The ferry problem may be viewed as generalizations of the classical wolf-goat-cabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.

This paper proves the local exact one-sided boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws with characteristics with constant multiplicity. This generalizes the results in [Li, T. and Yu, L., One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws, To appear in Journal de Mathématiques Pures et Appliquées, 2016.] for a class of strictly hyperbolic systems of conservation laws.

The authors identify the function space which is the tangent space to the integrable Teichmüller space. By means of quasiconformal deformation and an operator induced by a Zygmund function, several characterizations of this function space are obtained.

The authors establish the boundedness of the variation operators associated with the heat semigroup, Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schrödinger setting on the Morrey spaces.

The authors define strongly Gauduchon spaces and the class [inline-graphic not available: see fulltext], which are generalization of strongly Gauduchon manifolds in complex spaces. Comparing with the case of Kählerian, the strongly Gauduchon space and the class [inline-graphic not available: see fulltext] are similar to the Kähler space and the Fujiki class [inline-graphic not available: see fulltext] respectively. Some properties about these complex spaces are obtained. Moreover, the relations between the strongly Gauduchon spaces and the class [inline-graphic not available: see fulltext] are studied.

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property and the Gaussian upper bounds. Denote by Δ the Laplace-Beltrami operator and by ∇ the Riemannian gradient. In this paper, the author proves the weighted reverse inequality $\left\| {{\Delta ^{\frac{1}{2}}}f} \right\|_{L^p(w)}\leq C\left\| {|\nabla f|} \right\|_{L^p(w)}$, for some range of p determined by M and w. Moreover, a weak type estimate is proved when p = 1. Some weighted vector-valued inequalities are also established.

The authors get on Métivier groups the spectral resolution of a class of operators [inline-graphic not available: see fulltext], the joint functional calculus of the sub-Laplacian and Laplacian on the centre, and then give some restriction theorems together with their asymptotic estimates, asserting the mix-norm boundedness of the spectral projection operators $\mathcal{P}_\mu^m$ for two classes of functions m(a, b) = (a ^α + b ^β)^γ or (1 + a ^α + b ^β)^γ, with α, β > 0, γ ≠ 0.

Let P _r denote an almost prime with at most r prime factors, counted according to multiplicity. In the present paper, it is proved that for any sufficiently large even integer n, the equation $n = {x^3} + p_1^3 + p_2^3 + p_3^3 + p_4^3 + p_5^3 + p_6^4 + p_7^4$ has solutions in primes p _i with x being a P ₆. This result constitutes a refinement upon that of Hooley C.

Finite dimensional ribbon Hopf (super) algebras play an important role in constructing invariants of 3-manifolds. In the present paper, the authors give a necessary and sufficient condition for the Drinfel’d double of a finite dimensional Hopf superalgebra to have a ribbon element. The criterion can be seen as a generalization of Kauffman and Radford’s result in the non-super situation to the ℤ₂-graded situation, however, the derivation of the result in the ℤ₂-graded case will be much more complicated.

The authors prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, this result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.