By sharp maximal function, we establish a weighted estimate with multiple-weight for the multilinear singular integral operators with non-smooth kernels.

Let Y be a Gromov-Hausdorff limit of complete Riemannian n-manifolds with Ricci curvature bounded from below. A point in Y is called k-regular, if its tangent is unique and is isometric to a k-dimensional Euclidean space. By Cheeger-Colding and Colding-Naber, there is k>0 such that the set of all k-regular point R_k has a full renormalized measure. An open problem is if R_l= ∅ for all l<k? The main result in this paper asserts that if R₁≠∅, then Y is a one-dimensional topological manifold. Our result improves Honda’s result that under the assumption that 1≤dim_H(Y ) <2.

We study the asymptotic relation among the maximum of continuous weakly and strongly dependent stationary Gaussian process, the maximum of this process sampled at discrete time points, and the partial sum of this process. It is shown that these two extreme values and the sum are asymptotically independent if the grid of the discrete time points is sufficiently sparse and the Gaussian process is weakly dependent, and asymptotically dependent if the grid points are Pickands grids or dense grids.

Let G be a simple connected graph, and let di be the degree of its i-th vertex. The sum-connectivity index of the graph G is defined as χ(G)=Σvivj∈E(G)? (di+dj)−1/2. We discuss the effect on χ(G) of inserting an edge into a graph. Moreover, we obtain the relations between sum-connectivity index and Randić index.

We propose a characteristic finite element discretization of evolutionary type convection-diffusion optimal control problems. Nondivergence-free velocity fields and bilateral inequality control constraints are handled. Then some residual type a posteriori error estimates are analyzed for the approximations of the control, the state, and the adjoint state. Based on the derived error estimators, we use them as error indicators in developing efficient multi-set adaptive meshes characteristic finite element algorithm for such optimal control problems. Finally, one numerical example is given to check the feasibility and validity of multi-set adaptive meshes refinements.

This paper deals with anisotropic solutions u∈W1,(pi)(Ω,?N) to the nonlinear elliptic system

−Σi=1nDi(aiα(χ,Du(χ)))=−Σi=1nDiFiα(χ), α=1,2,...,N,

We present a monotonicity inequality for the matrix a=(aiα)∈?N×n,whichguarantees global pointwise bounds for anisotropic solutionsu.

Imaginary Verma modules, parabolic imaginary Verma modules,and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated,and several results are generalized from the affine Lie algebras. In particular,imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.

Let(Zt)t≥0be a one-dimensional symmetric α-stable process withα∈(0,2), and letσbe a bounded (from above and from below) and 1/(α∨1)-Hölder continuous function on?. Consider the stochastic differential equationdX=σ(Xt−)dZt,which admits a unique strong solution. By using thesplitting technique and the coupling method, we derive the Hölder continuity of the associated semigroup.

We study the uniqueness and existence of solutions of reflected G-stochastic differential equations (RGSDEs) with nonlinear resistance under an integral-Lipschitz condition of coefficients. Moreover, we obtain the comparison theorem for RGSDEs with nonlinear resistance.

Let (X, d, μ) be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions in the sense of Hytönen. Under this assumption, we prove thatθ-type Calderón-Zygmund operators which are bounded on L²(μ) are also bounded from L^∞(μ) into RBMO(μ) and from H^1,∞_at (μ) into L¹(μ).

In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp (p, p) estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on L^p(Hⁿ) is still p/(p−1). This goes some way to imply that the L^p norms of the Hardy operator are the same despite the domains are intervals on ℝ, balls in ℝⁿ, or ‘ellipsoids’ on the Heisenberg group Hⁿ. By constructing a special function, we find the best constant in the weak type (1,1) inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H¹ to L¹. Moreover, we describe the difference between M^p weights and A^p weights and obtain the characterizations of such weights using the weighted Hardy inequalities.

We prove the boundedness of fractional integral with a rough kernel on Triebel-Lizorkin spaces, where the rough kernel belongs to the block space and does not need to satisfy any moment conditions on the unit sphere.

This paper is concerned with the stability analysis of the exact and numerical solutions of the reaction-diffusion equations with distributed delays. This kind of partial integro-differential equations contains time memory term and delay parameter in the reaction term. Asymptotic stability and dissipativity of the equations with respect to perturbations of the initial condition are obtained. Moreover, the fully discrete approximation of the equations is given. We prove that the one-leg θ-method preserves stability and dissipativity of the underlying equations. Numerical example verifies the efficiency of the obtained method and the validity of the theoretical results.

Completely random allocation of the treatment combinations to the experimental units is appropriate only if the experimental units are homogeneous. Such homogeneity may not always be guaranteed when the size of the experiment is relatively large. Suitably partitioning inhomogeneous units into homogeneous groups, known as blocks, is a practical design strategy. How to partition the experimental units for a given design is an important issue. The blocked general minimum lower order confounding is a new criterion for selecting blocked designs. With the help of doubling theory and second order saturated design, we present a theory on constructing optimal blocked designs under the blocked general minimum lower order confounding criterion.

This paper establishes a lattice Boltzmann method (LBM) with two amending functions for solving partial differential equations (PDEs) arising in Asian and lookback options pricing. The time evolution of stock prices can be regarded as the movement of randomizing particles in different directions, and the discrete scheme of LBM can be interpreted as the binomial models. With the Chapman-Enskog multi-scale expansion, the PDEs are recovered correctly from the continuous Boltzmann equation and the computational complexity is O(N), where N is the number of space nodes. Compared to the traditional LBM, the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants. The stability of LBM is studied via numerical examples and numerical comparisons show that the LBM is as accurate as the existing numerical methods for pricing the exotic options and takes much less CPU time.