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In this paper, we study the directional entropy along irrational directions in $\mathbb {R}^2$ and present the structure of directional Pinsker $\sigma $-algebra of $\mathbb {Z}^2$-MPSs.
We study a cooperative mean field game problem arising from the production control for multiple firms with price stickiness in the commodity market. The price dynamics for each firm is described as a controlled jump-diffusion process with mean-field interaction. Each firm aims to maximize the so-called social rewards which is defined by the average of individual rewards for all firms. By solving the limiting control problem for the representative firm and an associated fixed-point problem, we construct an explicit approximating optimal strategy when the number of firms grows large.
Testing slope homogeneity is important in panel data modeling. Existing approaches typically take the summation over a sequence of test statistics that measure the heterogeneity of individual panels; they are referred to as Sum tests. We propose two procedures for slope homogeneity testing in large panel data models. One is called a Max test that takes the maximum over these individual test statistics. The other is referred to as a Combo test, which combines a certain Sum test (i.e., that of Pesaran and Yamagata in J Econom 142:50-93, 2008) and the proposed Max test together. We derive the limiting null distributions of the two test statistics, respectively, when both the number of individuals and temporal observations jointly diverge to infinity, and demonstrate that the Max test is asymptotically independent of the Sum test. Numerical results show that the proposed approaches perform satisfactorily.
In this paper, we show some vanishing theorems for harmonic p-forms on a locally conformally flat Riemannian manifold. In the concrete, provided that the integral of the traceless Ricci tensor has a suitable bound, we obtain a vanishing theorem for them without any scalar curvature conditions. Another theorem is also given under the condition on nonpositive scalar curvature, which improves and extends the ones previous.
When a parameterized probability density function is used to represent a landmark-based shape, the shape can be viewed as a point on the manifold that equips with a Riemannian metric corresponding to the mixture models. Hence, given two shapes parameterized by the same density model, the geodesic distance between them can be used for an appropriate shape distance measure. We provide a computational strategy, which is based on the cubic B-splines, to get geodesics and geodesic distances between plane shapes represented by the mixture of Gaussians. In contrast to the methods that discretize geodesic into a sequence of line segments, the proposed method is computationally efficient and numerically stable.
Lutwak et al. (Adv Math 329:85–132, 2018) introduced the $L_p$ dual curvature measure that unifies several other geometric measures in dual Brunn–Minkowski theory and Brunn–Minkowski theory. Motivated by works in Lutwak et al. (Adv Math 329:85–132, 2018), we consider the uniqueness and continuity of the solution to the $L_p$ dual Minkowski problem. To extend the important work (Theorem A) of LYZ to the case for general convex bodies, we establish some new Minkowski-type inequalities which are closely related to the optimization problem associated with the $L_p$ dual Minkowski problem. When $q< p$, the uniqueness of the solution to the $L_p$ dual Minkowski problem for general convex bodies is obtained. Moreover, we obtain the continuity of the solution to the $L_p$ dual Minkowski problem for convex bodies.
Let be a doubling metric measure space in the sense of R. R. Coifman and G. Weiss, L a non-negative self-adjoint operator on satisfying the Davies–Gaffney estimate, and a ball quasi-Banach function space on satisfying some extra mild assumptions. In this article, the authors introduce the Hardy type space by the Lusin area function associated with L and establish the atomic and the molecular characterizations of As an application of these characterizations of , the authors obtain the boundedness of spectral multiplies on . Moreover, when L satisfies the Gaussian upper bound estimate, the authors further characterize in terms of the Littlewood–Paley functions and and establish the boundedness estimate of Schrödinger groups on . Specific spaces to which these results can be applied include Lebesgue spaces, Orlicz spaces, weighted Lebesgue spaces, and variable Lebesgue spaces. This shows that the results obtained in the article have extensive generality.