We show that spatial zero extension across the boundary may fail in parabolic Sobolev spaces ℋ^1_p((0,T) × Ω), which can also be characterized as L_p(0,T;W^1_p(Ω))∩ W^1_p(0,T; W^-1_p(Ω)). More precisely, for any p∈ [1, ∞), we construct a function u∈ℋ^1_p((0,T)×ℝ^d_+) whose zero extension does not belong to ℋ^1_p((0,T)×ℝ^d). The obstruction occurs even for a flat boundary and is caused by a self-similar boundary layer concentrated at the initial-boundary corner, which produces a boundary supported normal flux defect after zero extension. We also discuss the suitability of various Sobolev-type spaces as solution spaces for parabolic equations in divergence form.

Many studies have focused on smart electronic noses combining machine learning and gas sensor arrays, but feature extraction for training has generally relied on dimensionality reduction techniques based on raw time-series data. These methods do not reflect the principles of sensor responses, limiting their applicability in diverse gas environments. In this study, we propose a new phase space, expressed through the first and second derivatives of dynamic response signals, to effectively characterize the nonlinear responses between gas sensors and gases. Sensing data transformed into a phase space showed unique patterns depending on the type and concentration of gases, and these were investigated for alkanes with various chain lengths (CH4, C3H8, C4H10). By applying these patterns as a preprocessing method, CNN-based gas identification machine learning achieved a high classification accuracy of 99.1% and a low concentration prediction error of 2.23 ppm using only a single sensor. Additionally, the algorithm was trained and validated across various regions of the phase space, identifying the minimum time and region required for simultaneous gas classification and concentration prediction. This study presents a novel strategy for the fast and accurate identification of gases within seconds and is expected to have significant scalability in diverse gas environments.

We establish the unique solvability of solutions in Sobolev spaces to linear parabolic equations in a more general form than those in the literature. A distinguishing feature of our equations is the inclusion of a half-order time derivative term on their right-hand side. We anticipate that such equations will prove useful in various problems involving time evolution terms. Notably, the coefficients of the equations exhibit significant irregularity, being merely measurable with respect to the temporal variable or one spatial variable.

This paper investigates weighted mixed-norm estimates for divergence-type parabolic equations on Reifenberg-flat domains with the conormal derivative boundary condition. The leading coefficients are assumed to be merely measurable in the time variable and to have small mean oscillations in the spatial variables. In deriving the boundary estimates, we overcome a regularity issue by employing half-time derivative estimates.

We establish trace and extension theorems for evolutionary equations with the Caputo fractional derivatives in (weighted) $L_p$ spaces. To achieve this, we identify weighted Sobolev and Besov spaces with mixed norms that accommodate solution spaces and their initial values well-suited for equations involving time-fractional derivatives. Our analysis encompasses both time-fractional sub-diffusion and super-diffusion equations. We also provide observations on the initial behavior of solutions to time-fractional equations.

Although many studies have explored 2D material-based field-effect transistor (FET)-type gas sensors to overcome relatively low gas responsivity, the role of gate bias remains unclear and understudied. In this work, the influence of gate-modulated channel states and thermally excited carriers on the responsivity of MoS2/h-BN FETs to NO2 gas is systematically investigated and a detailed mechanism is proposed. The electrical state of the channel is defined through electrical parameters such as threshold voltage, carrier density, and interface trap density, and the corresponding gas responsivity is evaluated using time-domain gas measurements. When the device operates in the subthreshold region, partial channel depletion leads to a weak conduction channel at the surface, where the gas reaction primarily occurs. With increasing temperature, thermally excited carriers further contribute to the gas reaction, resulting in a significant increase in responsivity, up to 2922%, without the need for surface functionalization. This study provides valuable insights into the gas sensing mechanisms of transition metal dichalcogenides-based FETs and proposes effective strategies for enhancing gas response under various conditions.

We construct Green functions of conormal derivative problems for the stationary Stokes system with measurable coefficients in a two-dimensional Reifenberg flat domain.

Field-effect transistors (FETs) using transition-metal dichalcogenides (TMDs) as channel materials are being extensively explored for future nanoelectronic applications. The mobility of MoS2 is significantly short of the theoretically anticipated value, primarily due to constraints imposed by charged impurities and phonon scattering. The interface states associated with charge trapping and detrapping can have a significant impact on mobility, necessitating a more comprehensive investigation and understanding. In this study, we fabricated MoS2 FET with hexagonal boron nitride (hBN) as a dielectric in a top-gate configuration and single pulse charge pumping (SPCP) method for different temperatures was introduced to evaluate interface trap densities. In pulse measurements, electrons move from the source to the gate during the rising phase and from the gate to source during the falling phase, manifesting as peaks in the current in time domain measurement. The interface trap densities obtained from SPCP revealed variations with temperature, observed to be comparable with those estimated from the subthreshold swing.

We present the unique solvability in Sobolev spaces of time fractional parabolic equations in divergence and non-divergence forms. The leading coefficients are merely measurable in (t,x1) for aij, 1≤i,j≤d, (i,j)≠(1,1). The coefficient a11 is merely measurable locally either in t or x1. As functions of the remaining variables, the coefficients have small mean oscillations. We consider mixed norm Sobolev spaces with Muckenhoupt weights. Our results generalize previous work on parabolic equations with time fractional derivatives to a much larger class of coefficients and solution spaces.

We prove the boundedness of the non-local operator ℒ^a u(x)=∫ _ℝ^d( u(x+y)-u(x)-χ _α (y) (∇ u(x),y )) a(x,y)dy/|y|^d+α from H_p,w^α (ℝ^d) to L_p,w(ℝ^d) for the whole range of p ∈ (1,∞ ) , where w is a Muckenhoupt weight. The coefficient a ( x , y ) is bounded, merely measurable in y , and Hölder continuous in x with an arbitrarily small exponent. We extend the previous results by removing the largeness assumption on p as well as considering weighted spaces with Muckenhoupt weights. Using the boundedness result, we prove the unique solvability in L_p spaces of the corresponding parabolic and elliptic non-local equations.

We prove the $$L_{p,q}$$ -solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the coefficients for the lower-order terms are not necessarily bounded. We study both the Dirichlet and conormal derivative boundary value problems on irregular domains. We also prove embedding results for parabolic Sobolev spaces, the proof of which is of independent interest.

We present weighted Sobolev spaces ℌ_p, θ^γ(S, T) and prove a trace theorem for the spaces. As an application, we discuss non-zero boundary value problems for parabolic equations. The weighted parabolic Sobolev spaces we consider are designed, in particular, for the regularity theory of stochastic partial differential equations on bounded domains.

We present a weighted \begin{document}$ L_p $\end{document}-theory of parabolic systems on a half space \begin{document}$ {\mathbb{R}}^d_+ $\end{document}. The leading coefficients are assumed to be only measurable in time \begin{document}$ t $\end{document} and have small bounded mean oscillations (BMO) with respect to the spatial variables \begin{document}$ x $\end{document}, and the lower order coefficients are allowed to blow up near the boundary.

We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm Ls,q-estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with s=q=2, the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for both divergence and non-divergence form parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.

A 300-GHz 4×4 multi-chip detector array based on a 65-nm CMOS technology has been developed and successfully applied to 3D terahertz imaging. The detector array with a modular assembly consists of 16 single-chip detector arrays in 4×4 format, each single chip including 7×7 unit pixels. The resultant large pixel count of 971 (including the virtual pixels) with the multi-chip array allows single-shot 2D images without raster scan, enabling a rapid 3D terahertz imaging with a significantly reduced acquisition time. In this work, 3D images of a metallic bolt were successfully acquired at 300 GHz with the image setup that employs the developed multi-chip array as the detector.

When thickness-dependent carrier mobility is coupled with Thomas-Fermi screening and interlayer resistance effects in two-dimensional (2D) multilayer materials, a conducting channel migrates from the bottom surface to the top surface under electrostatic bias conditions. However, various factors including (i) insufficient carrier density, (ii) atomically thin material thickness, and (iii) numerous oxide traps/defects considerably limit our deep understanding of the carrier transport mechanism in 2D multilayer materials. Herein, we report the restricted conducting channel migration in 2D multilayer ReS2 after a constant voltage stress of gate dielectrics is applied. At a given gate bias condition, a gradual increase in the drain bias enables a sensitive change in the interlayer resistance of ReS2, leading to a modification of the shape of the transconductance curves, and consequently, demonstrates the conducting channel migration along the thickness of ReS2 before the stress. Meanwhile, this distinct conduction feature disappears after stress, indicating the formation of additional oxide trap sites inside the gate dielectrics that degrade the carrier mobility and eventually restrict the channel migration. Our theoretical and experimental study based on the resistor network model and Thomas-Fermi charge screening theory provides further insights into the origins of channel migration and restriction in 2D multilayer devices.

We consider time fractional parabolic equations in divergence and non-divergence form when the leading coefficients a(ij) are measurable functions of (t, x(1)) except for a(11), which is a measurable function of either t or x(1). We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, and on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.

Field‐effect transistors (FETs), using transition metal dichalcogenides (TMD) as channels, have various types of interfaces, and their characteristics are sensitively changed in temperature and electrical stress. In this article, the effect of fast cyclic thermal stress on the performance of FETs using TMD as a channel is investigated and introduced. The Al2O3 passivation layer is deposited onto the TMD channel by atomic layer deposition process, and the hysteresis decreases and the direction changes from clockwise to counterclockwise. Applying cyclic thermal stress that rapidly heats and cools by 90 K in a 20 s cycle increases and decreases drain current repeatedly as charges move between the TMD channel and the interface traps. As cyclic thermal stress is applied, permanent interfacial damage occurs, resulting in increased interface trap density at the bottom and decreased hysteresis. These experimental results are also shown through technology computer‐aided design simulations. In addition, series resistance and mobility attenuation factor increase due to the concentration of the conduction paths at the bottom of the channel.

Through time-dependent defect spectroscopy and low-frequency noise measurements, we investigate and characterize the differences of carrier trapping processes occurred by different interfaces (top/sidewall) of the gate-all-around silicon nanosheet field-effect transistor (GAA SiNS FET). In a GAA SiNS FET fabricated by the top-down process, the traps at the sidewall interface significantly affect the device performance as the width decreases. Compare to expectations, as the width of the device decreases, the subthreshold swing (SS) increases from 120 to 230 mV/dec, resulting in less gate controllability. In narrow-width devices, the effect of traps located at the sidewall interface is significantly dominant, and the 1/f 2 noise, also known as generation-recombination (G-R) noise, is clearly appeared with an increased time constant (τ i ). In addition, the probability density distributions for the normalized current fluctuations (ΔI D) show only one Gaussian in wide-width devices, whereas they are separated into four Gaussians with increased in narrow-width devices. Therefore, fitting is performed through the carrier number fluctuation-correlated with mobility fluctuations model that separately considered the effects of sidewall. In narrow-width GAA SiNS FETs, consequently, the extracted interface trap densities (N T ) distribution becomes more dominant, and the scattering parameter ([Formula: see text]) distribution increases by more than double.