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Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial derivatives that allow for explosive behavior near the boundaries, the almost sure rate in Stute's representation of the empirical copula process can be recovered. The conditions are verified, for instance, in the case of the Gaussian copula with full-rank correlation matrix, many Archimedean copulas, and many extreme-value copulas.

In the setting of nonparametric multivariate regression with unknown error variance σ², we study asymptotic properties of a Bayesian method for estimating a regression function f and its mixed partial derivatives. We use a random series of tensor product of B-splines with normal basis coefficients as a prior for f, and σ is either estimated using the empirical Bayes approach or is endowed with a suitable prior in a hierarchical Bayes approach. We establish pointwise, L₂ and L∞-posterior contraction rates for f and its mixed partial derivatives, and show that they coincide with the minimax rates. Our results cover even the anisotropic situation, where the true regression function may have different smoothness in different directions. Using the convergence bounds, we show that pointwise, L₂ and L∞-credible sets for f and its mixed partial derivatives have guaranteed frequentisi coverage with optimal size. New results on tensor products of B-splines are also obtained in the course.

We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.

In this paper, we define the β-partial derivative as well as the β-directional derivative of a multi-variable function based on the β-difference operator, Dβ, which is defined by Dβf(t)=f(β(t))−f(t)/β(t)−t, where β is a strictly increasing continuous function. Some properties are proved. Furthermore, the β-gradient vector and the β-gradient directional derivative of a multi-variable function are introduced. Finally, we deduce the Hahn-partial and the Hahn-directional derivatives associated with the Hahn difference operator.

Recently, the conformable derivative and its properties have been introduced. In this work we have investigated in more detail some new properties of this derivative and we have proved some useful related theorems. Also, some new definitions have been introduced.

We consider smooth, infinitely divisible random fields (X(t), t ∈ M), M ⊂ ℝ d , with regularly varying Lévy measure, and are interested in the geometric characteristics of the excursion sets A u = {t ∈ M : X(t) > u} over high levels u. For a large class of such random fields, we compute the u → ∞ asymptotic joint distribution of the numbers of critical points, of various types, of X in A u , conditional on A u being nonempty. This allows us, for example, to obtain the asymptotic conditional distribution of the Euler characteristic of the excursion set. In a significant departure from the Gaussian situation, the high level excursion sets for these random fields can have quite a complicated geometry. Whereas in the Gaussian case nonempty excursion sets are, with high probability, roughly ellipsoidal, in the more general infinitely divisible setting almost any shape is possible.

We propose the use of Kernel Regularized Least Squares (KRLS) for social science modeling and inference problems. KRLS borrows from machine learning methods designed to solve regression and classification problems without relying on linearity or additivity assumptions. The method constructs a flexible hypothesis space that uses kernels as radial basis functions and finds the best-fitting surface in this space by minimizing a complexity-penalized least squares problem. We argue that the method is well-suited for social science inquiry because it avoids strong parametric assumptions, yet allows interpretation in ways analogous to generalized linear models while also permitting more complex interpretation to examine nonlinearities, interactions, and heterogeneous effects. We also extend the method in several directions to make it more effective for social inquiry, by (1) deriving estimators for the pointwise marginal effects and their variances, (2) establishing unbiasedness, consistency, and asymptotic normality of the KRLS estimator under fairly general conditions, (3) proposing a simple automated rule for choosing the kernel bandwidth, and (4) providing companion software. We illustrate the use of the method through simulations and empirical examples.

A theorem of Stepanoff claims that approximate differentiability almost everywhere of a function u is equivalent to existence almost everywhere of approximate partial derivatives of the function, while Whitney proved that approximate differentiability almost everywhere of u is equivalent to the following Lusin-type property: (*) Given ε > 0, there is a C1 function ν on ℝn such that |{x ∈ D : u(x) ≠ ν(x)}| < ε. Federer then established that (*) is equivalent to having u be approximately locally Lipschitz almost everywhere in the sense that $\\underset{\\mathrm{y}\\to \\mathrm{x}}{\\mathrm{a}\\mathrm{p} \\ \\mathrm{lim} \\ \\mathrm{sup}}\\frac{\\left|\\mathrm{u}\\left(\\mathrm{y}\\right)-\\mathrm{u}\\left(\\mathrm{x}\\right)\\right|}{|\\mathrm{y}-\\mathrm{x}|}<\\mathrm{\\infty }$ holds almost everywhere. This paper extends these results to the case of approximate differentiability of general order ɤ which is not necessarily an integer.

Constructing faradaic electrode with superior desalination performance is important for expanding the applications of capacitive deionization (CDI). Herein, a simple one‐step alkalized treatment for in situ synthesis of 1D TiO2 nanowires on the surface of 2D Ti3C2 nanosheets, forming a Ti3C2‐MXene partially derived hierarchical 1D/2D TiO2/Ti3C2 heterostructure as the cathode electrode is reported. Cross‐linked TiO2 nanowires on the surface help avoid layer stacking while acting as the protective layer against contact of internal Ti3C2 with dissolved oxygen in water. The inner Ti3C2 MXene nanosheets cross over the TiO2 nanowires can provide abundant active adsorption sites and short ion/electron diffusion pathways. . Density functional theory calculations demonstrated that Ti3C2 can consecutively inject electrons into TiO2, indicating the high electrochemical activity of the TiO2/Ti3C2. Benefiting from the 1D/2D hierarchical structure and synergistic effect of TiO2 and Ti3C2, TiO2/Ti3C2 heterostructure presents a favorable hybrid CDI performance, with a superior desalination capacity (75.62 mg g−1), fast salt adsorption rate (1.3 mg g−1 min−1), and satisfactory cycling stability, which is better than that of most published MXene‐based electrodes. This study provides a feasible partial derivative strategy for construction of a hierarchical 1D/2D heterostructure to overcome the restrictions of 2D MXene nanosheets in CDI. Cross‐linked 1D TiO2 nanowires on the surface of 2D Ti3C2 MXene nanosheets, which contribute to avoiding layer stacking, simultaneously act as the protective layer to prevent access to the dissolved oxygen in water and hinder the oxidation of Ti3C2 in the interior. TiO2/Ti3C2 heterostructure presents a favorable hybrid capacitive deionization performance, with a superior desalination capacity, fast salt adsorption rate, and satisfactory cycling stability.

We establish minimax lower bounds and maximum likelihood convergence rates of parameter estimation for mean-covariance multivariate Gaussian mixtures, shape-rate Gamma mixtures and some variants of finite mixture models, including the setting where the number of mixing components is bounded but unknown. These models belong to what we call \"weakly identifiable\" classes, which exhibit specific interactions among mixing parameters driven by the algebraic structures of the class of kernel densities and their partial derivatives. Accordingly, both the minimax bounds and the maximum likelihood parameter estimation rates in these models, obtained under some compactness conditions on the parameter space, are shown to be typically much slower than the usual n-½ or n-1/4 rates of convergence.