Smoothness Properties Research Articles

High-order accurate finite difference discretisations on fully unstructured dual quadrilateral meshes

We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which when differentiated provides the derivatives of the original function. The method generalises traditional finite difference methods to meshes of arbitrary topology in any number of dimensions for any order of derivative and accuracy. We demonstrate the accuracy of the numerical scheme using dual quadrilateral meshes and a refinement method based on subdivision surfaces. The scheme is applied to the solution of a range of partial differential equations, including both linear and nonlinear, second and fourth order equations, and a time-dependent first order equation.

Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus

In this paper we study the initial value problem of fully nonlinear third-order equations on the torus, that is ∂tu=F∂x3u,∂x2u,∂xu,u,x,t with F a smooth function depending on the space variable x, the time variable t, the first three derivatives of u with respect to x, and u. In particular we find conditions on u(0) and F for which one can construct a local and unique solution u. In particular if F and u(0) satisfy some conditions then the equation behaves like a diffusive one and it has a parabolic smoothing property: the solution is infinitely smooth in one direction of time and the problem is ill-posed in the other direction of time. If F and u(0) satisfy some other conditions then the equation behaves like a dispersive one. We also prove continuous dependence with respect to the data. The proof relies upon energy estimates combined with a gauge transformation [6–8] and the Bona-Smith argument (Bona and Smith, 1975).

Asymptotic analysis of the linearized Boltzmann collision operator from angular cutoff to non-cutoff

We give quantitative estimates on the asymptotics of the linearized Boltzmann collision operator and its associated equation from angular cutoff to non-cutoff. On one hand, the results disclose the link between the hyperbolic property resulting from Grad’s cutoff assumption and the smoothing property due to the long-range interaction. On the other hand, with the help of localization techniques in phase space, we observe some new phenomena in the asymptotic limit process.

On properties of univariate max functions at local maximizers

More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local maximizers, a property that is useful in several applications that we describe. We also investigate whether this smoothness property extends to max functions more generally. We show that the pointwise maximum of a finite set of q-times continuously differentiable univariate functions must have zero derivative at a maximizer for q=1, but arbitrarily close to the maximizer, the derivative may not be defined, even when q=3 and the maximizer is isolated.

High Dimensional Convolution Acceleration via Tensor Decomposition

The high-dimensional convolution, in either linear or nonlinear form, has been employed in a wide range of computer vision solutions due to its beneficial smoothing property. However, its full-kernel implementation is extremely slow. We do need a fast algorithm for this important operation. To solve this problem, we propose an acceleration pipeline assembled by three steps: [Formula: see text]-D nonlinear convolution [Formula: see text] [Formula: see text]-D linear convolution [Formula: see text] 1-D dimensional convolution [Formula: see text] 1-D recursive box filter. Thanks to the low computational complexity of box filtering, we speed up the computation significantly. Roughly speaking, our contribution is two-fold: (1) establishing the connection between the high-dimensional convolution acceleration algorithm and tensor decomposition; (2) propose total four acceleration technologies including demultiplexing–blurring–multiplexing framework, convolution decomposition, periodic tensorization and recursively box filtering to compose our acceleration pipeline under the line of the above connection. The effectiveness of these techniques is demonstrated in various comparisons and experiments. The running times of various applications are largely shortened from several minutes to fewer seconds or less.

On some regularity properties for the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov Equation

This work aims to study some smoothness properties concerning the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation. More precisely, we prove that the solutions to this model satisfy the so-called propagation of regularity. Roughly speaking, this principle states that if the initial data enjoys some extra smoothness prescribed on a family of half-spaces, then the regularity is propagated with infinite speed. In this sense, we prove that regardless of the scale measuring the extra regularity in such hyperplane collection, then all this regularity is also propagated by solutions of this model. Our analysis is mainly based on the deduction of propagation formulas relating homogeneous and non-homogeneous derivatives in certain regions of the plane.

Evaluation of lyophilized royal jelly and garlic extract emulgels using a murine model infected with methicillin-resistant Staphylococcus aureus

The limited therapeutic options associated with methicillin-resistant Staphylococcus aureus (MRSA) necessitate search for innovative strategies particularly, use of natural extracts such as lyophilized royal jelly (LRJ) and garlic extract (GE). Therefore, out study aimed to formulate emulgels containing different concentrations of both LRJ and GE and to evaluate their activities using a murine model infected with MRSA clinical isolate. Four plain emulgel formulas were prepared by mixing stearic acid/yellow soft paraffin-based O/W emulsion formulae based on Carbopol 940, Na alginate, Na carboxymethylcellulose or Hydroxypropyl methyl cellulose E4. Sodium alginate-based emulgel was selected for preparation of four medicated emulgel formulations combining LRJ and GE at four different concentrations. The selected medicated emulgels were used for the in vivo studies. The emulgel formulated with Na alginate and HPMC (MF3) exhibited optimum smooth homogeneous consistency, neutral pH, acceptable viscosity, spreadability, extrudability values and best storage stability properties. In vivo results revealed that, the wounds infected with MRSA isolate in rates were wet (oozing) and showed pus formation when compared to injured uninfected wounds. MF3 formula containing 4% LRJ and 50% GE showed the maximum wound healing properties, both in the apparent physical wound healing measurements and in the histopathological examination. In conclusion, the medicated emulgel formulation (MF3) prepared with Na alginate was found optimum for topical application. MF3 formula containing 4% LRJ and 50% GE has shown the highest in vivo wound healing capacities. Further clinical studies should be conducted to prove both its safety and efficacy and the potential use in human.

Effective Spatial Resolution of Photon Counting CT for Imaging of Trabecular Structures is Superior to Conventional Clinical CT and Similar to High Resolution Peripheral CT.

Photon counting computed tomography (PCCT) might offer an effective spatial resolution that is significantly improved compared with conventional state-of-the-art computed tomography (CT) and even provide a microstructural level of detail similar to high-resolution peripheral CT (HR-pQCT). The aim of this study was to evaluate the volumetric effective spatial resolution of clinically approved PCCT as an alternative to HR-pQCT for ex vivo or preclinical high-resolution imaging of bone microstructure. The experiment contained 5 human vertebrae embedded in epoxy resin, which were scanned 3 times each, and on 3 different clinical CT scanners: a PCCT (Naeotom Alpha), a dual-energy CT (Somatom Force [SF]), and a single-energy CT (Somatom Sensation 40 [S40]), all manufactured by Siemens Healthineers (Erlangen, Germany). Scans were performed with a tube voltage of 120 kVp and, to provide maximum scan performance and minimum noise deterioration, with exposures of 1500 mAs (SF), 2400 mAs (S40), and 4500 mAs (PCCT) and low slice increments of 0.1 (PCCT) and 0.3 mm (SF, S40). Images were reconstructed with sharp and very sharp bone kernels, Br68 and Br76 (PCCT), Br64 (SF), and B65s and B75h (S40). Ground truth information was obtained from an XtremeCT scanner (Scanco, Brüttisellen, Switzerland). Voxel-wise comparison was performed after registration, calibration, and resampling of the volumes to isotropic voxel size of 0.164 mm. Three-dimensional point spread- and modulation-transfer functions were calculated with Wiener's deconvolution in the anatomical trabecular structure, allowing optimum estimation of device- and kernel-specific smoothing properties as well as specimen-related diffraction effects on the measurement. At high contrast (modulation transfer function [MTF] of 10%), radial effective resolutions of PCCT were 10.5 lp/cm (minimum resolvable object size 476 μm) for kernel Br68 and 16.9 lp/cm (295 μm) for kernel Br76. At low contrast (MTF 5%), radial effective spatial resolutions were 10.8 lp/cm (464 μm) for kernel Br68 and 30.5 lp/cm (164 μm) for kernel Br76. Axial effective resolutions of PCCT for both kernels were between 27.0 (185 μm) and 29.9 lp/cm (167 μm). Spatial resolutions with kernel Br76 might possibly be still higher but were technically limited by the isotropic voxel size of 164 μm. The effective volumetric resolutions of PCCT with kernel Br76 ranged between 61.9 (MTF 10%) and 222.4 (MTF 5%) elements per cubic mm. Photon counting CT improved the effective volumetric resolution by factor 5.5 (MTF 10%) and 18 (MTF 5%) compared with SF and by a factor of 8.7 (MTF 10%) and 20 (MTF 5%) compared with S40. Photon counting CT allowed obtaining similar structural information as HR-pQCT. The effective spatial resolution of PCCT in trabecular bone imaging was comparable with that of HR-pQCT and more than 5 times higher compared with conventional CT. For ex vivo samples and when patient radiation dose can be neglected, PCCT allows imaging bone microstructure at a preclinical level of detail.

Substructured two-grid and multi-grid domain decomposition methods

Two-level Schwarz domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature, most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of Schwarz domain decomposition methods to define, for general elliptic problems, a new class of substructured two-level methods, for which both Schwarz smoothers and coarse correction steps are defined on the interfaces (except for the application of the smoother that requires volumetric subdomain solves). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require, like classical multi-grid methods, the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

Design of optimized interval type-2 fuzzy logic controller based on the continuity, monotonicity, and smoothness properties for a cart-pole inverted pendulum system

In this study, an interval type-2 fuzzy logic controller design method is proposed and validated on the real-time under-actuated nonlinear inverted pendulum system for swing-up and stabilization control problems. The swing-up algorithm is designed to give a faster response and the stabilization controller is designed based on continuity, monotonicity, and smoothness theorems for worst and best cases using the Mamdani fuzzy inference system in order to show the effectiveness of the proposed method. Outstanding type reduction methods are analyzed for the stabilization controller based on the design approach to determine the best type reduction algorithm for the effective and robust control performance. Real-time experiments are conducted to investigate the capability of the proposed controller in terms of adaptation performance and robustness ability. The controller also is able to handle structured and unstructured uncertainties such as measurement noise, external payload, undesirable internal/external disturbances, and parameter uncertainties. The results show that the proposed method clearly improves the control performance of the system in six experimental tasks conducted.

Searchlight asymptotics for high-frequency scattering by boundary inflection

The paper is devoted to an inner problem for a whispering gallery high-frequency asymptotic mode’s scattering by a boundary inflection. The related boundary-value problem for a Schrödinger equation on a half-line with a potential linear in both space and time turns out to be fundamental for describing transitions from modal to scattered asymptotic patterns, and despite having been intensively studied over several decades remains largely unsolved. The solution past the inflection point is shown to have a “searchlight” asymptotics corresponding to a beam concentrated near the limit ray. Certain decay and smoothness properties of the related searchlight amplitude are established. Further interpretations of the above result are also discussed: the existence of the associated generalized wave operator, and of a version of a unitary scattering operator connecting the modal and scattered asymptotic regimes.

Assessment of MEF2C as a novel myoepithelial marker using normal salivary gland and pleomorphic adenoma: An immunohistochemical study

The myoepithelium of the salivary gland demonstrates smooth muscle properties, and the myoepithelium-rich pleomorphic adenomas (PA) exhibit various cytological guises and frequently produce chondroid tissues. We recently reported that gene expression of myocyte enhancer factor 2 C (MEF2C) was up-regulated in myoepithelial cell lines established from mouse salivary glands. This transcription factor is known to regulate the development of smooth muscle cells and chondrocytes. These findings have led us to postulate that MEF2C may be a new marker of the myoepithelium and thus be of help for exploring myoepithelial participation in PA. The immunoreactivity of MEF2C was semi-quantitatively analyzed in normal salivary glands and in different types of cell in PA. The pattern of staining was also compared with that of α-smooth muscle actin (αSMA) and calponin. In the parenchyma of mouse and human submandibular glands, MEF2C expression was exclusive to the nuclei of myoepithelial cells. In the neoplastic myoepithelium of PA, epithelioid and stellate cells were consistently positive, but spindle and plasmacytoid cells exhibited selective staining. Although variability/loss of expression was evident, αSMA tended to show limited expression, whereas calponin tended to show extensive expression. Intense and uniform immunoreactivity for MEF2C was observed in lacunar cells in chondroid areas of PA. This is the first reported study to have demonstrated MEF2C immunoexpression in normal and neoplastic myoepithelial cells. Among the heterogeneously differentiated elements, MEF2C may be useful to define the myoepithelial/mesenchymal phenotypes of PA cells. Along with other relevant immunohistochemical markers, it may become another standard.

An efficient multigrid solver based on a four-color cell-block Gauss-Seidel smoother for 3D magnetotelluric forward modeling

Practical application of 3D magnetotelluric inversion requires efficient forward modeling of electromagnetic (EM) fields in the earth. To resolve realistic 3D structures, large computational domains and extremely large linear systems of equations are required. The iterative solvers, which are almost exclusively used to solve these systems, can be inefficient due to the abundant null space of the curl-curl operator. Multigrid (MG) solvers are considered a potentially efficient technique for solving such problems. However, due to the abundant null solution space and existence of the air layer, MG solvers can still converge slowly or even diverge. We have developed an efficient MG solver for finite-difference frequency-domain EM solution. In this algorithm, the excellent smoothing property of an efficient four-color cell-block Gauss-Seidel (GS) is exploited to remove the short-range errors effectively, and the interpolation and prolongation operators are used to handle the long-range errors. They work as a whole to speed the convergence of our algorithm remarkably. Because all of the nodes for the four-color cell-block GS are grouped into four colors and the edge components attached to different nodes in each color are completely decoupled, this can be used to develop a highly vectorized or parallelized algorithm. Another important property is that our algorithm is locally current divergence free, effectively eliminating spurious solutions in the null space of the curl-curl operator. The accuracy and efficiency of the algorithm are verified by comparing the numerical solutions obtained with our MG solver to those from the biconjugate gradient stabilized solver with different preconditioners based on synthetic models and a model from 3D inversion. Comparisons, in terms of iteration number and computational time, indicate that our algorithm is extremely stable and efficient relative to the other solvers. Our MG algorithm will be suitable for massively parallel computing as well.

Partial Smoothness and Constant Rank

The idea of partial smoothness in optimization blends certain smooth and nonsmooth properties of feasible regions and objective functions. As a consequence, the standard first-order conditions guarantee that diverse iterative algorithms (and post-optimality analyses) identify active structure or constraints. However, by instead focusing directly on the first-order conditions, the formal concept of partial smoothness simplifies dramatically: in basic differential geometric language, it is just a constant-rank condition. In this view, partial smoothness extends to more general mappings, such as saddlepoint operators underlying primal-dual splitting algorithms.

Improving photocatalytic activity under visible light over a novel food wastes biochar-based BiOBr nanocomposite

Biochar (C) applied in synthesizing photocatalysts to eliminate water pollution has been intensively investigated. Herein we report the first use of biochar pyrolyzed from food wastes at 400 °C (400C) and 700 °C to construct C/BiOBr composites via a facile hydrolysis approach. Photocatalytic performances could be significantly improved by choosing the appropriate carbonization temperature and adjusting the content of C in C/BiOBr composites. The prepared 1%400C/BiOBr exhibited the best photodegradation capacity towards methylene orange (20 mg/L) and tetracycline (50 mg/L). A series of characterization results illustrated that smooth structure and surface properties (oxygen functional groups and persistent free radicals) of 400C played an important role in enhancing the photocatalytic activities. Mechanism exploration suggested that h+ and ˙O2− were the main active species thus contributing to photodegradation. This study provided a new insight into utilization of biochar derived from food wastes in photocatalysis and environmental remediation.

Description of the smoothing effects of semigroups generated by fractional Ornstein–Uhlenbeck operators and subelliptic estimates

We study semigroups generated by general fractional Ornstein–Uhlenbeck operators acting on \(L^2({\mathbb {R}}^n)\). We characterize geometrically the partial Gevrey-type smoothing properties of these semigroups, and we sharply describe the blow-up of the associated seminorms for short times, generalizing the hypoelliptic and the quadratic cases. As a byproduct of this study, we establish partial subelliptic estimates enjoyed by fractional Ornstein–Uhlenbeck operators on the whole space by using interpolation theory.

Small screen-big information challenge for older adults: a study on visual momentum and gesture navigation

ABSTRACT The explosion of information and the miniaturisation of mobile devices seems to be an oxymoron. A large amount of information means a wider information space, and a smaller screen means a more limited display capacity. ‘Small screen-big information’ can often be disorienting, especially for older adults. Treating it as navigation in information space, we explored solutions in terms of spatial cognition and gesture motor control. First, we introduce a concept from cognitive psychology, visual momentum (VM). In terms of gesture control, we propose the device-based method that differs from traditional touch-based methods. To test their roles, two experiments were conducted among 30 older adults and comparatively tested 30 younger adults. Experiment 1 examined the contribution of high VM and a device-based method. Experiment 2 further examined an important property of view translation, smoothness, in touch-based and device-based methods. Results show that increasing the VM by adding grids and using a device-based method can help users find targets faster and more accurately, which is particularly effective for the older group. Meanwhile, differences in navigation strategies between touch-based and device-based methods were proposed. Finally, the smoothness of a device-based method in visual information processing is also demonstrated.

Smoothing effect and large time behavior of solutions to nonlinear elastic wave equations with viscoelastic term

The Cauchy problem for a nonlinear elastic wave equation with viscoelastic damping terms is considered on the 3 dimensional whole space. Decay and smoothing properties of the solutions are investigated when the initial data are sufficiently small; and asymptotic profiles as t→∞ are also derived.

Implementation of a Gaussian Markov random field sampler for forward uncertainty quantification in the Ice-sheet and Sea-level System Model v4.19

Abstract. Assessing the impact of uncertainties in ice-sheet models is a major and challenging issue that needs to be faced by the ice-sheet community to provide more robust and reliable model-based projections of ice-sheet mass balance. In recent years, uncertainty quantification (UQ) has been increasingly used to characterize and explore uncertainty in ice-sheet models and improve the robustness of their projections. A typical UQ analysis first involves the (probabilistic) characterization of the sources of uncertainty, followed by the propagation and sensitivity analysis of these sources of uncertainty. Previous studies concerned with UQ in ice-sheet models have generally focused on the last two steps but have paid relatively little attention to the preliminary and critical step of the characterization of uncertainty. Sources of uncertainty in ice-sheet models, like uncertainties in ice-sheet geometry or surface mass balance, typically vary in space and potentially in time. For that reason, they are more adequately described as spatio-(temporal) random fields, which account naturally for spatial (and temporal) correlation. As a means of improving the characterization of the sources of uncertainties for forward UQ analysis within the Ice-sheet and Sea-level System Model (ISSM), we present in this paper a stochastic sampler for Gaussian random fields with Matérn covariance function. The class of Matérn covariance functions provides a flexible model able to capture statistical dependence between locations with different degrees of spatial correlation or smoothness properties. The implementation of this stochastic sampler is based on a notable explicit link between Gaussian random fields with Matérn covariance function and a certain stochastic partial differential equation. Discretization of this stochastic partial differential equation by the finite-element method results in a sparse, scalable and computationally efficient representation known as a Gaussian Markov random field. In addition, spatio-temporal samples can be generated by combining an autoregressive temporal model and the Matérn field. The implementation is tested on a set of synthetic experiments to verify that it captures the desired spatial and temporal correlations well. Finally, we illustrate the interest of this stochastic sampler for forward UQ analysis in an application concerned with assessing the impact of various sources of uncertainties on the Pine Island Glacier, West Antarctica. We find that larger spatial and temporal correlations lengths will both likely result in increased uncertainty in the projections.

Symmetrical Acceptor–Donor–Acceptor Molecule as a Versatile Defect Passivation Agent toward Efficient FA0.85MA0.15PbI3 Perovskite Solar Cells

AbstractDespite the swift development in perovskite solar cells (PSCs), suppressing the ion defects in the perovskite bulk and further extending the long‐lasting stability of the cells remain the concerned issues that are yet to be solved. Here, a symmetrical organic acceptor−donor−acceptor (A−D−A) molecule with the core architecture of indaceno[1,2‐b:5,6‐b']dithiophene (IDT) and bilateral arms of oxindole, named IDT‐OD, as a versatile defect passivation agent, is adopted to inactivate the nonradiative recombination sites in the perovskite absorber. The S element in the IDT unit and carbonyl group CO in the bilateral acceptor unit as the Lewis‐base contributes to the passivation sites that are the under‐coordinated Pb2+ cation defects and the N−H group in oxindole unit interacts with halide dangling bonds. The molecular structure with its symmetrical double arms assists the formation of a superior perovskite layer with enlarged grain size, smooth surface topography, hydrophobic property, and low density of defect state. Consequently, the corresponding PSCs with the proper IDT‐OD additive yield a remarkable increase in efficiency from 22.77% to 24.04%, along with excellent long‐term environmental and thermal stabilities. This study offers a propitious approach for ionic defect passivation engineering toward high‐performance PSCs.