Norm Estimates Research Articles

How to Improve Immersive Experience?

With the explosive growth of online multi-modal applications that typically include audio, video, and haptic signals, immersive experience (IE) improvement has been broadly regarded as one of the most important tasks. Compared with traditional quality of experience (QoE) improvement for online audio/video applications, it highlights two sequential technical challenges to be resolved: i) much more stringent demand of real-time improvement due to the incorporation of delay-sensitive haptic signals, and ii) high-dimensional instead of existing one-dimensional (<i>i.e.,</i> network-level) paradigm for better online improvement. To get over this dilemma, this work systematically addresses the following three fundamental problems: i) which factors influence IE, ii) how to online improve IE, and iii) to what extent of the corresponding IE improvement can be achieved. To this end, we first comprehensively explore and categorize the influence factors on IE from various dimensions. Then, by combing network resource scheduling with the multi-domain collaboration of user profile, device specification, and application type, an online IE improvement strategy is proposed based on the efficient linear contextual bandit with the <inline-formula><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula>-norm estimation. Finally, we derive the theoretical bound of IE improvement, scaling at a poly-logarithmical function of data dimension. Numerical results on the practical system also demonstrate the remarkable improvement on IE.

A Reactive Approach to Handling Multirobot Collision Based on $p$-Norm Approximation

A Reactive Approach to Handling Multirobot Collision Based on $p$-Norm Approximation

Error estimate and superconvergence of a high-accuracy difference scheme for 2D heat equation with nonlocal boundary conditions

&lt;p&gt;In this work, we initially construct an implicit Euler difference scheme for a two-dimensional heat problem, incorporating both local and nonlocal boundary conditions. Subsequently, we harness the power of the discrete Fourier transform and develop an innovative transformation technique to rigorously demonstrate that our scheme attains the asymptotic optimal error estimate in the maximum norm. Furthermore, we derive a series of approximation formulas for the partial derivatives of the solution along the two spatial dimensions, meticulously proving that each of these formulations possesses superconvergence properties. Lastly, to validate our theoretical findings, we present two comprehensive numerical experiments, showcasing the efficiency and accuracy of our approach.&lt;/p&gt;

Noisy Tensor Completion via Low-Rank Tensor Ring.

Tensor completion is a fundamental tool for incomplete data analysis, where the goal is to predict missing entries from partial observations. However, existing methods often make the explicit or implicit assumption that the observed entries are noise-free to provide a theoretical guarantee of exact recovery of missing entries, which is quite restrictive in practice. To remedy such drawback, this article proposes a novel noisy tensor completion model, which complements the incompetence of existing works in handling the degeneration of high-order and noisy observations. Specifically, the tensor ring nuclear norm (TRNN) and least-squares estimator are adopted to regularize the underlying tensor and the observed entries, respectively. In addition, a nonasymptotic upper bound of estimation error is provided to depict the statistical performance of the proposed estimator. Two efficient algorithms are developed to solve the optimization problem with convergence guarantee, one of which is specially tailored to handle large-scale tensors by replacing the minimization of TRNN of the original tensor equivalently with that of a much smaller one in a heterogeneous tensor decomposition framework. Experimental results on both synthetic and real-world data demonstrate the effectiveness and efficiency of the proposed model in recovering noisy incomplete tensor data compared with state-of-the-art tensor completion models.

Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices

Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices

Emotion brain network topology in healthy subjects following passive listening to different auditory stimuli.

A large body of research establishes the efficacy of musical intervention in many aspects of physical, cognitive, communication, social, and emotional rehabilitation. However, the underlying neural mechanisms for musical therapy remain elusive. This study aimed to investigate the potential neural correlates of musical therapy, focusing on the changes in the topology of emotion brain network. To this end, a Bayesian statistical approach and a cross-over experimental design were employed together with two resting-state magnetoencephalography (MEG) as controls. MEG recordings of 30 healthy subjects were acquired while listening to five auditory stimuli in random order. Two resting-state MEG recordings of each subject were obtained, one prior to the first stimulus (pre) and one after the final stimulus (post). Time series at the level of brain regions were estimated using depth-weighted minimum norm estimation (wMNE) source reconstruction method and the functional connectivity between these regions were computed. The resultant connectivity matrices were used to derive two topological network measures: transitivity and global efficiency which are important in gauging the functional segregation and integration of brain network respectively. The differences in these measures between pre- and post-stimuli resting MEG were set as the equivalence regions. We found that the network measures under all auditory stimuli were equivalent to the resting state network measures in all frequency bands, indicating that the topology of the functional brain network associated with emotional regulation in healthy subjects remains unchanged following these auditory stimuli. This suggests that changes in the emotion network topology may not be the underlying neural mechanism of musical therapy. Nonetheless, further studies are required to explore the neural mechanisms of musical interventions especially in the populations with neuropsychiatric disorders.

Resting state connectivity biomarkers of seizure freedom after epilepsy surgery

Alterations in brain networks may cause the lowering of the seizure threshold and hypersynchronization that underlie the recurrence of unprovoked seizures in epilepsy. The aim of this work is to estimate functional network characteristics, which may help predicting outcome of epilepsy surgery.Twenty patients were studied (11 females, 9 males, mean age 33 years) with scalp-recorded HD-EEG in resting state (eyes closed, no interictal discharges) before intracranial evaluation, which allowed the precise determination of the epileptogenic zone. Dipole source time courses in the brain were estimated using Weighted Minimum Norm Estimate based on HD-EEG signals. Information inflow and outflow of atlas-based brain regions were computed using partial directed connectivity. A set of graph measures for pairwise connections in standard EEG frequency bands was calculated.After epilepsy surgery 10 patients were seizure-free (Engel 1a) and 10 patients continued suffering from seizures (Engel outcome worse than 1a). Inflow of the regions containing the epileptogenic zone in the beta and delta frequency bands was significantly lower in patients who achieved seizure-freedom after surgery, compared with patients who continued to have seizures (p = 0.012, and p = 0.026, respectively). Average path length in the beta frequency band was significantly higher in patients who achieved seizure freedom (p = 0.012). In the delta frequency band, local efficiency and clustering coefficient were significantly higher in patients who achieved seizure freedom (0.033, 0.046).In patients who achieved seizure freedom after surgery, the preoperative analysis of the epileptic network exhibited stronger separation of the region containing the seizure onset zone, with less inflow of information. In contrast, shorter paths within the epileptic network may facilitate hypersynchronous neuronal activity and thus the recurrence of seizures in non-seizure free patients. This study supports the hypothesis that epileptic network properties might help to define suitable candidates for epilepsy surgery.

Effect of Inverse Solutions, Connectivity Measures, and Node Sizes on EEG Source Network: A Simultaneous EEG Study.

Brain network provides an essential perspective for studying normal and pathological brain activities. Reconstructing the brain network in the source space becomes more needed, for example, as a target in non-invasive neuromodulation. Precise estimating source activities from the scalp EEG is still challenging because it is an ill-posed question and because of the volume conduction effect. There is no consensus on how to reconstruct the EEG source network. This study uses simultaneous scalp EEG and stereo-EEG to investigate the effect of inverse solutions, connectivity measures, and node sizes on the reconstruction of the source network. We evaluated the performance of different methods on both source activity and network. Numerical simulation was also carried out for comparison. The weighted phase-lag index (wPLI) method achieved significantly better performance on the reconstructed networks in source space than five other connectivity measures (directed transfer function (DTF), partial directed coherence (PDC), efficient effective connectivity (EEC), Pearson correlation coefficient (PCC), and amplitude envelope correlation (AEC)). There is no significant difference between the inverse solutions (standardized low-resolution brain electromagnetic tomography (sLORETA), weighted minimum norm estimate (wMNE), and linearly constrained minimum variance (LCMV) beamforming) on the reconstructed source networks. The source network based on signal phases can fit intracranial activities better than signal waveform properties or causality. Our study provides a basis for reconstructing source space networks from scalp EEG, especially for future neuromodulation research.

Norm and numerical radius inequalities for operator matrices

Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices with positive real and imaginary parts will be discussed, and sharper bounds will be shown for such classes.

Solution of the EEG inverse problem by random dipole sampling

Electroencephalography (EEG) source imaging aims to reconstruct brain activity maps from the neuroelectric potential difference measured on the skull. To obtain the brain activity map, we need to solve an ill-posed and ill-conditioned inverse problem that requires regularization techniques to make the solution viable. When dealing with real-time applications, dimensionality reduction techniques can be used to reduce the computational load required to evaluate the numerical solution of the EEG inverse problem. To this end, in this paper we use the random dipole sampling method, in which a Monte Carlo technique is used to reduce the number of neural sources. This is equivalent to reducing the number of the unknowns in the inverse problem and can be seen as a first regularization step. Then, we solve the reduced EEG inverse problem with two popular inversion methods, the weighted Minimum Norm Estimate (wMNE) and the standardized LOw Resolution brain Electromagnetic TomogrAphy (sLORETA). The main result of this paper is the error estimates of the reconstructed activity map obtained with the randomized version of wMNE and sLORETA. Numerical experiments on synthetic EEG data demonstrate the effectiveness of the random dipole sampling method.

An enriched cut finite element method for Stokes interface equations

In this paper, we consider an enriched cut finite element method (ECFEM) with interface-unfitted meshes for solving Stokes interface equations consisting of two incompressible fluids with different viscosities. By approximating the velocity with the enriched P1 element and the pressure with the P0 element, and stabilizing the Galerkin variational formulation with suitable ghost penalty terms, we propose the new ECFEM and prove that it is well-posed and has the optimal a priori error estimate in the energy norm. All derived results are independent of the interface position. Moreover, compared with other conforming finite element methods with the optimal rate in convergence, the proposed scheme here not only has the minimum degrees of freedom, but also avoids using the derivative of the pressure in the penalty term. The presented numerical examples validate the theoretical predictions.

Numerical solutions for Biharmonic interface problems via weak Galerkin finite element methods

This paper focuses on the advancement of weak Galerkin (WG) finite element methods for addressing two-dimensional and three-dimensional Biharmonic interface problems with polygonal/polyhedral meshes. The WG method has been demonstrated to be accurate and efficient, providing optimal order error estimates in discrete H2 and standard L2 norms. A series of extensive numerical tests are conducted to validate the WG solutions, showcasing the flexibility, stability, and robustness of the proposed method for handling both smooth and complicated interfaces.

Minimal [formula omitted]-solutions to singular sublinear elliptic problems

We solve the existence problem for the minimal positive solutions u∈Lp(Ω,dx) to the Dirichlet problems for sublinear elliptic equations of the form Lu=σuq+μinΩ,lim infx→yu(x)=0y∈∂∞Ω,where 0<q<1 and Lu≔−div(A(x)∇u) is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient σ and data μ are nonnegative Radon measures on an arbitrary domain Ω⊂Rn with a positive Green function associated with L. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inequalities, and norm estimates in terms of generalized energy.

Solution to delayed linear discrete system with constant coefficients and second‐order differences and application to iterative learning control

SummaryIn this article, we give an alternative representation and prior estimation of solution to delayed linear discrete systems with a second‐order difference with the help of the notation of delayed discrete matrix functions via their norm estimations. As an application, this solution is used to study iterative learning control under the suitable updating laws and sufficient conditions to guarantee that asymptotic convergence tracking results are presented. Finally, two numerical examples are given to verify the theoretical results.

On existence, uniqueness and stability of solutions to Cahn–Hilliard/Allen–Cahn systems with cross-kinetic coupling

A system of phase-field equations with strong-coupling through state and gradient dependent non-diagonal mobility matrices is studied. Existence of weak solutions is established by the Galerkin approximation and a-priori estimates in strong norms. Relative energy estimates are used to derive a general nonlinear stability estimate. As a consequence, a weak–strong uniqueness principle is obtained and stability with respect to model parameters is investigated.

Bayesian Inference for High Dimensional Cox Models with Gaussian and Diffused-Gamma Priors: A Case Study of Mortality in COVID-19 Patients Admitted to the ICU.

Bayesian approaches have been utilized to address the challenge of variable selection and statistical inference in high-dimensional survival analysis. However, the discontinuity of the -norm prior, including the useful spike-and-slab prior, may lead to computational and implementation challenges, potentially limiting the widespread use of Bayesian methods. The Gaussian and diffused-gamma (GD) prior has emerged as a promising alternative due to its continuous-and-differentiable -norm approximation and computational efficiency in generalized linear models. In this paper, we extend the GD prior to semi-parametric Cox models by proposing a rank-based Bayesian inference procedure with the Cox partial likelihood. We develop a computationally efficient algorithm based on the iterative conditional mode (ICM) and Markov chain Monte Carlo methods for posterior inference. Our simulations demonstrate the effectiveness of the proposed method, and we apply it to an electronic health record dataset to identify risk factors associated with COVID-19 mortality in ICU patients at a regional medical center.

Racial and ethnic disparities in self-reported sleep duration: Roles of subjective socioeconomic status and sleep norms

ObjectivesThere are racial and ethnic disparities in sleep duration, with members of historically marginalized groups typically reporting shorter sleep than White Americans. This study examines subjective social status (SSS) as a moderator, and variation in ideal sleep norms as a mediator, of differences in sleep duration between racial/ethnic groups. MethodsAsian, Black, Hispanic, and non-Hispanic White respondents in an online survey reported their typical weeknight and weekend-night sleep duration, along with estimates of ideal sleep duration norms. Objective and subjective indicators of socioeconomic status were also assessed. A conditional process analysis was used to examine whether racial or ethnic differences in sleep duration were a) moderated by SSS and b) mediated by ideal sleep duration norms. ResultsRacial/ethnic disparities in sleep duration varied by group. Hispanic participants reported shorter weeknight sleep than White participants. In Asian and Black participants, shorter weeknight sleep relative to White participants was only observed at medium (Black) or high (Black and Asian) levels of SSS. Shorter norms for ideal sleep duration partially mediated differences in sleep duration between Black and White adults, but not the other racial/ethnic groups. There was no evidence of moderated mediation. Neither income nor education moderated racial/ethnic disparities in sleep duration. ConclusionsRacial and ethnic disparities in sleep duration may partially depend on SSS. Continued research into moderators and mediators of racial/ethnic differences in sleep duration is warranted.

Control system design for a class of non-cascade nonlinear under-actuated systems with an application to a rotary crane system

For cascade under-actuated systems, several effective control methods have been established. For non-cascade under-actuated systems with one under-actuation degree, several control methods, such as cascade transformation and energy-based control, have been studied. For non-cascade nonlinear under-actuated systems with two or more under-actuation degrees, approximate linearization is necessary to apply these methods, thus only stability at the equilibrium point has been guaranteed. In this paper, control system design and stability analysis are presented for a class of non-cascade nonlinear under-actuated systems with two or more under-actuated degrees. The stability condition obtained in the analysis is a conservative sufficient condition because norm estimation is used. Based on the sufficient condition, we propose a stabilization parameter search algorithm to numerically obtain the practical control parameters. The proposed algorithm is applied to a rotary crane system as one of the nonlinear systems with two under-actuated degrees. The effectiveness of the proposed control system design is verified by numerical simulations and experiments.

End-point norm estimates for Cesàro and Copson operators

End-point norm estimates for Cesàro and Copson operators

Corrigendum: Sharp norm estimates for composition operators and Hilbert‐type inequalities

AbstractWe correct mistakes in Lemma 8 and in the proof of Lemma 10 from (Bull. Lond. Math. Soc. 49 (2017), no. 6, 965–978).