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We study a non-parametric multi-armed bandit problem with stochastic covariates, where a key complexity driver is the smoothness of payoff functions with respect to covariates. Previous studies have focused on deriving minimax-optimal algorithms in cases where it is a priori known how smooth the payoff functions are. In practice, however, the smoothness of payoff functions is typically not known in advance, and misspecification of smoothness may severely deteriorate the performance of existing methods. In this work, we consider a framework where the smoothness of payoff functions is not known, and study when and how algorithms may adapt to unknown smoothness. First, we establish that designing algorithms that adapt to unknown smoothness of payoff functions is, in general, impossible. However, under a self-similarity condition (which does not reduce the minimax complexity of the dynamic optimization problem at hand), we establish that adapting to unknown smoothness is possible, and further devise a general policy for achieving smoothness-adaptive performance. Our policy infers the smoothness of payoffs throughout the decision-making process, while leveraging the structure of off-the-shelf non-adaptive policies. We establish that for problem settings with either differentiable or non-differentiable payoff functions, this policy matches (up to a logarithmic scale) the regret rate that is achievable when the smoothness of payoffs is known a priori.

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.

When handling occlusion in unsupervised stereo matching, existing methods tend to neglect the supportive role of occlusion and to perform inappropriate disparity smoothing around the occlusion. To address these problems, we propose an occlusion-aware stereo network that contains a specific module to first estimate occlusion as an additional depth cue. In the occlusion inference module, a pixel is classified with a three-category label based on whether an area is occluded by an object on the left, occluded by an object on the right, or unoccluded. After the occluders are detected, we introduce a directed disparity smoothing loss that allows valid disparity estimates to be propagated to fill the occluded region, while ambiguous matches in the occluded region do not affect other regions. Disparity and occlusion are trained alternately in an unsupervised manner with detached backpropagation to enable the directed smoothness. Experiments show that our method achieves 3-pixel threshold error rates of 6.51% and 5.69% on the KITTI 2015 and KITTI 2012 validation sets, state-of-the-art results among unsupervised learning networks at the time of submission.

Single image haze removal, which is to recover the clear version of a hazy image, is a challenging trask in computer vision. In this paper, an additive haze model is proposed to approximate the hazy image formation process. In contrast with the traditional optical model, it regards the haze as an additive layer to a clean image. The model thus avoids estimating the medium transmission rate and the global atmospherical light. In addition, based on a critical observation that haze changes gradually and smoothly accross the image, a haze smoothness prior is proposed to constrain this model. This prior assumes that the haze layer is much smoother than the clear image. Benefiting from this prior, we can directly separate the clean image from a single hazy image. Experimental results and comparisons with synthetic images and real-world images demonstrate that the proposed method outperforms state-of-the-art single image haze removal algorithms.

The edge information plays a key role in the restoration of a depth map. Most conventional methods assume that the color image and depth map are consistent in edge areas. However, complex texture regions in the color image do not match exactly with edges in the depth map. In this paper, firstly, we point out that in most cases the consistency between normal map and depth map is much higher than that between RGB-D pairs. Then we propose a dual normal-depth regularization term to guide the restoration of depth map, which constrains the edge consistency between normal map and depth map back and forth. Moreover, considering the bimodal characteristic of weight distribution that exists in depth discontinuous areas, a reweighted graph Laplacian regularizer is proposed to promote this bimodal characteristic. And this regularization is incorporated into a unified optimization framework to effectively protect the piece-wise smoothness(PWS) characteristics of depth map. By treating depth image as graph signal, the weight between two nodes is adapted according to its content. The proposed method is tested for both noise-free and noisy cases, and is compared against the state-of-the-art methods on both synthesis and real captured datasets. Extensive experimental results demonstrate the superior performance of our method compared with most state-of-the-art works in terms of both objective and subjective quality evaluations. Specifically, our method is more effective on edge areas and more robust to noises.

To compare the effects of single- versus dual-task balance training on the gait smoothness and balance of community-dwelling older adults, 69 volunteers were randomized to single-, dual-task training, and control (no intervention) groups. Exercises were received in 18 sessions through 6 weeks. The gait smoothness was measured by the harmonic ratio of trunk accelerations using a triaxial accelerometer. Balance performance was assessed through the Fullerton Advanced Balance scale, Timed Up and Go test, Activities-specific Balance Confidence, and gait speed. After the trial, all variables improved significantly in the training groups. Moreover, differences in the mean change of all variables, except the Timed Up and Go test, were statistically significant between the interventional groups and the control group, but no significant difference was reported between the two training groups. This study suggests that balance training can improve gait smoothness as well as balance status in healthy older adults.

Empirical distributions in a range of fields are often substantially non-Gaussian but smooth enough to suggest that the underlying population distribution has at least several derivatives. Linear combinations of many random variables often have smooth densities even if the random variables are not independent. The main result here generalizes the elementary result that a convolution of densities has as many derivatives as the sum of the number of derivatives of each component to certain nonlinear functions of random vectors whose components may not be independent. Linearity is replaced by an assumption that the nonlinear function has positive first partial derivatives bounded away from 0 along at least some coordinates and independence is replaced by an assumption that the joint density has certain mixed partial derivatives. This approach to justifying the smoothness of empirical distributions is contrasted with other possible approaches, including solutions of stochastic partial differential equations and ergodic distributions for chaotic dynamical systems.

Hyperspectral images provide plentiful spectral-spatial information regarding the nature of different materials, leading to the potential of more efficient detection in diverse areas. However, the high volume of spectral bands, along with limited reference data, leads to many challenges. To alleviate these issues, we propose a superpixel-correlation-based multiview classification approach. Here, the spectral-spatial multiple views are generated via multiscale superpixel segmentation and correlation-based spectral band clustering. A random forest classifier in conjunction with Markov random field regularization is used as the backend classifier of each view. In particular, an energy function with improved metrics of smoothness is introduced. For decision fusion, the pixelwise weight maps of the views are generated based on both classification certainty and neighboring smoothness. The proposed approach is evaluated on three widely used hyperspectral data sets, and the experimental results demonstrate that the proposed method can achieve a competitive performance compared with other existing methods.

We investigate distribution of eigenvalues of growing size Toeplitz matrices [ a n + k − j ] 1 ≤ j , k ≤ n as n → ∞ , when the entries { a j } are “smooth” in the sense, for example, that for some α > 0 , a j − 1 a j + 1 a j 2 = 1 − 1 α j ( 1 + o ( 1 ) ) , j → ∞ . Typically they are Maclaurin series coefficients of an entire function. We establish that when suitably scaled, the eigenvalue counting measures have limiting support on [ 0 , 1 ] , and under mild additional smoothness conditions, the universal scaled and weighted limit distribution is | π log ⁡ t | − 1 / 2 d t on [ 0 , 1 ] .

The stability and smoothness of rolling stock running could be defined accurately by universal Sperling's comfort index. The divergences of variation of Sperling's comfort index of a passenger car under specific operating conditions of running gear are examining in this paper. Numerical simulations of a passenger car running with independently rotating wheels under various conditions have been performing. Gained results showed that divergences of the Sperling's comfort index variation are particularly significant due to running gear component oscillations in the horizontal plane (lateral direction). A field experiment of a passenger car with a solid (traditional) wheelset with a flat running surface proved this hypothesis. The obtained results of this experiment confirmed this assumption. Therefore, the study of the regularities of lateral oscillations of a passenger car is the logical direction of further research.

This research aims to use the (Whittaker - Henderson) method, which is an (Non parametric Method) for graduate, the crude death rates to obtain the smoothness and fitness death rates, because it is one of the most common methods for smoothing the crude death rates, for crude mortality rates. It has the capacity to explicitly balance the smoothness as well as the fitness of the smoothed curve relative to the crude rates through its parameters by minimizing a difference equation. Non-parametric methods depend only on the nature of the observed data. The crude central death rates for the single ages were smoothed and fitted using Whittaker-Henderson graduation technique by four assumptions through the program (R) and the third hypothesis was the best assumption of graduated for both males and females. The study recommended the important of graduation for the crude death rates and their smoothing to rid them of the irregularities resulting from the randomness in the 2 exposure and the number of deaths in order to facilitate dealing with these rates to complete the process of establishing death tables.

3D mesh subdivision is essential for geometry modeling of complex surfaces, which benefits many important applications in the fields of multimedia such as computer animation. However, in the ordinary adaptive subdivision, with the deepening of the subdivision level, the benefits gained from the improvement of smoothness cannot keep pace with the cost caused by the incremental number of faces. To mitigate the gap between the smoothness and the number of faces, this paper devises a novel improved mesh subdivision method to coordinate the smoothness and the number of faces in a harmonious way. First, this paper introduces a variable threshold, rather than a constant threshold used in existing adaptive subdivision methods, to reduce the number of redundant faces while keeping the smoothness in each subdivision iteration. Second, to achieve the above goal, a new crack-solving method is developed to remove the cracks by refining the adjacent faces of the subdivided area. Third, as a result, the problem of coordinating the smoothness and the number of faces can be formulated as a multi-objective optimization problem, in which the possible threshold sequences constitute the solution space. Finally, the Non-dominated sorting genetic algorithm II (NSGA-II) is improved to efficiently search the Pareto frontier. Extensive experiments demonstrate that the proposed method consistently outperforms existing mesh subdivision methods in different settings.

We consider the second boundary-value problem for a second-order differential-difference equation with variable coefficients on the interval (0,d). We investigate the existence of a generalized solution and obtain conditions on the right-hand side of the equation which ensure the smoothness of generalized solutions on the entire interval (0,d).

In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl equations with low regularity (see Oleinik and Samokhin [9] , P.21, Theorem 2.1.1). Due to the degeneracy of the equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5] . In this paper, we prove that Oleinik's solutions are smooth up to the boundary y = 0 for any x > 0 , using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x = 0 , our result implies instant smoothness (in the steady case, x = 0 is often considered as initial time).

The aim of this paper is to investigate a parabolic-elliptic system, which was proposed by Albi et al (2016) to describe the evolution of ion transport networks. Our first result asserts that the corresponding Neumann-Dirichlet initial-boundary value problem possesses a global weak solution for general large data in the two-dimensional setting. Our second result further reveals that such solution at least eventually becomes bounded and smooth, and approaches spatial equilibria in the large time limit.

This paper introduces an ensemble-based field inversion framework to augment the turbulence models by incorporating prior physical knowledge. Different types of prior knowledge such as smoothness, ...

With the increasing load and speed of trains, the problems caused by various random excitations (such as safety and passenger comfort) have become more prominent and thus arises the necessity to analyze stochastic dynamical systems, which is important in both academic and engineering circles. The existing analysis methods are inadequate in terms of computational accuracy, computational efficiency, and applicability in solving complex problems. For that, a new efficient and accurate method is used in this paper, suitable for linear and nonlinear random vibration analysis of large structures as well as static and dynamic reliability assessment. It is the direct probability integration method, which is extended and applied to the random vibration reliability analysis of dynamical systems. Dynamical models of the dynamic system and coupled system “three-car vehicle-rail-bridge” are established, the time-varying differential equations of motion are derived in detail, and the dynamic response of the system is calculated using the explicit Newmark algorithm. The simulation results show the influence of the number of representative points on the smoothness of the image of the probability density function and the accuracy of the calculation results.

A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we investigate in new detail the residuals in Tikhonov regularization viewed as functions of the regularization parameter. We show that the residual carries, with some restrictions, the information on both the unknown solution and the noise level. By calculating approximate solutions for a large range of regularization parameters, we can extract both parameters from the residual given only one set of noisy data and the forward operator. The smoothness in the residual allows to revisit parameter choice rules and relate a-priori, a-posteriori, and heuristic rules in a novel way that blurs the lines between the classical division of the parameter choice rules. All results are accompanied by numerical experiments.

It can be challenging to pick high-quality first arrivals on noisy seismic data sets. The stability and smoothness criteria of the picked first arrival are not satisfied for data sets with shingles and interference from unexpected and backscattered events. To improve first-arrival picking, we have adopted an automatic first-arrival picking workflow using global path tracing (GPT) to find a global solution for first-arrival picking with the condition of smoothness of the traced path. Our methodology is composed of data preconditioning, GPT, and a final addition of traced and piloted traveltimes to compute the total picked traveltime. We propose several ways to precondition the data set, including the use of the amplitude and amplitude ratio with and without a pilot. A 2D GPT is composed of two steps, namely, accumulation of energy on the potential path and backtracking of the optimal path with a strain factor for smoothness. For higher dimensional data sets, two strategies were adopted. One was to split the higher dimension data into subdomains of two dimensions to which 2D GPT was applied. The alternative method was to smooth the preconditioned data set in directions except for the one used to trace the path before applying 2D GPT. We discuss the importance of choosing the proper parameters for data preconditioning and for constraining GPT. We demonstrate the robustness and stability of our automatic first-arrival picking via GPT using synthetic and field data examples.

It is known, in harmonic analysis theory, that maximal operators measure local smoothness of L p functions. These operators are used to study many important problems of function theory such as the embedding theorems of Sobolev type and description of Sobolev space in terms of the metric and measure. We study the Sobolev-type embedding results on weighted Besov–Triebel–Lizorkin spaces via the sharp maximal functions. The purpose of this paper is to study the extent of smoothness on weighted function spaces under the condition M α # f ∈ L p , μ , where μ is a lower doubling measure, M α # f stands for the sharp maximal function of f , and 0 ≤ α ≤ 1 is the degree of smoothness.

Although there is an increase in the amount of seismic data acquired with wide-azimuth geometry, it is difficult to achieve regular data distributions in spatial directions owing to limitations imposed by the surface environment and economic factors. To address this issue, interpolation is an economical solution. The current state-of-the-art methods for seismic data interpolation are iterative methods. However, iterative methods tend to incur high computational costs, which restricts their application in cases of large, high-dimensional data sets. Hence, we have developed a two-step noniterative method to interpolate nonstationary seismic data based on streaming prediction filters (SPFs) with varying smoothness in the time-space domain and we extend these filters to two spatial dimensions. Streaming computation, which is the kernel of the method, directly calculates the coefficients of nonstationary SPF in the overdetermined equation with local smoothness constraints. In addition to the traditional streaming prediction-error filter, we adopt a similarity matrix to improve the constraint condition in which the smoothness characteristics of the adjacent filter coefficient change with the varying data. We also design noncausal-in-space filters for interpolation by using several neighboring traces around the target traces to predict the signal; this is performed to obtain more accurate interpolated results than those from the causal-in-space version. Compared to the Fourier projection onto a convex sets interpolation method, our method has the advantages of fast computational speed and nonstationary event reconstruction. Application of our method on synthetic and nonstationary field data finds that it can successfully interpolate high-dimensional data with low computational cost and reasonable accuracy even in the presence of aliased and conflicting events.

In this paper, we summarize results on convergence rates of various kernel based non- and semiparametric estimators, focusing on the impact of insufficient distributional smoothness, possibly unknown smoothness and even non-existence of density. In the presence of a possible lack of smoothness and the uncertainty about smoothness, methods of safeguarding against this uncertainty are surveyed with emphasis on nonconvex model averaging. This approach can be implemented via a combined estimator that selects weights based on minimizing the asymptotic mean squared error. In order to evaluate the finite sample performance of these and similar estimators we argue that it is important to account for possible lack of smoothness.

Trimap generation is currently playing an important role in the field of image matting. The current trimap generation methods cannot adaptively adjust the classification thresholds, and ignore the nonlocal correlations between pixels. Besides, they also need finely provided trimaps. This paper raises a dual-level trimap generation approach which requires sparse user inputs. The first level introduces a unary data term from adaptive learned thresholds combined with local spatial constraints to effectively overcome the drawback of fixed threshold methods. The second level introduces a smoothness term from a nonlocal affinity for foreground probability estimation , and a pixel correlation constraint is computed and added to the data term. The foreground probability is finally solved in a closed form solution and truncated to generate the final trimap. In the experiments, firstly, this approach raises a novel trimap evaluation criterion according to the two classic terms of missing and false classifications. Based on this criterion, the final trimaps generated by our approach are better than those generated by other methods. Secondly, in the final matting process from four typical matting algorithms, the matting results from the trimaps by our approach are still better than those from the trimaps by other trimap generation methods.

In this paper, we generalize the typical geometric constants of Banach spaces to modular spaces. We study the equivalence between the convexity of modular and normed spaces, and obtain the relationship between ρ-Neumann-Jordan constant and ρ-James constant. In particular, we extend the convexity and smoothness modular, and obtain the criterion theorems of the uniform convexity and strict convexity.

We study the smoothness of a solution to the Cauchy problem for a one-dimensional parabolic equation with Dini–continuous coefficients in the Dini space. We obtain estimates for the volume parabolic potential and its spatial first order derivative in the Dini space.

Under a suitable bunching condition, we establish that stable holonomies inside center-stable manifolds for $C^{1+\beta }$ diffeomorphisms are uniformly bi-Lipschitz and, in fact, $C^{1+\mathrm {H}\ddot{\rm o}\mathrm {lder}}$ . This verifies the ergodicity of suitably center-bunched, essentially accessible, partially hyperbolic $C^{1+\beta }$ diffeomorphisms and verifies that the Ledrappier–Young entropy formula holds for $C^{1+\beta }$ diffeomorphisms of compact manifolds.

Smoother surfaces after laser vision correction have been widely accepted as a factor for improving visual recovery regardless of the used technique (PRK, LASIK, or even SMILE). We tested the impact of laser beam truncation, dithering (expressing a continuous profile on a basis of lower resolution causing pixels to round up/down the number of pulses to be placed), and jitter (a controlled random noise (up to ±20 µm in either direction) added to the theoretical scanner positions) on residual smoothness after Poly(methyl methacrylate) (PMMA) ablations, using a close-to-Gaussian beam profile. A modified SCHWIND AMARIS system has been used providing a beam profile with the following characteristics: close-to-Gaussian beam profile with full width at half maximum (FWHM) of 540 µm, 1050 Hz. Laser parameters have been optimized following Invest. Ophthalmol. Vis. Sci., vol. 58, no. 4, pp. 2021–2037, 2017, the pulse energy has been optimized following Biomed. Opt. Express vol. 4, pp. 1422–1433, 2013. For the PMMA ablations, two configurations (with a 0.7 mm pinhole and 0.75 mJ and without pinhole and 0.9 mJ (for fluences of 329 mJ/cm2 and 317 mJ/cm2 and corneal spot volumes of 174 and 188 pl)) were considered, along with two types of lattices (with and without ordered dithering to select the optimum pulse positions), and two types of spot placement (with and without jitter). Real ablations on PMMA (ranging from −12D to +6D with and without astigmatism of up to 3D) completed the study setup. The effect of the 2 × 2 × 2 different configurations was analyzed based on the roughness in ablation estimated from the root mean square error in ablation. Truncation of the beam is negatively associated to a higher level of residual roughness; ordered dithering to select the optimum pulse positions is positively associated to a lower level of residual roughness; jitter is negatively associated to a higher level of residual roughness. The effect of dithering was the largest, followed by truncation, and jitter had the lowest impact on results. So that: Dithering approaches help to further minimize residual roughness after ablation; minimum (or no) truncation of the beam is essential to minimize residual roughness after ablation; and jitter shall be avoided to minimize residual roughness after ablation. The proposed model can be used for optimization of laser systems used for ablation processes at relatively low cost and would directly improve the quality of results. Minimum (or no) truncation of the beam is essential to minimize residual roughness after ablation. Ordered dithering without jitter helps to further minimize residual roughness after ablation. Other more complex dithering approaches may further contribute to minimize residual roughness after ablation.

Nonnegative matrix factorization methods have been widely used in many applications in recent years. However, the clustering performances of such methods may deteriorate dramatically in the presence of non-Gaussian noise or outliers. To overcome this problem, in this paper, we propose correntropy-based dual graph regularized NMF with LP smoothness (CDNMFS) for data representation. Specifically, we employ correntropy instead of the Euclidean norm to measure the incurred reconstruction error. Furthermore, we explore the geometric structures of both the input data and the feature space and impose an Lp norm constraint to obtain an accurate solution. In addition, we introduce an efficient optimization scheme for the proposed model and present its convergence analysis. Experimental results on several image datasets demonstrate the superiority of the proposed CDNMFS method.

The article deals with some approaches to solving convex problems of the min-min type with smoothness and strong convexity in only one of the two groups of variables. It is shown that the proposed approaches based on Vaidya’s method, the fast gradient method, and the accelerated gradient method with variance reduction have linear convergence. It is proposed to use Vaidya’s method to solve the exterior problem and the fast gradient method to solve the interior (smooth and strongly convex) one. Due to its importance for applications in machine learning, the case where the objective function is the sum of a large number of functions is considered separately. In this case, the accelerated gradient method with variance reduction is used instead of the fast gradient method. The results of numerical experiments are presented that illustrate the advantages of the proposed procedures for a logistic regression problem in which the a priori distribution for one of the two groups of variables is available.

The skewing method, which has been originally proposed as a bias correction device for local linear regression estimation using standard symmetric kernels, is extended to the cases of asymmetric kernels. The method is defined as a convex combination of three local linear estimators. It is demonstrated that the skewed estimator using asymmetric kernels with properly chosen weights can accelerate the bias convergence from O(b) to O(b2) as b → 0 under sufficient smoothness of the unknown regression curve while not inflating the variance in an order of magnitude, where b is the smoothing parameter and the regressor is assumed to have at least one boundary. As a consequence, the estimator has optimal pointwise convergence of n − 4 / 9 when best implemented, where n is the sample size. It is noteworthy that these properties are the same as those for a local cubic regression estimator. Finite-sample properties of the skewed estimator are assessed in comparison with local linear and local cubic estimators. An application of the skewed estimation to real data is also considered.

Exact constants are found in inequalities of the Jackson–Stechkin type for smoothness characteristics $\Lambda_{m}(f),\,m \in\mathbb{N}$ determined by averaging the norm in $L_{2}$ of the mth order finite differences of the functions f. For classes of functions defined by the smoothness characteristic $\Lambda_{m}(f)$ , and the majorants $\Phi$ satisfying a certain condition, the exact values of different n-widths are calculated.

We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for the approximation of functions from the H\"older-Zygmund space of mixed smoothness defined on the $d$-dimensional unit cube when the dimension $d$ may be very large. The approximation error is measured in the norm of isotropic Sobolev space. For every function $f$ from the H\"older-Zygmund space of mixed smoothness, we explicitly construct a deep ReLU neural network having an output that approximates $f$ with a prescribed accuracy $\varepsilon$, and prove tight dimension-dependent upper and lower bounds of the computation complexity of this approximation, characterized as the size and the depth of this deep ReLU neural network, explicitly in $d$ and $\varepsilon$. The proof of these results are in particular, relied on the approximation by sparse-grid sampling recovery based on the Faber series., Comment: 5 figures

Interior-penalized weak Galerkin (IPWG) finite element methods are proposed and analyzed for solving second order elliptic equations. The new methods employ the element $(\mathbb{P}_{k}, \mathbb{P}_{k}, \mathcal{RT}_{k})$, with dimensions of space $d = 2, 3$, and the optimal a priori error estimates in discrete $H^1$-norm and $L^2$-norm are established. Moreover, provided enough smoothness of the exact solution, superconvergence in $H^1$ and $L^2$ norms can be derived. Some numerical experiments are presented to demonstrate flexibility, effectiveness and reliability of the IPWG methods. In the experiments, the convergence rates of the IPWG methods are optimal in $L^2$-norm, while they are suboptimal for NIPG and IIPG if the polynomial degree is even.

We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are {\em sufficiently close} to critical points of a suitable non-local potential. We then consider the fractional perimeter in half-spaces. We prove the existence of a minimizer under fixed volume constraint, showing some of its properties such as smoothness and symmetry, being a graph in the $x_N$-direction, and characterizing its intersection with the hyperplane $\{x_N=0\}$., 22 pages, 3 figures

In this study, the optical flow method for the skin-friction-stress estimation from the unsteady oil film distribution was reformulated based on the variational method, and the validity of the proposed method was verified in comparison with the conventional method. The regularization is proposed to be added directly to the skin-friction-stress field in the proposed method while the regularization is added to the amount of oil movement in the conventional method. As a result, the smoothness of the skin-friction-stress field can be controlled by adjusting the regularization parameter in the proposed method whereas it was difficult in the conventional method. The performance of the proposed method was evaluated to be superior to the previous method through the numerical and experimental data.

We study a family of non-convex functionals $\{\mathcal{E}\}$ on the space of measurable functions$u: \Omega_1\times\Omega_2 \subset \mathbb{R}^{n_1}\times\mathbb{R}^{n_2} \to \mathbb{R}$. These functionals vanish on the non-convex subset $S(\Omega_1\times\Omega_2)$ formed by functions of the form $u(x_1,x_2)=u_1(x_1)$ or $u(x_1,x_2)=u_2(x_2)$. We investigate under which conditions the converse implication "$\mathcal{E}(u) = 0 \Rightarrow u \in S(\Omega_1\times\Omega_2)$" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) $\mathcal{E}(u)$ controls in a strong sense the distance of $u$ to $S(\Omega_1\times\Omega_2)$.

The motion smoothness of 3C locking robot directly affects the machining performance. Improving the motion smoothness can optimize the motion trajectory and reduce the processing time. In this paper, a novel machining path optimization model including motion smoothness is built by employing the coordinate boundary of velocity and acceleration after evaluating the machining motion smoothness of the 3C locking robot. Secondly, based on the creation of the ant colony of adaptive function algorithm, the optimization model of the 3C locking robot in the situation of fixed bolt hole position and floating bolt hole position is resolved. Lastly, the proposed approach collects and analyses a huge amount of data to enable robots to make on-the-fly decisions in the middle of production, even when faced with unexpected circumstances. In the Spark distributed environment, we use the conventional K clustering technique to improve the final output utilizing clustering means. The results show that the machining path optimization of fixed hole considering the motion smoothness improves the smoothness but extends the machining path; the cooperative machining path optimization of multiregion floating bolt holes can significantly improve the motion smoothness and effectively reduce the length of the path. The research results provide theoretical support and design guidance for designers.

In a recent paper, Yu (2021) define a k=(k1,k2,…,kd)-th order derivative of self-intersection local times with respect to d-dimensional fractional Brownian motion. By the existence and Holder conti...

We propose wavelet-L1-estimation for non parametric location-scale models from multiple subjects. The advantage of the wavelet is to avoid the restrictive smoothness requirement for the location an...

Fractal interpolation is a modern technique to fit and analyze scientific data. We develop a new class of fractal interpolation functions which converge to a data generating (original) function for any choice of the scaling factors. Consequently, our method offers an alternative to the existing fractal interpolation functions (FIFs). We construct a sequence of α-FIFs using a suitable sequence of iterated function systems (IFSs). Without imposing any condition on the scaling vector, we establish constrained interpolation by using fractal functions. In particular, the constrained interpolation discussed herein includes a method to obtain fractal functions that preserve positivity inherent in the given data. The existence of $${{\cal C}^r} - \alpha - {\rm{FIFs}}$$ is investigated. We identify suitable conditions on the associated scaling factors so that α-FIFs preserve r-convexity in addition to the $${{\cal C}^r} - {\rm{smoothness}}$$ of original function.

The nondispersive per-sample channel model for the optical fiber channel is considered. Under certain smoothness assumptions, the problem of finding the minimum amount of noise energy that can render two different input points indistinguishable is formulated. This minimum noise energy is then taken as a measure of distance between the points in the input alphabet. Using the machinery of optimal control theory, necessary conditions that describe the minimum-energy noise trajectories are stated as a system of nonlinear differential equations. It is shown how to find the distance between two input points by solving this system of differential equations. The problem of designing signal constellations with the largest minimum distance subject to a peak power constraint is formulated as a clique-finding problem. As an example, a 16-point constellation is designed and compared with conventional quadrature amplitude modulation. A computationally efficient approximation for the proposed distance measure is provided. It is shown how to use this approximation to design large constellations with large minimum distances. Based on the control-theoretic viewpoint of this paper, a new decoding scheme for such nonlinear channels is proposed.

Exact constants in Jackson–Stechkin type inequalities are found for the smoothness characteristics $$\Lambda_m (f)$$ , $$ m\in\mathbb N$$ , determined by averaging the norm of finite differences of $$m$$ th order of functions $$ f \in L_2$$ . A solution is given of the extremal problem of finding the supremum for best joint polynomial approximations of functions and their successive derivatives on some classes of functions from $$L_2$$ whose averaged norms of finite differences are bounded above by $$1$$ .

Refinements of some relations between mixed moduli of smoothness of fractional orders in the metrics $$L_p$$ and $$L_\infty$$ are obtained.

Among the finitely generated groups of diffeomorphisms of the line and the circle, groups that act freely on the orbit of almost every point of the line (circle) are allocated. The paper is devoted to the study of the structure of the set of finitely generated groups of orientation-preserving diffeomorphisms of the line and the circle of C(1) smoothness with a given number of generators and the property noted above. It is shown that such a set contains a massive subset (contains a countable intersection of open everywhere dense subsets). Such a result for finitely generated groups of orientation-preserving diffeomorphisms of the circle, in the case of C(2) smoothness, was obtained by the author earlier.

In this article we introduce rather general notion of the stationary solution of the bistable equation which allows to treat discontinuous density dependent diffusion term with singularities and degenerations, as well as degenerate or non-Lipschitz balanced bistable reaction term. We prove the existence of new-type solutions which do not occur in case of the ``classical'' setting of the bistable equation. In the case of the power-type behavior of the diffusion and bistable reaction terms near the equilibria we provide detailed asymptotic analysis of the corresponding solutions and illustrate the lack of smoothness due to the discontinuous diffusion.

In this paper we present a bilinear estimate for commutators on Triebel–Lizorkin spaces with variable smoothness and integrability, and under no vanishing assumptions on the divergence of vector fields.

Let p ( ζ ) p(\zeta ) be a positive function defined on the unit circle T \mathbb {T} and satisfying the condition | p ( ζ 2 ) − p ( ζ 1 ) | ≤ c 0 log ⁡ e | ζ 2 − ζ 1 | , ζ 1 , ζ 2 ∈ T , \begin{equation*} |p(\zeta _2)-p(\zeta _1)|\le \frac {c_0}{\log \frac {e} {|\zeta _2-\zeta _1|}}, \quad \zeta _1,\zeta _2\in \mathbb {T}, \end{equation*} p − = min ζ ∈ T p ( ζ ) p_-=\min _{\zeta \in \mathbb {T}}p(\zeta ) . Futhermore, let 0 > α > 1 0>\alpha >1 , r ≥ 0 r\ge 0 , r ∈ Z r\in \mathbb {Z} , and assume that p − > 1 α p_->\frac {1}{\alpha } . Define a class of analytic functions in the unit disk D \mathbb {D} as follows: f ∈ H r + α p ( ⋅ ) f\in H^{p(\,\cdot \,)}_{r+\alpha } if sup 0 > ρ > 1 sup 0 > | θ | > π ∫ 0 2 π | f ( r ) ( ρ e i ( λ + θ ) ) − f ( r ) ( ρ e i λ ) | θ | α | p ( e i λ ) d λ > ∞ . \begin{equation*} \sup _{0>\rho >1}\,\sup _{0>|\theta |>\pi } \int ^{2\pi }_0 \bigg |\frac {f^{(r)}(\rho e^{i(\lambda +\theta )})-f^{(r)}(\rho e^{i\lambda })} {|\theta |^{\alpha }}\bigg |^{p(e^{i\lambda )}}\,d\lambda >\infty . \end{equation*} The following main results are proved. Theorem 1. Let f ∈ H r + α p ( ⋅ ) , f\in H^{p(\,\cdot \,)}_{r+\alpha }, and let I I be an inner function, f / I ∈ H 1 f/I\in H^1 . Then f / I ∈ H r + α p ( ⋅ ) f/I\in H^{p(\,\cdot \,)}_{r+\alpha } . Theorem 2. Let f ∈ H r + α p ( ⋅ ) , f\in H^{p(\,\cdot \,)}_{r+\alpha }, and let I I be an inner function, f / I ∈ H ∞ f/I\in H^{\infty } . Assume that the multiplicity of every zero of f f in D \mathbb {D} is at least r + 1 r+1 . Then f I ∈ H r + α p ( ⋅ ) fI\in H^{p(\,\cdot \,)}_{r+\alpha } .

A NiCr film is introduced in magnetic tunnel junction (MTJ) seed layer to improve the smoothness of multilayer and to enhance the perpendicular magnetic anisotropy (PMA) of the pinning layer. Espec...