Consider a multivariate function f that maps an open region of Rp into Rp, together with its inverse function. This paper presents a general method for obtaining the partial derivatives of the inverse function, of any order, expressed in terms of the partial derivatives of the inverse of the Jacobian matrix of f. The results are local. They require only that f have continuous partial derivatives up to the order of interest in a neighbourhood of a point, and a nonsingular Jacobian at that point, so global one-to-one-ness is not needed. A recursion for the derivatives of the inverse is proved by induction. Explicit formulas are derived for all partial derivatives up to the fourth order, both for the inverse function and for the inverse Jacobian. The symmetry of the mixed partial derivatives of the inverse is established. The computational cost is discussed: the explicit formulas grow factorially with the order, whereas a Leibniz-type recursion gives polynomial cost for numerical evaluation. A recursive scheme implemented in Python using symbolic computation is provided for derivatives of arbitrary order, together with a program that verifies the explicit formulas independently against direct differentiation. The univariate case, published previously by Withers, Nadarajah and Teal in the journal Axioms, is reviewed first, recovering known results expressed in terms of Bell polynomials. The method is illustrated by worked examples in two dimensions. Its use in mathematical statistics is shown by casting maximum likelihood estimation in canonical exponential families, including a canonical regression setting, in the framework of the paper, and by deriving explicitly the bias-corrected maximum likelihood estimator with bias On−2, which is checked against an exact calculation for the normal distribution.
In this paper, an efficient iterative method based on the generalized multiscale finite element (Iter-GMs) method is applied to solve the distributed optimal control problems governed by stochastic parabolic equations. Within this framework, we firstly reformulate the time-dependent optimality system into an equivalent form, which contains the parameter-independent diffusion coefficients and the parameter-dependent right-hand side. Then, the fixed-point iteration method is used to solve the reformulated optimality system. Quantifying the statistics of stochastic optimal control problems requires solving a coupled optimality system across numerous samples in the stochastic space, which incurs a prohibitively high computational cost and may even trigger the so-called “curse of dimensionality.” To mitigate this issue, we exploit the multiscale and high-contrast nature of the physical parameters by invoking the generalized multiscale finite element method (GMsFEM) to construct a reduced-order space. We divide the proposed iterative method into two stages: an offline stage and an online stage. In the offline stage, we construct local multiscale basis functions for each coarse-grid region based on the deterministic multiscale optimality system, and then build the corresponding offline spaces. In the online stage, we project the reformulated model onto the reduced multiscale space and apply the fixed-point iteration method to obtain the optimal solutions of the reduced system. Finally, two numerical experiments are presented to validate the rationality and effectiveness of the proposed method.
We determine the exact operator norms of the weighted Hardy operator Uψ and the weighted Cesàro operator Vψ on Total Morrey spaces Lp,λ,μ(Rn). We also study their commutators with symbols b∈BMO(Rn) satisfying the additional dilation condition sup0
We developed a random-field theory for the Euler–Poisson–Darboux equation on Euclidean space under additive Gaussian forcing that was white in the evolution variable and spatially homogeneous. An explicit two-parameter Bessel Green kernel provided the basic deterministic propagator. Sharp high-frequency bounds for its integrated energy yielded a necessary and sufficient Dalang criterion for the existence of the stochastic convolution and showed that the singular Darboux coefficient did not change the wave-type spectral threshold. A dissipative energy identity for the Green kernel provided temporal increment estimates which, together with the spatial Fourier bounds, led to joint space–time Hölder regularity under a natural strengthened spectral moment. We also distinguished between the two initial traces: the Dalang condition guaranteed the mean-square displacement trace, whereas a mean-square zero-velocity trace required additional spectral regularity; finite total spectral mass provided a simple sufficient condition. Space–time white noise and Riesz-colored noise were treated explicitly. Finally, the mean solution was identified with the Darboux spherical mean, Brownian first-exit sampling, and the Pizzetti expansion.
In this paper, we investigate wave equations with a general class of Lipschitz nonlinearities combining power and logarithmic terms. Local existence and uniqueness of weak solutions are established via the Galerkin method. For nonlinearities of defocusing type, we prove global existence of the weak solutions for arbitrary initial energy. In particular, we provide examples of globally defined solutions with negative initial energy.
