This paper deals with the existence and uniqueness of mild solutions to neutral stochastic delay functional integro-differential equations perturbed by a fractional Brownian motion B ^H, with Hurst parameter H ∈ (1/2, 1). We use the theory of resolvent operators developed by R. Grimmer to show the existence of mild solutions. An example is provided to illustrate the results of this work.

We define an integral approximation for the modulus of the gradient |∇u(x)| for functions f: Ω ⊂ ℝ ⁿ → ℝ by modifying a classical result due to Calderon and Zygmund. Our integral approximations are more stable than the pointwise defined derivatives when applied to numerical differentiation for discrete data. We apply our results to design and analyse neighborhood filters. These filters correspond to well-behaved nonlinear heat equations with the conductivity decreasing with respect to the modulus of gradient |∇u(x)|. We also show some numerical experiments and evaluate the effectiveness of our filters.

We introduce a space [graphic not available: see fulltext], where [graphic not available: see fulltext] is the testing function space whose functions are infinitely differentiable and have bounded support, and [graphic not available: see fulltext] is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from [graphic not available: see fulltext] onto itself.

Let G be a circuit graph of a connected matroid. P. Li and G. Liu [Comput. Math. Appl., 2008, 55: 654–659] proved that G has a Hamilton cycle including e and another Hamilton cycle excluding e for any edge e of G if G has at least four vertices. This paper proves that G has a Hamilton cycle including e and excluding e′ for any two edges e and e′ of G if G has at least five vertices. This result is best possible in some sense. An open problem is proposed in the end of this paper.

This paper is interested in solving a multidimensional backward stochastic differential equation (BSDE) whose generator satisfies the Osgood condition in y and the Lipschitz condition in z. We establish an existence and uniqueness result of solutions for this kind of BSDEs, which generalizes some known results.

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L²(H¹) and L²(L²) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition k_n ⩾ ch², which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.

The two-level penalty mixed finite element method for the stationary Navier-Stokes equations based on Taylor-Hood element is considered in this paper. Two algorithms are proposed and analyzed. Moreover, the optimal stability analysis and error estimate for these two algorithms are provided. Finally, the numerical tests confirm the theoretical results of the presented algorithms.

Suppose that G is a finite group and H is a subgroup of G. H is said to be s-permutably embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup H_se of G contained in H such that G = HT and H ∩ T ⩽ H_se. In this paper, we continue the work of [Comm. Algebra, 2009, 37: 1086–1097] to study the influence of the weakly s-permutably embedded subgroups on the structure of finite groups, and we extend some recent results.

Let f(z) be a Hecke-Maass cusp form for SL₂(ℤ), and let L(s, f) be the corresponding automorphic L-function associated to f. For sufficiently large T, let N(σ, T) be the number of zeros ρ = β +iγ of L(s, f) with |γ| ⩽ T, β ⩾ σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4 ⩽ σ ⩽ 1 − ɛ, there exists a constant C = C(ɛ) such that N(σ,T) ≪ T^2(1−σ)/σ(logT) ^C, which improves the previous results.

We introduce the generalized Jacobi-Gauss-Lobatto interpolation involving the values of functions and their derivatives at the endpoints, which play important roles in the Jacobi pseudospectral methods for high order problems. We establish some results on these interpolations in non-uniformly weighted Sobolev spaces, which serve as the basic tools in analysis of numerical quadratures and various numerical methods of differential and integral equations.

Let µ be a nonnegative Radon measure on ℝ ^d which satisfies the polynomial growth condition that there exist positive constants C₀ and n ∈ (0, d] such that, for all x ∈ ℝ ^d and r > 0, µ(B(x, r)) ⩽ C₀ r ⁿ, where B(x, r) denotes the open ball centered at x and having radius r. In this paper, we show that, if µ(ℝ ^d) < ∞, then the boundedness of a Calderón-Zygmund operator T on L²(µ) is equivalent to that of T from the localized atomic Hardy space h¹(µ) to L^1,∞(µ) or from h¹(µ) to L¹(µ).

We determine the derivation algebra and the automorphism group of the generalized topological N = 2 superconformal algebra.

Let {X_i, π_kⁱ, ω} be an inverse sequence and $X = \mathop {\lim }\limits_ \leftarrow \left\{ {X_i ,\pi _k^i ,\omega } \right\}$. If each X_i is hereditarily (resp. metaLindelöf, σ-metaLindelöf, σ-orthocompact, weakly suborthocompact, δθ-refinable, weakly θ-refinable, weakly δθ-refinable), then so is X.