta _{q}),$ $c(\Delta _{q})$ or $c_{0}(\Delta _{q})$. The aim of this paper is to establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the extended spaces and also to characterize some classes of compact operators by using the Hausdorff measure of noncompactness.

Aspects of the Kahane–Salem–Zygmund inequalities in Banach spaces

The main aim of this work is to discuss several different approaches to the celebrated Kahane–Salem–Zygmund inequalities. In particular, we prove estimates for exponential Orlicz norms of averages sup _{1le jle N}Big |sum _{i=1}^K a_i(j) gamma _iBig |,, where (a_i(j)) in ell _infty ^N, , 1 le i le K and the (gamma _i) form a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get some new Kahane–Salem–Zygmund type inequalities—in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces, and also for subgaussian random Dirichlet polynomials.

The supercloseness property of the Stoke projection for the transient Navier–Stokes equations and global superconvergence analysis

In this paper, we derive the supercloseness properties and global superconvergence results for the implicit Euler scheme of the transient Navier–Stokes equations. Using a prior estimate of finite element solutions, the properties of the Stokes projection and Stokes operator, the derivative transforming skill and the \(H^{-1}\)-norm estimate, we deduce the supercloseness properties of the Stokes projection for the velocity in \(L^\infty (H^1)\)-norm and pressure in \(L^\infty (L^2)\)-norm. Then the supercloseness properties of the interpolation operators are obtained for two pairs of rectangular element: the bilinear-constant element and the Bernadi–Raugel element. Finally, by the interpolation postprocessing technique, we obtain the global superconvergent results. Compared with previous results, no time step restriction is required in the analysis, and moreover, the supercloseness analysis is based on the Stokes projection, which makes the proof more concise.

The unresolved tension between the approximation of criminal norms in the EU and the question of national discretion

The unresolved tension between the approximation of criminal norms in the EU and the question of national discretion

A fast converging sampling operator

We construct a sampling operator with the property that the smoother a function is, the faster its approximation is. We establish a direct estimate and a weak converse estimate of its rate of approximation in the uniform norm by means of a modulus of smoothness and a $K$-functional. The case of weighted approximation is also considered. The weights are positive and power-type with non-positive exponents at infinity. This sampling operator preserves every algebraic polynomial.

Distributed Resource Management in Downlink Cache-Enabled Multi-Cloud Radio Access Networks

In light of the premises of beyond fifth generation (B5G) networks, the need for better exploiting the capabilities of cloud-enabled networks arises, so as to cope with the large-scale interference resulting from the massive increase of data-hungry systems. A compound of several clouds, jointly managing inter-cloud and intra-cloud interference, constitutes a practical solution to account for the requirements of B5G networks. This paper considers a multi-cloud radio access network model (MC-RAN), where each cloud is connected to a distinct set of base stations (BSs) via limited capacity fronthaul links. The BSs are equipped with local cache storage and baseband processing capabilities, as a means to alleviate the fronthaul congestion problem. The paper then investigates the problem of jointly assigning users to clouds and determining their beamforming vectors so as to maximize the network-wide energy efficiency (EE) subject to fronthaul capacity and transmit power constraints. This paper solves such a mixed discrete-continuous, non-convex optimization problem using fractional programming (FP) and successive inner-convex approximation (SICA) techniques to deal with the non-convexity of the continuous part of the problem, and $l_0$-norm approximation to account for the binary association part. A highlight of the proposed algorithm is its capability of being implemented in a distributed fashion across the network's multiple clouds through a reasonable amount of information exchange. The numerical simulations illustrate the pronounced role the proposed algorithm plays in alleviating the interference of large-scale MC-RANs, especially in dense networks.

ABOUT DATA PROTECTION STANDARDS AND INTELLECTUAL PROPERTY REGULATION IN THE DIGITAL ECONOMY: KEY ISSUES FOR UKRAINE

Changes that are constantly taking place in the digital economy cause increasing instability of legislation in the field of data protection and security. For example, in Ukraine, under martial law, there is an urgent need to adapt the legal regulation to European data protection standards (in terms of personal data processing). First of all, the correlation between EU law, national law of the EU Member States and national legislation of the EU candidate countries results in the principle of direct effect of EU law. In addition, EU data protection law has become an essential source for EU Member States in regulating artificial intelligence (AI), e-commerce and the Internet of Things (IoT). The article considers the specific topic of the conditions of approximation of international norms and legislation of Ukraine to EU law, trying to answer the questions of personal data protection in the conditions of martial law that have arisen. This work is based on a comparative analysis of the General Data Protection Regulation 2016/679 and internal data protection rules in Ukraine. At present, the research purpose of the article is to reveal the fact that data protection is a specific category of procedural law based on the principles of intellectual property law regarding data access rights and data ownership rights in the digital economy.

Tight computationally efficient approximation of matrix norms with applications

We address the problems of computing operator norms of matrices induced by given norms on the argument and the image space. It is known that aside of a fistful of ``solvable cases,'' most notably, the case when both given norms are Euclidean, computing operator norm of a matrix is NP-hard. We specify rather general families of norms on the argument and the images space (``ellitopic'' and ``co-ellitopic,'' respectively) allowing for reasonably tight computationally efficient upper-bounding of the associated operator norms. We extend these results to bounding ``robust operator norm of uncertain matrix with box uncertainty,'' that is, the maximum of operator norms of matrices representable as a linear combination, with coefficients of magnitude $\leq1$, of a collection of given matrices. Finally, we consider some applications of norm bounding, in particular, (1) computationally efficient synthesis of affine non-anticipative finite-horizon control of discrete time linear dynamical systems under bounds on the peak-to-peak gains, (2) signal recovery with uncertainties in sensing matrix, and (3) identification of parameters of time invariant discrete time linear dynamical systems via noisy observations of states and inputs on a given time horizon, in the case of ``uncertain-but-bounded'' noise varying in a box.

Error analysis of Petrov-Galerkin immersed finite element methods

This paper designs and analyzes a new and stable Petrov–Galerkin (PG) immersed finite element method (IFEM) for the second-order elliptic interface problems by introducing stabilization terms based on the classical PG-IFEM, which lacks the local positivity. The Petrov–Galerkin immersed finite element method uses the immersed finite element functions for the trial space and the standard finite element functions for the test space. Both the a priori and a posteriori error estimates of the method are analyzed in this paper. We prove the continuity and inf-sup condition and the a priori error estimate of the energy norm. The proposed a posteriori error estimator is proved to be both reliable and efficient, with both reliability and efficiency constants independent of the location of the interface. Extensive numerical results confirm the numerical scheme’s optimal convergence and indicate the robustness with respect to the interface-mesh intersection and the coefficient contrast, despite the robustness of the inf-sup constant with respect to the interface-mesh intersection has yet been theoretically proved.

Quadratic Sparse Domination and Weighted Estimates for Non-integral Square Functions

We prove a quadratic sparse domination result for general non-integral square functions S. That is, for p_0 in [1,2) and q_0 in (2,infty ], we prove an estimate of the form where q_{0}^{*} is the Hölder conjugate of q_{0}/2, M is the underlying doubling space and {mathcal {S}} is a sparse collection of cubes on M. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace–Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space L^{p}(w).

Downlink Resource Allocation With Pilot Length Optimization for User-Centric Cell-Free MIMO Networks

Pilot length is a crucial parameter in cell-free multiple-input multiple-output (MIMO) networks, which could significantly affect the downlink throughput. In this letter, we jointly optimize the user scheduling, power allocation and pilot length for user-centric (UC) cell-free MIMO networks. We consider the frequency band is uniformly split into several sub-channels. As the number of users scheduled on one sub-channel is commonly small, orthogonal pilot sequences are adopted among users on each sub-channel to avoid pilot contamination. To achieve user fairness, we aim to maximize the minimum ergodic user rate in the downlink transmission, yielding a challenging mix-integer non-convex optimization. Inspired by the weighted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula> -norm approximation in compressive sensing together with successive convex approximation (SCA), an iterative scheme is then proposed to solve the problem. Simulation results demonstrate that the proposed scheme outperforms its state-of-the-art counterpart, which is a three-step algorithm including user grouping, sub-channel allocation, and power optimization.

Norm estimates for the pseudo-differential operator involving fractional Hankel-like transform on $${\mathcal {S}}$$-type spaces

Norm estimates for the pseudo-differential operator involving fractional Hankel-like transform on $${\mathcal {S}}$$-type spaces

Circuit Design and Analysis of Smoothed $${l}_0$$ Norm Approximation for Sparse Signal Reconstruction

Circuit Design and Analysis of Smoothed $${l}_0$$ Norm Approximation for Sparse Signal Reconstruction

Global well-posedness and large time behavior of epitaxy thin film growth model

We consider the global well-posedness and large time behavior of solutions for epitaxy thin film growth model in mathbb{R}^{d} with the dimensional dgeq 3. First, using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution under the condition that the initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.