We construct superprocesses with dependent spatial motion (SDSMs) in Euclidean spaces ℝd with d ≥ 1 and show that, even when they start at some unbounded initial positive Radon measure such as Lebesgue measure on ℝd, their local times exist when d ≤ 3. A Tanaka formula of the local time is also derived.
In this paper, we study a family of Hartogs domains fibred over Hermitian symmetric manifolds being a unit ball in ℂm. The aim of the present study is to establish the rigidity results about proper holomorphic mappings between two equidimensional Hartogs domains over Hermitian symmetric manifolds. In particular, we can fully determine its biholomorphic equivalence and automorphism group.
In this paper, we consider the following perturbed fractional Hamiltonian systems
$$left{ {matrix{{_tD_infty ^alpha {(_{ - infty }}D_t^alpha u(t)) + L(t)u(t) = {nabla _u}W(t,u(t)) + {nabla _u}G(t,u(t)),} hfill & {t in mathbb{R},} hfill cr {u in {H^alpha }(mathbb{R},{mathbb{R}^N}),} hfill & {} hfill cr } } right.$$where (alpha in (1/2,1],,,L in C(mathbb{R},{mathbb{R}^{N times N}})) is symmetric and not necessarily required to be positive definite, (W in {C^1}(mathbb{R}times {mathbb{R}^N},mathbb{R})) is locally subquadratic and locally even near the origin, and perturbed term (G in {C^1}(mathbb{R} times {mathbb{R}^N},mathbb{R})) maybe has no parity in u. Utilizing the perturbed method improved by the authors, a sequence of nontrivial homoclinic solutions is obtained, which generalizes previous results.
In this paper we focus on the boundary regularity for a class of k-Hessian equations which can be degenerate and (or) singular on the boundary of the domain. Motivated by the case of Monge–Ampère equations, we first construct sub-solutions, then apply the characteristic of the global Hölder continuity for convex functions, and finally use the maximum principle to obtain the boundary Hölder continuity for the solutions of the k-Hessian equations. However, finding such sub-solutions is very difficult due to the complexity of the k-Hessian operator. In particular, we employ the symmetric mean to overcome the difficulties.
We establish new identities for Moore–Penrose inverses of some operator products, and prove their associated reverse-order laws. Moreover, our results concerning the Moore–Penrose inverse of a product of two operators lead in finding a relation between the operators in the case where Greville’s inclusions are made into equalities.
We study the following Schrödinger equation with variable exponent
where (epsilon > 0,,,1 < p < {{N + 2} over {N - 2}},,,a(x) in {C^1}({mathbb{R}^N}) cap {L^infty }({mathbb{R}^N}),,,N ge 3) Under certain assumptions on a vector field related to a(x), we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem. We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.
Multiple change-points estimation for functional time series is studied in this paper. The change-point problem is first transformed into a high-dimensional sparse estimation problem via basis functions. Group least absolute shrinkage and selection operator (LASSO) is then applied to estimate the number and the locations of possible change points. However, the group LASSO (GLASSO) always overestimate the true points. To circumvent this problem, a further Information Criterion (IC) is applied to eliminate the redundant estimated points. It is shown that the proposed two-step procedure estimates the number and the locations of the change-points consistently. Simulations and two temperature data examples are also provided to illustrate the finite sample performance of the proposed method.
A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G. A graph G is acyclically k-choosable if for any list assignment L = {L(v): v ∈ V(G)} with ∣L(v)∣ ≥ k for each vertex v ∈ V(G), there exists an acyclic proper vertex coloring ϕ of G such that ϕ(v) ∈ L(v) for each vertex v ∈ V(G). In this paper, we prove that every graph G embedded on the surface with Euler characteristic number ε = −1 is acyclically 11-choosable.
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