A system of linear conditions is presented for Wronskian and Grammian solutions to a (3+1)-dimensional generalized vcKP equation. The formulations of these solutions require a constraint on variable coefficients.

It is proved that, any finite dimensional complex Lie algebra L = [L, L], hence, over a field of characteristic zero, any finite dimensional Lie algebra L = [L, L] possessing a basis with complex structure constants, should be a weak co-split Lie algebra. A class of non-semi-simple co-split Lie algebras is given.

In this paper, a multiscale problem arising in material science is considered. The problem involves a random coefficient which is assumed to be a perturbation of a deterministic coefficient, in a sense made precisely in the body of the text. The homogenized limit is then computed by using a perturbation approach. This computation requires repeatedly solving a corrector-like equation for various configurations of the material. For this purpose, the reduced basis approach is employed and adapted to the specific context. The authors perform numerical tests that demonstrate the efficiency of the approach.

The authors consider a non-Newtonian fluid governed by equations with p-structure in a cubic domain. A fluid is said to be shear thinning (or pseudo-plastic) if 1 < p < 2, and shear thickening (or dilatant) if p > 2. The case p > 2 is considered in this paper. To improve the regularity results obtained by Crispo, it is shown that the secondorder derivatives of the velocity and the first-order derivative of the pressure belong to suitable spaces, by appealing to anisotropic Sobolev embeddings.

The authors introduce a kind of slowly increasing cohomology HS* (X) for a discrete metric space X with polynomial growth, and construct a character map from the slowly increasing cohomology HS* (X) into HC*_cont(S(X)), the continuous cyclic cohomology of the smooth subalgebra S(X) of the uniform Roe algebra B*(X). As an application, it is shown that the fundamental cocycle, associated with a uniformly contractible complete Riemannian manifold M with polynomial volume growth and polynomial contractibility radius growth, is slowly increasing.

The authors study the regular submanifolds in the conformal space ℚ _p ⁿ and introduce the submanifold theory in the conformal space ℚ _p ⁿ. The first variation formula of the Willmore volume functional of pseudo-Riemannian submanifolds in the conformal space ℚ _p ⁿ is given. Finally, the conformal isotropic submanifolds in the conformal space ℚ _p ⁿ are classified.

In this paper, quasi-almost-Einstein metrics on complete manifolds are studied. Two examples are given and several formulas are established. With the help of these formulas, the author proves rigid results on compact or noncompact manifolds, in which some basic tools, such as the weighted volume comparison theorem and the weak maximum principle at infinity, are used. A lower bound estimate for the scalar curvature is also obtained.

For the weighted approximation in L _p-norm, the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiener space. By this result, it is known that in the sense of information-based complexity, if permissible information functionals are Hermite data, then the p-average errors of this sequence are weakly equivalent to those of the corresponding sequence of the minimal p-average radius of nonadaptive information.

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I = (a, b), a function G ∈ S(w):= {f: Σ _I|f(x)|w(x)dx < ∞} satisfying the conditions G ^(2j)(x) ≥ 0, x ∈ (a, b), j = 0, 1, …, and growing as fast as possible as x → a+ and x → b−, plays an important role. But to find such a function G is often difficult and complicated. This implies that to prove convergence of Gaussian quadrature formulas, it is enough to find a function G ∈ S(w) with G ≥ 0 satisfying \mathop {\sup }\limits_n \sum\limits_{k = 1}^n {\lambda _{0kn} G(x_{kn} ) < \infty } instead, where the x _kn’s are the zeros of the nth power orthogonal polynomial with respect to the weight w and λ _0kn’s are the corresponding Cotes numbers. Furthermore, some results of the convergence for Gaussian quadrature formulas involving the above condition are given.

Consider the following Cauchy problem: \begin{gathered} u_t = div(|\nabla u^m |^{p - 2} \nabla u^m ),(x,t) \in S_T = \mathbb{R}^N \times (0,T), \hfill \\ u(x,0) = \mu ,x \in \mathbb{R}^N \hfill \\ \end{gathered} where 1 < p < 2, 1 < m < \tfrac{1}{{p - 1}}, and µ is a σ-finite measure in ℝ ^N. By the Moser’s iteration method, the existence of the weak solution is obtained, provided that \tfrac{{(m + 1)N}}{{mN + 1}} < p. In contrast, if \tfrac{{(m + 1)N}}{{mN + 1}} \geqslant p, there is no solution to the Cauchy problem with an initial value δ(x), where δ(x) is the classical Dirac function.

The author studies the linkage between the standardness and the standard automorphisms of Chevalley groups over rings. It is proved that if H is any standard subgroup of G(R), then each of its automorphisms can be extended to an automorphism of G(R, I), restricted to an automorphism of E(R, I), and an automorphism of E(R, I) can be extended to one of G(R, I). The case of Chevalley groups of rank at least two is treated in this paper. Further results about the case of Chevalley groups of rank one, the case of non-commutative ground ring and some others exceptions will appear elsewhere.