Norm Estimates Research Articles

Volterra-type operator on the subclasses of univalent functions

UDC 517.5 In this article, we examine the necessary and sufficient conditions for a member to belong to the class of starlike and convex functions of complex order and spirallike functions of type with the complex order We obtain sharp estimates for the coefficient of the second term in the Taylor series of functions belonging to the mentioned classes. In the main part of this paper, we obtain the necessary and sufficient conditions of boundedness for the image of the open unit disk under the action of a Volterra-type operator and the product of the composition operator and Volterra-type operator in the space of univalent functions and its subspace. Finally, we obtain an estimate of the Schwartzian norm of the above operators in these spaces.

Sharpening Bounds for Multilinear Schur Multipliers

In this paper we make a brief survey of major estimates for norms and traces of multilinear Schur multipliers. We also discuss several open problems in operator theory and mathematical physics whose resolution could be reduced to sharpening existing bounds for Schur multipliers.

Transforming and comparing data between standard SQUID and OPM-MEG systems.

Optically pumped magnetometers (OPMs) have recently become so sensitive that they are suitable for use in magnetoencephalography (MEG). These sensors solve operational problems of the current standard MEG, where superconducting quantum interference device (SQUID) gradiometers and magnetometers are being used. The main advantage of OPMs is that they do not require cryogenics for cooling. Therefore, they can be placed closer to the scalp and are much easier to use. Here, we measured auditory evoked fields (AEFs) with both SQUID- and OPM-based MEG systems for a group of subjects to better understand the usage of a limited sensor count OPM-MEG. We present a theoretical framework that transforms the within subject data and equivalent simulation data from one MEG system to the other. This approach works on the principle of solving the inverse problem with one system, and then using the forward model to calculate the magnetic fields expected for the other system. For the source reconstruction, we used a minimum norm estimate (MNE) of the current distribution. Two different volume conductor models were compared: the homogeneous conducting sphere and the three-shell model of the head. The transformation results are characterized by a relative error and cross-correlation between the measured and the estimated magnetic field maps of the AEFs. The results for both models are encouraging. Since some commercial OPMs measure multiple components of the magnetic field simultaneously, we additionally analyzed the effect of tangential field components. Overall, our dual-axis OPM-MEG with 15 sensors yields similar information to a 62-channel SQUID-MEG with its field of view restricted to the right hemisphere.

Decomposed structured subsets for semidefinite and sum-of-squares optimization

Semidefinite programs (SDPs) are standard convex problems that are frequently found in control and optimization applications. Interior-point methods can solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the size of matrix variables and the number of constraints increases. To improve scalability, SDPs can be approximated with lower and upper bounds through the use of structured subsets (e.g., diagonally-dominant and scaled-diagonally dominant matrices). Meanwhile, any underlying sparsity or symmetry structure may be leveraged to form an equivalent SDP with smaller positive semidefinite constraints. In this paper, we present a notion of decomposed structured subsets to approximate an SDP with structured subsets after an equivalent conversion. The lower/upper bounds found by approximation after conversion become tighter than the bounds obtained by approximating the original SDP directly. We apply decomposed structured subsets to semidefinite and sum-of-squares optimization problems with examples of H∞ norm estimation and constrained polynomial optimization. An existing basis pursuit method is adapted into this framework to iteratively refine bounds.

Embeddings of function spaces via the Caffarelli–Silvestre extension, capacities and Wolff potentials

Let Pαf(x,t) be the Caffarelli–Silvestre extension of a smooth function f(x):Rn→R+n+1≔Rn×(0,∞). The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the Lebesgue spaces Lp(Rn) to the Lq(R+n+1,μ). On one hand, these embeddings will be characterized by using a newly introduced Lp−capacity associated with the Caffarelli–Silvestre extension. In doing so, the mixed norm estimates of Pαf(x,t), the dual form of the Lp−capacity, the Lp−capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when p>q>1, these embeddings will also be characterized in terms of the Hedberg–Wolff potential of μ. Secondly, we characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the homogeneous Sobolev spaces Ẇβ,p(Rn) to the Lq(R+n+1,μ) in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.

Lp Quasi Norm State Estimator for Power Systems

This paper proposes an L<inf>p</inf> (0&#x003C;p&#x003C;1) quasi norm state estimator for power system static state estimation. Compared with the existing L, and L<inf>2</inf> norm estimators, the proposed estimator can suppress the bad data more effectively. The robustness of the proposed estimator is discussed, and an analysis shows that its ability to suppress bad data increases as <tex>$p$</tex> decreases. Moreover, an algorithm is suggested to solve the non-convex state estimation problem. By introducing a relaxation factor in the mathematical model of the proposed estimator, the algorithm can prevent the solution from converging to a local optimum as much as possible. Finally, simulations on a 3-bus DC system, the IEEE 14-bus and IEEE 300-bus systems as well as a 1204-bus provincial system verify the high computation efficiency and robustness of the proposed estimator.

Conditional square functions, the sine-cosine decomposition for Hardy martingales and dyadic perturbation

We prove that the $\mathcal P $-norm estimate between a Hardy martingale and its cosine part are stable under dyadic perturbations, and show how dyadic-stability of the $\mathcal P $-norm estimate is used in the proof that $L^1$ embeds into $L^1/H^1$.

A Novel Family of Sparsity-Aware Robust Adaptive Filters Based on a Logistic Distance Metric

In the recent past, logarithmic hyperbolic cosine based-cost function has been widely applied in adaptive filters as it offers robust performance against outliers. However, the performance of such adaptive filters suffers from high steady-state misalignment due to its significant weight update, even in the presence of outliers. This paper proposes a logistic distance metric-based novel robust cost function, and the corresponding logistic distance metric adaptive filter (LDMAF) has been developed. The proposed LDMAF provides negligible weight update when the desired signals are affected by significant outliers, resulting in low steady-state misalignment. The bound on learning rate has been estimated, and computational complexity comparison of the proposed and other existing algorithms has also been carried out. To further exploit the system's sparse nature and robustness against the outliers, zero attraction-based LDMAF (ZA-LDMAF) and re-weighted zero attraction-based LDMAF (RZA-LDMAF) algorithms have also been developed in this paper. In addition, a new sparse penalty function based on a generalized multivariate Geman-McClure function has been introduced, which provides smooth <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{0}$</tex-math></inline-formula> -norm approximation over other existing functions. Based on this new sparse penalty function, this paper has also developed a novel sparsity-aware robust adaptive filter called generalized Geman-McClure LDMAF (GGM-LDMAF). Simulation studies confirmed the improved convergence behaviour achieved by the proposed algorithms over other existing algorithms for system identification and acoustic echo cancellation scenarios.

Hyperspectral Image Denoising Using Factor Group Sparsity-Regularized Nonconvex Low-Rank Approximation

Hyperspectral image (HSI) mixed noise removal is a fundamental problem and an important preprocessing step in remote sensing fields. The low-rank approximation-based methods have been verified effective to encode the global spectral correlation for HSI denoising. However, due to the large scale and complexity of real HSI, previous low-rank HSI denoising techniques encounter several problems, including coarse rank approximation (such as nuclear norm), the high computational cost of singular value decomposition (SVD) (such as Schatten <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -norm), and adaptive rank selection (such as low-rank factorization). In this article, two novel factor group sparsity-regularized nonconvex low-rank approximation (FGSLR) methods are introduced for HSI denoising, which can simultaneously overcome the mentioned issues of previous works. The FGSLR methods capture the spectral correlation via low-rank factorization, meanwhile utilizing factor group sparsity regularization to further enhance the low-rank property. It is SVD-free and robust to rank selection. Moreover, FGSLR is equivalent to Schatten <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -norm approximation ( <xref ref-type="theorem" rid="theorem1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorem 1</xref> ), and thus FGSLR is tighter than the nuclear norm in terms of rank approximation. To preserve the spatial information of HSI in the denoising process, the total variation regularization is also incorporated into the proposed FGSLR models. Specifically, the proximal alternating minimization is designed to solve the proposed FGSLR models. Experimental results have demonstrated that the proposed FGSLR methods significantly outperform existing low-rank approximation-based HSI denoising methods.

Sparse Reconstruction Using Block Sparse Bayesian Learning With Fast Marginalized Likelihood Maximization for Near-Infrared Spectroscopy

The absorption peak contains a great amount of important chemical information that is critical for the qualitative/quantitative analysis of organic compounds in high-dimensional near-infrared (NIR) spectral data. Sparse Bayesian learning (SBL) algorithm can capture key information from the underlying signal with only a small number of measurements. The computational cost is high because the inverse of a large matrix must be calculated in each iteration with the use of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> -norm approximation or least squares. To overcome this problem, an improved block SBL (BSBL) is adopted to first reconstruct the absorption peak variables by determining the position and width of the limited peaks autonomously. Then, the fast-marginalized likelihood maximization (FMLM) is used to obtain the optimal hyperparameters of the cost function and realize a fast convergence rate. The proposed method is verified using two NIR spectral datasets and compared with the full-spectral-based partial least squares (PLS) and the other three models. The results demonstrate that the proposed method has a low computational cost without losing prediction performance for the NIR spectroscopy (NIRS).

On multipliers for Fourier series in Sobolev orthogonal polynomials

Multipliers are studied for Fourier series in polynomials orthogonal in continuous-discrete Sobolev spaces. For the multiplier operator, existence results and norm estimates are obtained. The proofs are based on a representation of the Fejér kernel, the construction of a ‘humpbacked majorant’ and estimates for the norms of maximal functions. Bibliography: 45 titles.

Berezin–Toeplitz quantization in real polarizations with toric singularities

On a compact Kahler manifold $X$, Toeplitz operators determine a deformation quantization $(\operatorname{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$ with separation of variables [10] with respect to transversal complex polarizations $T^{1, 0}X, T^{0, 1}X$ as $\hbar \to 0^+$ [15]. The analogous statement is proved for compact symplectic manifolds with transversal non-singular real polarizations [13]. In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localization of our Toeplitz operators are essential. Via norm estimations, we show that these Toeplitz operators determine a star product on $X$ as $\hbar \to 0^+$.

The SIPG method of Dirichlet boundary optimal control problems with weakly imposed boundary conditions

&lt;abstract&gt;&lt;p&gt;In this paper, we consider the symmetric interior penalty Galerkin (SIPG) method which is one of Discontinuous Galerkin Methods for the Dirichlet optimal control problems governed by linear advection-diffusion-reaction equation on a convex polygonal domain and the difficulties which we faced while solving this problem numerically. Since standard Galerkin methods have failed when the boundary layers have occurred and advection diffusion has dominated, these difficulties can occur in the cases of higher order elements and non smooth Dirichlet data in using standard finite elements. We find the most convenient natural setting of Dirichlet boundary control problem for the Laplacian and the advection diffusion reaction equations.After converting the continuous problem to an optimization problem, we solve it by "discretize-then-optimize" approach. In final, we estimate the optimal priori error estimates in suitable norms of the solutions and then support the result and the features of the method with numerical examples on the different kinds of domain.&lt;/p&gt;&lt;/abstract&gt;

Lowest-order equivalent nonstandard finite element methods for biharmonic plates

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the C0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side F ∈ H−2(Ω) replaced by F ○ (JIM) and then are quasi-optimal in their respective discrete norms. The smoother JIM is defined for a piecewise smooth input function by a (generalized) Morley interpolation IM followed by a companion operator J. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj, ESAIM: M2AN 49 (2015) 977–990.] without data oscillations. This paper extends the work [Veeser and Zanotti, SIAM J. Numer. Anal. 56 (2018) 1621–1642] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.

A new self-adaptive inertial algorithm with $ W $-mapping for solving split feasibility problem in Banach spaces

&lt;abstract&gt;&lt;p&gt;In this paper, the split feasibility problem is studied in real Banach spaces. Through the $ W $-mapping, a new iterative algorithm with the inertial technique for solving the split feasibility problem is proposed, which the step size is self-adaptive and no prior estimation of operator norm is required. We prove that the proposed algorithm converges weakly to a solution of the split feasibility problem under some mild conditions. Finally, the effectiveness of the proposed algorithm is indicated by numerical experiments. Our results are innovative and can enrich recently announced related results in the literature.&lt;/p&gt;&lt;/abstract&gt;

Convergence analysis for split hierachical monotone variational inclusion problem in Hilbert spaces

Abstract In this paper, we introduce a new iterative algorithm for approximating a common solution of Split Hierarchical Monotone Variational Inclusion Problem (SHMVIP) and Fixed Point Problem (FPP) of k-strictly pseudocontractive mappings in real Hilbert spaces. Our proposed method converges strongly, does not require the estimation of operator norm and it is without imposing the strict condition of compactness; these make our method to be potentially more applicable than most existing methods in the literature. Under standard and mild assumption of monotonicity of the SHMVIP associated mappings, we establish the strong convergence of the iterative algorithm.We present some applications of our main result to approximate the solution of Split Hierarchical Convex Minimization Problem (SHCMP) and Split Hierarchical Variational Inequality Problem (SHVIP). Some numerical experiments are presented to illustrate the performance and behavior of our method. The result presented in this paper extends and complements some related results in literature.

Compact matrix operators on Banach space of absolutely $k$-summable series

In a more recent paper [23], Banach spaces of absolutely $k$-summable series $\left\vert T_{\phi }\right\vert _{k}$ have been introduced and studied by Sarıgöl and Agarwal. In this paper, we characterize matrix and compact operators from the space $\left\vert T_{\phi }\right\vert _{k}$ to the classical spaces $c_{0},c$ and $l_{\infty },$ by determining identities or estimates for operator norms and the Hausdorff measure of noncompactness.

Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities

&lt;p style='text-indent:20px;'&gt;We consider the initial value problem (IVP) associated to a coupled system of modified Kawahara/KdV type equations with polynomials nonlinearities. For the model in question, the Cauchy problem is of interest, and is shown to be well-posed for given data in a Gevrey spaces. Our results make use of techniques presented in Grujić and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view generalizes the system of modified Kawahara/KdV type equations studied by Kondo and Pes, which contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important.&lt;/p&gt;

A priori estimates for the free boundary problem of incompressible inviscid Boussinesq and MHD-Boussinesq equations without heat diffusion

&lt;abstract&gt;&lt;p&gt;For all physical spatial dimensions $ n = 2 $ and $ 3 $, we establish a priori estimates of Sobolev norms for free boundary problem of inviscid Boussinesq and MHD-Boussinesq equations without heat diffusion under the Taylor-type sign condition on the initial free boundary. It is different from MHD equations because the energy of the system is not conserved.&lt;/p&gt;&lt;/abstract&gt;

ПІДВИЩЕННЯ ФІНАНСОВОЇ ВІДКРИТОСТІ УКРАЇНИ НА ШЛЯХУ ДО ЄВРОПЕЙСЬКОЇ ІНТЕГРАЦІЇ

Introduction. The article analyzes the changes in approaches to the small economies financial openness improvement. It takes into account the new challenges of global economic instability and the contradictions of the European integration requirements of financial openness in the conditions of global economic destabilization and highlights possible ways of overcome them. Materials and methods. To disclose the theoretical model of the interconnection between financial openness, financial stability and economic growth has been used system-functional and comparative economic-statistical analysis methods based on World Bank and IMF statistic data. Results and discussion. It is emphasized that there is a trend towards strengthening control over capital flows at the global level and the removal of restrictions on the movement of international capital flows is one of the key requirements for the approximation of regulatory norms in the field of financial sector for countries going to join the EU with the regulatory framework of the latter. Given the current conditions of global uncertainty and the threat of increasing financial instability, the EU's position on the free movement of capital flows at this stage may be significantly revised. Due to the current conditions, the requirement of free movement of capital flows contradicts with a list of requirements for the approximation of the characteristics of the financial sector of the candidate countries and the EU. The need for a capital flow control policy for small open economies wishing to join the EU at this stage is not just only the need of ensuring macroeconomic stability, but also to the convergence criteria of systems, in particular in terms of joining to ERM II and BEPS. Conclusions. In overall case of small open economies, in the context of the implementation of European norms and the approximation of regulatory framework in the financial sector, it is necessary to talk about the feasibility of controlling the international capital flows at the stage of building adequate institutional environment as a support measure for further transition to free exchange rate and capital flows liberalization. As a proposal, it is stated that the elimination of identified contradictions and the transition to the free capital movement in the interests of Ukraine's European integration requires proper development of the country's financial sector and, above all, its institutional environment. The implementation of appropriate practical measures for the institutional provision of financial openness should include a policy of control over the movement of capital flows, which combines the potential of fiscal, monetary and regulatory policies in different segments of financial markets as part of the macroprudential policy.