We introduce the p-adic weighted multilinear Hardy-Cesàro operator. We also obtain the necessary and sufficient conditions on weight functions to ensure the boundedness of that operator on the product of Lebesgue spaces, Morrey spaces, and central bounded mean oscillation spaces. In each case, we obtain the corresponding operator norms. We also characterize the good weights for the boundedness of the commutator of weighted multilinear Hardy-Cesàro operator on the product of central Morrey spaces with symbols in central bounded mean oscillation spaces.

We introduce two exponentially stochastic differentiability conditions to study moderate deviations for M-estimators. Under a generalized exponentially stochastic differentiability condition, a moderate deviation principle is established. Some sufficient conditions of the exponentially stochastic differentiability and examples are also given.

Recently, T. K. Chandra, T. -C. Hu and A. Rosalsky [Statist. Probab. Lett., 2016, 116: 27–37] introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. We introduce a weaker definition on uniform nonintegrability (W-UNI) of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of WUNI, one of which is a W-UNI analogue of the celebrated de La Vallée Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised by Chandra et al.

We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the ‘resolution theorem’ in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different algebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-finite Gorenstein algebras.

We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter, and lower bounded injectivity radius.

We are concerned with a class of neutral stochastic partial differential equations driven by Rosenblatt process in a Hilbert space. By combining some stochastic analysis techniques, tools from semigroup theory, and stochastic integral inequalities, we identify the global attracting sets of this kind of equations. Especially, some sufficient conditions ensuring the exponent p-stability of mild solutions to the stochastic systems under investigation are obtained. Last, an example is given to illustrate the theory in the work.

We investigate stable homology of modules over a commutative noetherian ring R with respect to a semidualzing module C, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces C to be trivial and R to be Gorenstein.

An (a, d)-edge-antimagic total labeling of a graph G is a bijection f from V (G) ∪ E(G) onto {1, 2, . . . , |V (G)| +|E(G)|} with the property that the edge-weight set {f(x) + f(xy) + f(y) | xy ∈ E(G)} is equal to {a, a + d, a + 2d, . . . , a + (|E(G)| − 1)d} for two integers a>0 and d≥0. An (a, d)-edgeantimagic total labeling is called super if the smallest possible labels appear on the vertices. In this paper, we completely settle the problem of the super (a, d)-edge-antimagic total labeling of the complete bipartite graph Km,n and obtain the following results: the graph K_m,n has a super (a, d)-edge-antimagic total labeling if and only if either (i) m = 1, n = 1, and d≥0, or (ii) m = 1, n≥2 (or n= 1 and m≥2), and d ∈ {0, 1, 2}, or (iii) m = 1, n = 2 (or n = 1 and m = 2), and d = 3, or (iv) m, n≥2, and d= 1.

We obtain some De Lellis-Topping type inequalities on the smooth metric measure spaces, some of them are as generalization of De Lellis-Topping type inequality that was proved by X. Cheng [Ann. Global Anal. Geom., 2013, 43: 153–160].

We consider a real-valued doubly-perturbed stochastic differential equation driven by a subordinated Brownian motion. By using classic Malliavin calculus, we prove that the law of the solution is absolutely continuous with respect to the Lebesgue measure on ℝ.

Let A be a path A_∞-algebra over a positively graded quiver Q. We prove that the derived category of A is triangulated equivalent to the derived category of kQ, which is viewed as a DG algebra with trivial differential. The main technique used in the proof is Koszul duality for DG algebras.

We study a strongly elliptic partial differential operator with timevarying coefficient in a parabolic diagonalizable stochastic equation driven by fractional noises. Based on the existence and uniqueness of the solution, we then obtain a kernel estimator of time-varying coefficient and the convergence rates. An example is given to illustrate the theorem.

A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete cavitation solution is obtained, and the convergence of the method is proved. An algorithm combined a gradient type method with a damped quasi-Newton method is applied to solve the discretized nonlinear equilibrium equations. Numerical experiments show that the Fourier-Chebyshev spectral method is efficient and capable of producing accurate numerical cavitation solutions.

We consider Hamiltonian partial differential equations u_tt +|∂_x|u+ σu = f(u), x ∈ T, t ∈ℝ, with periodic boundary conditions, where f(u) is a real-analytic function of the form f(u) = u⁵ + o(u⁵) near u = 0, σ ∈ (0, 1) is a fixed constant, and \mathbb{T}=ℝ/\mathrm{2}\pi \mathbb{Z}T= R/2πZ. A family of quasi-periodic solutions with 2-dimensional are constructed for the equation above with σ ∈ (0, 1)\ \mathbb{Q}. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.