Degree Of Smoothness Research Articles

A note on the L1 discretization error for the Caputo derivative in Hölder spaces

We establish uniform error bounds of the L1 discretization of the Caputo derivative of Hölder continuous functions. The result can be understood as: error = degree of smoothness - order of the derivative. We present an elementary proof and illustrate its optimality with numerical examples.

Photocatalyst immobilisation on cylindrical surface: Scale-up potential of continuous photocatalytic reactor

The use of continuous photocatalytic reactor is important for sustainable water treatment technology. Photocatalyst immobilisation on supports (of varying shapes and sizes) having the desired degree of uniformity and smoothness is challenging, and is evolving. The major concerns are the geometry of the surface (which restricts the scalability of such systems) and the appropriate adhesion of the photocatalyst. Flat surface has an inherent limitation of low photocatalyst area per floor space, and poor light distribution, whereas cylindrical surfaces can overcome all these limitations. In this work, we have developed and assessed the photocatalyst coating methodology over cylindrical (quartz) surface and have prepared a modular (continuous) photocatalytic reactor. We have immobilised titanium-dioxide on the outer surface of a cylindrical quartz tube using a rotating (coating) mechanism, using the viscous sol prepared from the sol-gel method followed by heat treatment, and not with any binder to the as-synthesized powder. Various characterization techniques, including ultraviolet-diffuse reflectance spectroscopy, energy-dispersive spectroscopy, x-ray diffraction, field-emission scanning electron microscopy, optical surface profilometry, and ultraviolet–visible spectroscopy, are used to evaluate the impact of process conditions – number of layers, rotational speed, and sol viscosity, on the coating uniformity, smoothness, and layer thickness. The performance of the custom-made (modular-type) reactor is evaluated for degradation of the model molecule (pollutant). The reactor performance and capacity can be easily scaled up by parallel connections or introducing multiple coated tubes (to increase the photocatalyst area). The outcome of the work will be of immense value in scaling up the continuous large scale photocatalytic operation and treatment process.

The Noether–Lefschetz locus of surfaces in P3${\mathbb {P}}^3$ formed by determinantal surfaces

AbstractWe compute the dimension of certain components of the family of smooth determinantal degree surfaces in , and show that each of them is the closure of a component of the Noether–Lefschetz locus . Our computations exhibit that smooth determinantal surfaces in of degree 4 form a divisor in with five irreducible components. We will compute the degrees of each of these components: , and .

Generalized Additive Models

Generalized additive models are generalized linear models in which the linear predictor includes a sum of smooth functions of covariates, where the shape of the functions is to be estimated. They have also been generalized beyond the original generalized linear model setting to distributions outside the exponential family and to situations in which multiple parameters of the response distribution may depend on sums of smooth functions of covariates. The widely used computational and inferential framework in which the smooth terms are represented as latent Gaussian processes, splines, or Gaussian random effects is reviewed, paying particular attention to the case in which computational and theoretical tractability is obtained by prior rank reduction of the model terms. An empirical Bayes approach is taken, and its relatively good frequentist performance discussed, along with some more overtly frequentist approaches to model selection. Estimation of the degree of smoothness of component functions via cross validation or marginal likelihood is covered, alongside the computational strategies required in practice, including when data and models are reasonably large. It is briefly shown how the framework extends easily to location-scale modeling, and, with more effort, to techniques such as quantile regression. Also covered are the main classes of smooths of multiple covariates that may be included in models: isotropic splines and tensor product smooth interaction terms.

A New Detailed Global Map of Lunar Light Plains

Lunar light plains (LPs) are globally distributed, smooth and flat geologic units occurring in crater floors and topographic lows, similar in morphology to mare basalts, but with moderate to high albedo. Despite numerous studies investigating LPs, their emplacement mechanism(s), ages, and origins remain highly debated. To aid in deciphering their origin(s), we produced a new independent global map of the extent of the LPs. Thus, we (1) evaluated the morphological and compositional definitions of LP; (2) tested the reproducibility of previous work; and (3) extended the previously mapped extents of LPs. We find morphological similarities between near- and farside LPs with regional variations in albedo and FeO contents. We were able to reproduce the majority of previous LP maps, with some notable exceptions. We find ∼13% of the lunar surface is covered by LPs, which is slightly higher than previous studies, likely due to our inclusion of smaller deposits. Overall, the distribution of LPs on the nearside appears to be denser around mare regions and exhibits more widely varying degrees of smoothness, size, albedo, and FeO contents compared to the farside highlands. In contrast, on the farside, we find a large number of LPs with similar morphologies and FeO contents, in ray-like patterns likely related to the Orientale impact event. Beyond Orientale’s influence, the occurrences of LPs are of lower albedo and FeO contents are fewer and sparser, with similar morphologies to the nearside deposits. Thus, our study indicates that an impact-related origin for the majority of LP deposits is most likely.

A Review of Research on Cold Chain Logistics and Distribution of Fresh Agricultural Products in China

At present, China, as the second largest economy in the world, is in the new development pattern, that is, to build a new development pattern with the domestic macro-cycle as the main body and the domestic and international double-cycle promoting each other, which has high requirements for the smoothness of the circulation system, urgently needs the circulation system to have a high degree of smoothness, and urgently needs to accelerate the upgrading of the logistics distribution industry, so as to realise the synchronous development of the logistics distribution industry and the economy of the country. Moreover, under the background of China's rapid development, people's needs for fresh agricultural products and other cold-chain products, as well as product quality requirements have gradually improved, therefore, under the background of rapid changes in the market, the management of China's cold-chain logistics should also be further improved to meet the market requirements. Based on this, this paper takes cold chain logistics and distribution as an entry point, and through the current development status of cold chain logistics and distribution of fresh agricultural products, the distribution mode status quo, the current research status of the constraints of fresh agricultural products logistics and distribution, as well as distribution optimisation strategy research, etc., it is hoped to draw the research direction and research focus of future cold chain logistics and distribution of fresh agricultural products, with a view to providing reference opinions to promote China's fresh agricultural products in the cold chain logistics development and improvement, so as to meet the market's requirements. In order to provide reference opinions to promote the development and improvement of the cold chain logistics of fresh agricultural products in China, so as to meet the higher requirements of the people for fresh agricultural products.

An adaptive functional regression framework for locally heterogeneous signals in spectroscopy

Abstract In recent years, there has been growing attention towards food nutritional properties, traceability, and production systems prioritizing environmental sustainability. Consequently, there is a rising demand for tools evaluating food quality and authenticity, with mid-infrared (MIR) spectroscopy techniques playing a pivotal role to collect vast amounts of data. These data pose some challenges that existing methods struggle to address, thus necessitating the development of new statistical techniques. We introduce an adaptive functional regression framework allowing for the definition of a flexible estimator accommodating different degrees of smoothness. We provide an optimization procedure handling both Gaussian and non-Gaussian responses, and allowing for the inclusion of scalar covariates. Our proposal is applied to MIR spectroscopy data, providing excellent performances when predicting milk composition and cows’ dietary regimens. Furthermore, the developed inferential routine enhances the interpretability of the results, providing valuable insights leading to a deeper understanding of the relation between specific wavenumbers and milk characteristics.

Low-degree Hurwitz stacks in the Grothendieck ring

For $2 \leq d \leq 5$, we show that the class of the Hurwitz space of smooth degree $d$, genus $g$ covers of $\mathbb {P}^1$ stabilizes in the Grothendieck ring of stacks as $g \to \infty$, and we give a formula for the limit. We also verify this stabilization when one imposes ramification conditions on the covers, and obtain a particularly simple answer for this limit when one restricts to simply branched covers.

Fuel property modeling by high-speed gas chromatography coupled with partial least squares data analysis

Partial least squares (PLS) regression is a valuable chemometric tool for property prediction when coupled with gas chromatography (GC). Since the separation run time and stationary phase selection are crucial for effective PLS modeling, we study these GC parameters on the prediction of viscosity, density and hydrogen content for 50 aerospace fuels. Due to the diversity of compounds in the fuels (primarily alkanes, cycloalkanes, and aromatics), we explore both polar and non-polar stationary phase columns. The robustness for the PLS models was evaluated by their normalized root mean square error of cross-validation (NRMSECV). PLS models built for viscosity across 1-min, 3-min, 7-min, and 10-min time window (TW) high-speed GC separations produced nearly the same NRMSECV with the polar column data with an average (standard deviation) of 4.41 % (0.34 %) versus the non-polar column data of 4.69 % (0.15 %). In contrast, while the NRMSECV of density modeling with the polar column data varied more than the viscosity models, averaging 7.54 % (0.67 %), the non-polar column data produced a significantly higher average NRMSECV of 10.06 % (0.35 %). Similarly, for hydrogen content, the NRMSECV with the polar column data averaged 9.50 % (0.87 %), which was significantly lower than the NRMSECV with the non-polar column data averaging 12.10 % (0.88 %). We also investigated the impact of smoothing the GC data on the corresponding PLS models. By applying varying degrees of smoothing, we can effectively obtain similar chromatographic peak patterns in a shorter TW. For example, a 10-min smoothed chromatogram appears like the 1-min separation with no smoothing but resulted in nearly the same NRMSECV. Overall, the fast separation with a 1-min TW produced robust PLS models for viscosity with either stationary phase column, whereas for density and hydrogen content the polar stationary phase column produced superior PLS models, thus with proper stationary phase selection, a fast separation run time could be readily applied with optimal PLS property modeling results.

Approximation of the Hilbert transform on the half–line

The paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule.

Third Order Polynomial Bundle Adjustment For Optical Calibration Refinement

We propose a refinement and validity region expansion process of a 3rd order polynomial calibration based on bundle adjustment. The present methodology can be expanded to any model as long as the mapping is a smooth function. Both methods are tested with a tomographic Particle Tracking Velocimetry measurement whose results are compared in terms of the smoothness and length of the retrieved tracks. The refinement method shows a significant improvement in both the associated PDFs. Similarly, the expansion shows a corresponding behaviour to the original calibration even though a higher degree of smoothness of the tracks is achieved.

THE INFLUENCE OF SURFACE EDDY CURRENT PROBE SIGNAL SMOOTHING ON ASSESSING STRESS-CORROSION CRACKS DEPTH IN MAIN GAS PIPELINE

The paper presents the results on the influence of signals smoothing of absolute and differential surface eddy current probes (ECP) above an individual crack and a group of stress-corrosion cracks on the accuracy of estimating the depths of stress-corrosion cracks during eddy current flaw detection of a main gas pipeline (MGP). Mathematical models demonstrate that the degree of signal smoothing of the surface ECP over an individual crack is higher than the degree of smoothing of the ECP signal over a group of stress-corrosion cracks in the MGP. Smoothing results in a decreased resolution of the surface ECP signal. The differential surface ECP is characterized by a higher resolution than the similar absolute surface ECP. The study demonstrates the reduction in amplitude difference between the actual signals of the absolute surface ECP over an individual crack and a group of stress-corrosion cracks, compared to the smoothed signal of the surface ECP. In the estimates of depths in the group of stress-corrosion cracks in MGP, obtained from the amplitude of the actual signal of the absolute surface ECP, where smoothing of the ECP signal is excluded, a significant reduction in the relative error is observed.

Facile, inexpensive, and reliable morphological characterization of microplastics using Optical Microscopy images

A morphological classification method was developed and applied to model contaminants of polyethylene microplastics (MPs) obtained in the laboratory by simple procedures. Primary MPs were produced by pump-assisted aqueous isolation from a commercial cosmetic cream, and secondary MPs were obtained by mechanical abrasion from a domestic plastic can. The MPs were investigated by optical microscopy and their images were analyzed according to forms and types of edge by means of a self-written Python-based script. Four simple geometrical forms were employed to classify the MPs, i.e., fiber, ellipse, triangle, and quadrilateral shapes, while their edges were defined as plain, wavy, lobed, and filiform, quantifying each type of edge on every MP. In addition, the numerical results provided by the computational analysis were employed to develop two complimentary indexes, R and I, associated to the degree of surface roughness and smoothness, ranging from 0 to 1 for the quantitative morphological characterization of the regularity and irregularity, respectively, of the particles. Thus, an inexpensive and simple but rigorous detailed physical characterization for MPs is provided, which can be particularly helpful for monitoring the changes that these particles may undergo in the environment.

Improvement of Fresnel Diffraction Convolution Algorithm

With the development of digital holography, the accuracy requirements for the reconstruction phase are becoming increasingly high. The transfer function of the double fast transform (D-FFT) algorithm is distorted when the diffraction distance is larger than the criterion distance dt, which reduces the accuracy of solving the phase. In this paper, the Fresnel diffraction integration algorithm is improved by using the low-pass Tukey window to obtain more accurate reconstructed phases. The improved algorithm is called the D-FFT (Tukey) algorithm. The D-FFT (Tukey) algorithm adjusts the degree of edge smoothing of the Tukey window, using the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM) to remove the ringing effect and obtain a more accurate reconstructed phase. In a simulation of USAF1951, the longitudinal resolution of the reconstructed phase obtained by D-FFT (Tukey) reached 1.5 μm, which was lower than the 3 μm obtained by the T-FFT algorithm. The results of Fresnel holography experiments on lung cancer cell slices also demonstrated that the phase quality obtained by the D-FFT (Tukey) algorithm was superior to that of the T-FFT algorithm. D-FFT (Tukey) algorithm has potential applications in phase correction, structured illumination digital holographic microscopy, and microscopic digital holography.

Robust adversarial learning model to segment non-speckle regions in blood flow echo

In our previous study, we analyzed the contrast of blood flow echo, and non-speckle regions were more frequently detected in the porcine blood with the high flow velocity. However, this contrast method is dependent on the degree of smoothing and threshold for outliers. This study developed a new U-Net model incorporating domain adaptation with both in silico and experimental data. This model segments blood flow echo into speckle and non-speckle regions. The performance of the developed U-Net model with several conditions of scatterer number density from 0.1 to 1.5 scatterers mm−3 and scatterer amplitude from 2 to 50 times against the speckle component was assessed using in silico data and experimental data with blood-mimicking fluid. The results indicated that the developed U-Net model with adversarial learning could stably detect non-speckle regions compared to the model without the adversarial learning and the contrast analysis method, in both in silico and experimental data.

How many neurons do we need? A refined analysis for shallow networks trained with gradient descent

We analyze the generalization properties of two-layer neural networks in the neural tangent kernel (NTK) regime, trained with gradient descent (GD). For early stopped GD we derive fast rates of convergence that are known to be minimax optimal in the framework of non-parametric regression in reproducing kernel Hilbert spaces. On our way, we precisely keep track of the number of hidden neurons required for generalization and improve over existing results. We further show that the weights during training remain in a vicinity around initialization, the radius being dependent on structural assumptions such as degree of smoothness of the regression function and eigenvalue decay of the integral operator associated to the NTK.

On Conic Equations Under Bernstein Operators

One of the most important problems in approximation theory in mathematical analysis is the determination of sequences of polynomials that converge to functions and have the same geometric properties. This type of approximation is called the shape-preserving approximation. These types of problems are usually handled depending on the convexity of the functions, the degree of smoothness depending on the order of differentiability, or whether it satisfies a functional equation. The problem addressed in this paper belongs to the third class. A quadratic bivariate algebraic equation denotes geometrically some well-known shapes such as circles, ellipses, hyperbolas and parabolas. Such equations are known as conic equations. In this study, it is investigated whether conic equations transform into a conic equation under bivariate Bernstein polynomials, and if so, which conic equation it transforms into.

A flexible modulated pesticide release platform through poly(urethane-urea) microcapsules: effect of different crosslinkers compositions.

Polymeric microcapsules (MCs) have become an important issue and have attracted increasing attention because of their tunable physical and chemical properties. Diverse shell structures can confer multiple properties on MCs. Different polyols (1,4-butanediol and glycerin) and polyamines (triethylenetetramine and isophorondiamine) were selected as crosslinkers to obtain emamectin benzoate (EB)-loaded poly(urethane-urea) MCs (PU-MCs) by interfacial polymerization. The four obtained PU-MCs showed sphericity with different degrees of smoothness on their surfaces, and displayed a uniform size distribution ranging from 500 to 700 nm. Moreover, transmission electron microscopy showed that the shell thickness was roughly uniform, and was greatly influenced by the type and structure of the crosslinker. GI-MCs, prepared using glycerin and isophorondiamine, had the largest shell thickness. GT-MCs, obtained using glycerin and triethylenetetramine, had the highest encapsulation efficiency and drug-loading content, and BT-MCs, obtained using mixtures of 1,4-butanediol and triethylenetetramine, had the fastest release behavior. Thermogravimetric analysis revealed that the greater the degree of shell crosslinking, the higher decomposition temperature and the greater the thermal stability. A BT-MC suspension had the lowest viscosity and contact angle with the best wettability. Bioassay experiments showed that BT-MCs exhibited good insecticidal activity against Plutella xylostella larvae with a half-maximal lethal concentration of 4.19 mg/L. Furthermore, a BT-MC suspension showed good thermal and light stability, with potential applications in minimizing the toxicity of EB through sustained release. Various properties of EB-loaded PU-MCs were modulated through simple selection of different polyols and polyamines during fabrication, which might have an important role in constructing the pesticide delivery system and improving pesticide utilization. © 2024 Society of Chemical Industry.

Analysis of Physical and Mechanical Properties of Universal Composites under Different Types of Polishing before and after Acid Challenge.

This study aimed to evaluate in vitro the degree of surface smoothness provided by two different polishing techniques and the effect of acid challenge on the alteration of surface roughness (Ra), microhardness (Knoop), and color (ΔE00) of three nanoparticulate composites, simulating 1 year of exposure to hydrochloric acid (HCl). Eighty specimens for each composite were divided into four groups (n = 240), being control without polishing, control with wear, WPC (wear + polishing with Cosmedent Kit), and WPB (wear + BisCover LV liquid polish). Repeated measures ANOVA was applied for Ra and Knoop Microhardness. For the color (ΔE) three-way ANOVA was applied. In cases of statistically significant the Tukey posttest was applied (α = 0.05). Both types of polishing tested resulted in a surface smoothness below the critical value established by the studies (Ra ≥ 0.2 μm), even after immersion. The microhardness of all composite resins decreased after the challenges. The specimens immersed in HCl showed a lower microhardness (42.2 Kgf/mm2) when compared to the specimens immersed in artificial saliva (44.7 Kgf/mm2). Regarding the color change, the composites presented values compatible with clinical acceptability, with a statistically significant difference only between the control group and the other types of polishing for the Z350 XT resin (ΔE00 = 3.78). It was concluded that both mechanical and chemical polishing produced a satisfactory surface smoothness, even after immersions in artificial saliva and HCl. The microhardness of the composites was affected by the challenges and the composites tested were within clinical acceptability with regard to color change.

Existence and regularity of steady-state solutions of the Navier-Stokes equations arising from irregular data

We analyze the forced incompressible stationary Navier-Stokes flow in R+n, n>2. Existence of a unique solution satisfying a global integrability property measured in a scale of tent spaces is established for small data in homogeneous Sobolev space with s=−12 degree of smoothness. The velocity field is shown to be locally Hölder continuous while the pressure belongs to Llocp for any p∈(1,∞). Our approach is based on the analysis of the inhomogeneous Stokes system for which we derive a new solvability result involving Dirichlet data in Triebel-Lizorkin classes with negative amount of smoothness and is of independent interest.