Norm Estimates Research Articles

Language-related brain areas in childhood epilepsy with centrotemporal spikes studied with MEG

ObjectiveChildren with self-limited epilepsy with centrotemporal spikes (SeLECTS) typically indicate cognitive impairment with widespread speech impairment. We explored how epilepsy affects language-related brain areas and areas in their vicinity. MethodsTwenty-two children with SeLECTS and declined verbal comprehension (DVC), 21 with SeLECTS and normal verbal comprehension (NVC), and 23 healthy controls (HCs) underwent high-sampling magnetoencephalography recordings. According to a previous study, 24 language-related regions of interest were selected bilaterally, and the relative spectral power was estimated using a minimum norm estimate. ResultsThe highest mean power spectral density was observed in the delta band for the DVC group, in the theta band for the NVC group, and in the alpha band for HCs within language-specific brain regions. The distinctions between the DVC and NVC groups in the delta and theta frequency bands were primarily concentrated in the right linguistic brain area. ConclusionsChildren with SeLECTS may have developmental problems in language-related brain areas, with different developmental levels observed in the DVC, NVC, and HC groups. The DVC group could have inferior speech comprehension due to a more significant number of seizures and more left-sided spike locations. SignificanceChildren having SeLECTS showed impaired brain maturation, leading to associated language impairment.

Generalized Upper Bounds Estimation of Numerical Radius and Norm for the Sum of Operators

Generalized Upper Bounds Estimation of Numerical Radius and Norm for the Sum of Operators

Joint BS-RIS-User Association and Beamforming Design for RIS-Assisted Cellular Networks

Reconfigurable intelligent surface (RIS) is a revolutionary technology for sixth-generation (6G) networks owing to its ability to manipulate wireless environments. As a frequency-selective device, RIS can only effectively shape the propagation of signals within a certain frequency band. Due to this frequency-selective property, the deployment of RIS in cellular networks will introduce a complicated base station (BS)-RIS-user association issue since adjacent BSs operate at different frequency bands. In this paper, with the consideration of the frequency-selective characteristics of RIS, we aim to jointly optimize BS-RIS-user association, active beamforming at BSs, and passive beamforming of RIS to maximize the sum-rate of a RIS-assisted cellular network. We first leverage <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 to efficiently integrate BS-RIS-user association with active and passive beamforming. Then, we adopt fractional programming (FP) and block coordinate descent (BCD) methods to deal with logarithmic and fractional parts and decouple the joint association and beamforming design problem into several sub-problems. Efficient algorithms which combine <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, majorization-minimization (MM), and alternating direction method of multipliers (ADMM) are developed to alternately solve the sub-problems. Extensive simulation results illustrate the importance of BS-RIS-user association optimization in RIS-assisted cellular networks and verify the effectiveness of the proposed joint association and beamforming design algorithm.

Volterra operator norms : a brief survey

Abstract In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.

Resource-Aware Video Delivery in Fog Radio Access Networks: A Joint QoE and QoS Perspective

In this paper, the video delivery strategy is studied under the cooperative framework of cache-enabled fog radio access networks (F-RAN), where multicast beamforming and video-quality selection are both taken into account in terms of different scenarios. In the user-centered scenario, an optimization problem regarding to the network quality of experience (QoE) improvement is first formulated under the essential quality of service (QoS) requirements of the users. Furthermore, an energy efficiency maximization scheme is proposed in the user-dense scenario to improve the resource utilization when network resources may be insufficient to guarantee the QoS requirements of all the users. In particular, a user scheduling scheme is studied in this case. More specifically, the non-convex constraints and fractional object functions in the aforementioned problems are dealt with by successive convex approximation (SCA) technique and the Dinkelbach approach. In order to solve the <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 problem caused by user scheduling, we introduce the <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 to improve the sparsity of the solution. Finally, the effectiveness of the proposed schemes is validated by extensive simulations.

Conditioning Theory for Generalized Inverse CA‡ and Their Estimations

The conditioning theory of the generalized inverse CA‡ is considered in this article. First, we introduce three kinds of condition numbers for the generalized inverse CA‡, i.e., normwise, mixed and componentwise ones, and present their explicit expressions. Then, using the intermediate result, which is the derivative of CA‡, we can recover the explicit condition number expressions for the solution of the equality constrained indefinite least squares problem. Furthermore, using the augment system, we investigate the componentwise perturbation analysis of the solution and residual of the equality constrained indefinite least squares problem. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator to devise the first algorithm and the small-sample statistical condition estimation method for the other two algorithms. In the end, the numerical examples illuminate the obtained results.

Discontinuous Galerkin Approximation of the Fully Coupled Thermo-poroelastic Problem

We present and analyze a discontinuous Galerkin method for the numerical modeling of the nonlinear fully coupled thermo-poroelastic problem. For the spatial discretization, we design a high-order discontinuous Galerkin method on polygonal and polyhedral grids based on a novel four-field formulation of the problem. To handle the nonlinear convective transport term in the energy conservation equation we adopt a fixed-point linearization strategy. We perform a robust stability analysis for the linearized semidiscrete problem under mild requirements on the problem’s physical parameters. A priori hp-version error estimates in suitable energy norms are also derived. A complete set of numerical simulations is presented in order to validate the theoretical analysis, to inspect numerically the robustness properties, and to test the capability of the proposed method in a practical scenario inspired by a geothermal problem.

Norm approximation for the Fröhlich dynamics in the mean-field regime

We study the time evolution of the Fröhlich Hamiltonian in a mean-field limit in which many particles weakly couple to the quantized phonon field. Assuming that the particles are initially in a Bose-Einstein condensate and that the excitations of the phonon field are initially in a coherent state we provide an effective dynamics which approximates the time evolved many-body state in norm, provided that the number of particles is large. The approximation is given by a product state which evolves according to the Landau–Pekar equations and which is corrected by a Bogoliubov dynamics. In addition, we extend the results from Leopold et al. (2021) [27] about the approximation of the time evolved many-body state in trace-norm topology to a larger class of many-body initial states with an improved rate of convergence.

A new formalization of Dirichlet-type spaces

Applications and findings involving the norm of the weighted Dirichlet space Dα are steadily being discovered. There are notable characterizations of Dα through estimations of its norm. Applications include determining the conditions for boundedness of a Bergman type operator, and characterizing functions in the Drury–Arveson space that induce bounded Hankel operators. This paper improves the norm estimation by extending the domain of relevant parameters, and establishing the asymptotic formula with precise constants.

Evaluation of Norm of (p, q)-Bernstein Operators

Abstract In this paper, we aim to study about the estimation of norm of (p, q)-Bernstein operators B p , q n $\mathcal{B}_{p,q}^{n}$ in C[0,1] for the case q &gt; p &gt; 1 by applying (p, q)-calculus and divided difference analogue of (p, q)-Bernstein operators. Some basic theorem and related results are also discussed in this paper. Here, the extra parameter p shows more flexibility by choosing the value of p.

Two infinity norm bounds for the inverse of Nekrasov matrices

Nekrasov matrices play an important role in various scientific disciplines. The estimation of infinity norm bounds for the inverse of Nekrasov matrices brings a lot of convinces in many fields. In this paper, we introduce two new bounds for the inverse of Nekrasov matrices. The advantages of our bounds and numerical examples are also presented.

A two-grid immersed finite element method with the Crank-Nicolson time scheme for semilinear parabolic interface problems

In this paper, we propose and analyze a new two-grid partially penalized immersed finite element method for solving the semilinear parabolic interface problems with meshes independent of the coefficient discontinuity. Based on the corresponding time-discrete system, we can unconditionally derive the optimal error estimates in both the L2 norm and semi-H1 norm, while previous works always require the coupling condition of time step and space size (e.g. condition τ=O(H)). Then, we design a two-grid algorithm based on Newton iteration to deal with nonlinear source term. It is shown, both theoretically and numerically, that the algorithm can achieve asymptotically optimal approximation in L2 norm (or semi-H1 norm) when the mesh size satisfies H=O(h3/2) (or H=O(h3)).

Norm estimates of the partial derivatives and Schwarz lemma for α-harmonic functions

Suppose -1 $ ]]> α > − 1 and 1 ≤ p ≤ ∞ . Let f = P α [ F ] be an α-harmonic mapping on D with the boundary F being absolute continuous and F ˙ ∈ L p ( 0 , 2 π ) , where F ˙ ( e iθ ) := d d θ F ( e iθ ) . In this paper, we investigate the membership of f z and f z ¯ in the space H G p ( D ) , the generalized Hardy space. We prove, if 0 $ ]]> α > 0 , then both f z and f z ¯ are in H G p ( D ) . If α < 0 , then f z and f z ¯ ∈ H G p ( D ) if and only if f is analytic. Finally, we investigate a Schwarz Lemma for α-harmonic functions.

Pointwise error estimates and local superconvergence of Jacobi expansions

As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] revealed that the Chebyshev interpolation of | x − a | |x-a| (with | a | &gt; 1 |a|&gt;1 ) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about 95 95% range of [ − 1 , 1 ] [-1,1] except for a small neighbourhood near the singular point x = a . x=a. In this paper, we rigorously show that the Jacobi expansion for a more general class of Φ \Phi -functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired log ⁡ n \log n -factor in the pointwise error estimate for the Legendre expansion recently stated in Babus̆ka and Hakula [Comput. Methods Appl. Mech Engrg. 345 (2019), pp. 748–773] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates.

Altered neuromagnetic activity in default mode network in childhood absence epilepsy.

The electrophysiological characterization of resting state oscillatory functional connectivity within the default mode network (DMN) during interictal periods in childhood absence epilepsy (CAE) remains unclear. Using magnetoencephalographic (MEG) recordings, this study investigated how the connectivity within the DMN was altered in CAE. Using a cross-sectional design, we analyzed MEG data from 33 children newly diagnosed with CAE and 26 controls matched for age and sex. The spectral power and functional connectivity of the DMN were estimated using minimum norm estimation combined with the Welch technique and corrected amplitude envelope correlation. Default mode network showed stronger activation in the delta band during the ictal period, however, the relative spectral power in other bands was significantly lower than that in the interictal period (p corrected < 0.05 for DMN regions, except bilateral medial frontal cortex, left medial temporal lobe, left posterior cingulate cortex in the theta band, and the bilateral precuneus in the alpha band). It should be noted that the significant power peak in the alpha band was lost compared with the interictal data. Compared with controls, the interictal relative spectral power of DMN regions (except bilateral precuneus) in CAE patients was significantly increased in the delta band (p corrected < 0.01), whereas the values of all DMN regions in the beta-gamma 2 band were significantly decreased (p corrected < 0.01). In the higher frequency band (alpha-gamma1), especially in the beta and gamma1 band, the ictal node strength of DMN regions except the left precuneus was significantly higher than that in the interictal periods (p corrected < 0.01), and the node strength of the right inferior parietal lobe increased most significantly in the beta band (Ictal: 3.8712 vs. Interictal: 0.7503, p corrected < 0.01). Compared with the controls, the interictal node strength of DMN increased in all frequency bands, especially the right medial frontal cortex in the beta band (Controls: 0.1510 vs. Interictal: 3.527, p corrected < 0.01). Comparing relative node strength between groups, the right precuneus in CAE children decreased significantly (β: Controls: 0.1009 vs. Interictal: 0.0475; γ 1: Controls:0.1149 vs. Interictal:0.0587, p corrected < 0.01) such that it was no longer the central hub. These findings indicated DMN abnormalities in CAE patients, even in interictal periods without interictal epileptic discharges. Abnormal functional connectivity in CAE may reflect abnormal anatomo-functional architectural integration in DMN, as a result of cognitive mental impairment and unconsciousness during absence seizure. Future studies are needed to examine if the altered functional connectivity can be used as a biomarker for treatment responses, cognitive dysfunction, and prognosis in CAE patients.

Effect of channel density, inverse solutions and connectivity measures on EEG resting-state networks reconstruction: A simulation study

Along with the study of brain activity evoked by external stimuli, the past two decades witnessed an increased interest in characterizing the spontaneous brain activity occurring during resting conditions. The identification of connectivity patterns in this so-called "resting-state" has been the subject of a great number of electrophysiology-based studies, using the Electro/Magneto-Encephalography (EEG/MEG) source connectivity method. However, no consensus has been reached yet regarding a unified (if possible) analysis pipeline, and several involved parameters and methods require cautious tuning. This is particularly challenging when different analytical choices induce significant discrepancies in results and drawn conclusions, thereby hindering the reproducibility of neuroimaging research. Hence, our objective in this study was to shed light on the effect of analytical variability on outcome consistency by evaluating the implications of parameters involved in the EEG source connectivity analysis on the accuracy of resting-state networks (RSNs) reconstruction. We simulated, using neural mass models, EEG data corresponding to two RSNs, namely the default mode network (DMN) and dorsal attentional network (DAN). We investigated the impact of five channel densities (19, 32, 64, 128, 256), three inverse solutions (weighted minimum norm estimate (wMNE), exact low-resolution brain electromagnetic tomography (eLORETA), and linearly constrained minimum variance (LCMV) beamforming) and four functional connectivity measures (phase-locking value (PLV), phase-lag index (PLI), and amplitude envelope correlation (AEC) with and without source leakage correction), on the correspondence between reconstructed and reference networks. We showed that, with different analytical choices related to the number of electrodes, source reconstruction algorithm, and functional connectivity measure, high variability is present in the results. More specifically, our results show that a higher number of EEG channels significantly increased the accuracy of the reconstructed networks. Additionally, our results showed significant variability in the performance of the tested inverse solutions and connectivity measures. Such methodological variability and absence of analysis standardization represent a critical issue for neuroimaging studies that should be prioritized. We believe that this work could be useful for the field of electrophysiology connectomics, by increasing awareness regarding the challenge of variability in methodological approaches and its implications on reported results.

Spectral and norm estimates for matrix-sequences arising from a finite difference approximation of elliptic operators

When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for matrix-sequences arising from the approximation of the Laplacian via ad hoc finite differences. The analysis involves several tools from matrix theory and in particular from the setting of Toeplitz operators and Generalized Locally Toeplitz matrix-sequences. Several numerical experiments are conducted, which confirm the correctness of the theoretical findings.

A class of three dimensional Cahn-Hilliard equation with nonlinear diffusion

In this paper, we study the well-posedness and asymptotic behavior for a class of Cahn-Hilliard equation with nonlinear diffusion in R3. In order to overcome the difficulties caused by the derivatives of multi-well potential and the nonlinear terms, we “borrow” a linear principle part from the derivatives of multi-well potential, rewrite the equation as an equivalent equation with linear principle part. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Moreover, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the H4-norm of initial data is sufficiently small. In the end, based on parabolic interpolation inequality, bootstrap argument and some weighted estimates, we also establish the space-time decay estimates of strong solutions.

Two-weight inequalities for multilinear maximal functions in Orlicz spaces

In this note, we provide various two-weight norm estimates of the multi-linear fractional maximal function and weighted maximal function between different Orlicz spaces. More precisely, we obtain Sawyer-type characterizations and norm estimates for these operators.

ASYMPTOTIC NORMALITY OF L1-APPROACH A KERNEL ESTIMATOR OF CONDITIONAL CUMULATIVE DISTRIBUTION FUNCTION IN THE FUNCTIONAL SINGLE INDEX MODEL

The main of this paper is to treat the estimation of conditional distribution function for functional data. We defined the L1 norm estimator. Under some assumption in functional data analysis the asymptotic normality of estimator is established.