Convergence properties of generalized Lupaş-Kantorovich operators

In the present paper, we consider the Kantorovich modification of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing and unbounded function $\rho$. For these new operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation.

A Note on New Construction of Meyer-König and Zeller Operators and Its Approximation Properties

The topic of approximation with positive linear operators in contemporary functional analysis and theory of functions has emerged in the last century. One of these operators is Meyer–König and Zeller operators and in this study a generalization of Meyer–König and Zeller type operators based on a function τ by using two sequences of functions will be presented. The most significant point is that the newly introduced operator preserves {1,τ,τ2} instead of classical Korovkin test functions. Then asymptotic type formula, quantitative results, and local approximation properties of the introduced operators are given. Finally a numerical example performed by MATLAB is given to visualize the provided theoretical results.

The Fractal Geometry of TiAlNiAu Thin Film Metal System and Its Sheet Resistance (Lateral Size Effect)

This paper investigates the relation between the geometry of metric space of a TiAlNiAu thin film metal system and the geometry of normed functional space of its sheet resistances (functionals), which are elements of the functional space. The investigation provides a means to describe a lateral size effect that involves a dependency in local approximation of sheet resistance Rsq of TiAlNiAu metal film on its lateral linear dimensions (in (x,y) plane). This dependency is defined by fractal geometry of dendrites, or, more specifically, it is a power-law dependency on fractal dimension Df value. The revealed relation has not only fundamental but also a great practical importance both for a precise calculation of thin film metal system Rsq values in designing discreet devices and ICs, and for controlling results at micro- and nanoscale in producing workflow for thin metal films and systems based on them.

Gradient Regularization as Approximate Variational Inference

We developed Variational Laplace for Bayesian neural networks (BNNs), which exploits a local approximation of the curvature of the likelihood to estimate the ELBO without the need for stochastic sampling of the neural-network weights. The Variational Laplace objective is simple to evaluate, as it is the log-likelihood plus weight-decay, plus a squared-gradient regularizer. Variational Laplace gave better test performance and expected calibration errors than maximum a posteriori inference and standard sampling-based variational inference, despite using the same variational approximate posterior. Finally, we emphasize the care needed in benchmarking standard VI, as there is a risk of stopping before the variance parameters have converged. We show that early-stopping can be avoided by increasing the learning rate for the variance parameters.

Identifying the microtearing modes in the pedestal of DIII-D H-modes using gyrokinetic simulations

Recent evidence points toward the microtearing mode (MTM) as an important fluctuation in the H-mode pedestal for anomalous electron heat transport. A study of the instabilities in the pedestal region carried out using gyrokinetic simulations to model an ELMy H-mode DIII-D discharge (USN configuration, 1.4 MA plasma current, and 3 MW heating power) is presented. The simulations produce MTMs, identified by predominantly electromagnetic heat flux, small particle flux, and a substantial degree of tearing parity. The magnetic spectrogram from Mirnov coils exhibits three distinct frequency bands---two narrow bands at lower frequency ($\sim$35-55 kHz and $\sim$70-105 kHz) and a broader band at higher frequency ($\sim$300-500 kHz). Global linear GENE simulations produce MTMs that are centered at the peak of the $\omega_*$ profile and correspond closely with the bands in the spectrogram. The three distinctive frequency bands can be understood from the basic physical mechanisms underlying the instabilities. For example (i) instability of certain toroidal mode numbers (n) is controlled by the alignment of their rational surfaces with the peak in the $\omega^*$ profile, and (ii) MTM instabilities in the lower n bands are the conventional collisional slab MTM, whereas the higher n band depends on curvature drive. While many features of the modes can be captured with the local approximation, a global treatment is necessary to quantitatively reproduce the detailed band gaps of the low-n fluctuations. Notably, the transport signatures of the MTM are consistent with careful edge modeling by SOLPS.

Influence of Spatial Dispersion on Propagation Properties of Waveguides Based on Hyperbolic Metamaterial

In this work, we study the effect of spatial dispersion on propagation properties of planar waveguides with the core layer formed by hyperbolic metamaterial (HMM). In our case, the influence of spatial dispersion was controlled by changing the unit cell’s dimensions. Our analysis revealed a number of new effects arising in the considered waveguides, which cannot be predicted with the help of local approximation, including mode degeneration (existence of additional branch of TE and TM high-β modes), power flow inversion, propagation gap, and plasmonic-like modes characterized with long distance propagation. Additionally, for the first time we reported unusual characteristic points appearing for the high-β TM mode of each order corresponding to a single waveguide width for which power flow tends to zero and mode stopping occurs.