The reverse order law for {1, 3, 4}-inverse of the product of two matrices

The relationship between {1, 3, 4}-inverses of AB and the product of {1, 3, 4}-inverses of A and B have been studied in this paper. The necessary and sufficient conditions for B{1,3,4}A{1,3,4}⊆(AB){1,3,4}, B{1,3,4}A{1,3,4}⊇(AB){1,3,4} and B{1,3,4}A{1,3,4}=(AB){1,3,4} are given.     

On transmission time through k minimal paths of a capacitated-flow network

The quickest path problem is to minimize the transmission time for sending a specified amount of data through a single minimal path. Two deterministic attributes are involved herein; the capacity and the lead time. However, in many real-life networks such as computer systems, urban traffic systems, etc., the arc capacity should be multistate due to failure, maintenance, etc. Such a network is named a capacitated-flow network. The minimum transmission time is thus not a fixed number. This paper is mainly to evaluate system reliability that d units of data can be transmitted through k minimal paths under time constraint T. A simple algorithm is proposed to generate all minimal capacity vectors meeting the demand and time constraints. The system reliability is subsequently computed in terms of such vectors. The optimal k minimal paths with highest system reliability can further be derived.     

Calculation of minimal capacity vectors through k minimal paths under budget and time constraints

Reducing the transmission time is an important issue for a flow network to transmit a given amount of data from the source to the sink. The quickest path problem thus arises to find a single path with minimum transmission time. More specifically, the capacity of each arc is assumed to be deterministic. However, in many real-life networks such as computer networks and telecommunication networks, the capacity of each arc is stochastic due to failure, maintenance, etc. Hence, the minimum transmission time is not a fixed number. Such a network is named a stochastic-flow network. In order to reduce the transmission time, the network allows the data to be transmitted through k minimal paths simultaneously. Including the cost attribute, this paper evaluates the probability that d units of data can be transmitted under both time threshold T and budget B. Such a probability is called the system reliability. An efficient algorithm is proposed to generate all of lower boundary points for (d, T, B), the minimal capacity vectors satisfying the demand, time, and budget requirements. The system reliability can then be computed in terms of such points. Moreover, the optimal combination of k minimal paths with highest system reliability can be obtained.     

Halphen pencils on quartic threefolds

For any smooth quartic threefold in P⁴ we classify pencils on it whose general element is an irreducible surface birational to a surface of Kodaira dimension zero.     

Spherical homogeneous spaces of minimal rank

Let G be a connected reductive algebraic group over an algebraically closed field K of characteristic zero. Let G/B denote the complete flag variety of G. A G-homogeneous space G/H is said to be spherical if H has finitely many orbits in G/B. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the group G (viewed as a G×G-homogeneous space) has particularly nice properties. Namely, the pair (G,H) is called a spherical pair of minimal rank if there exists x in G/B such that the orbit H.x of x by H is open in G/B and the stabilizer Hx of x in H contains a maximal torus of H. In this article, we study and classify the spherical pairs of minimal rank.     

Darboux-Jouanolou-Ghys integrability for one-dimensional foliations on toric varieties

We use the existence of homogeneous coordinates for simplicial toric varieties to prove a result analogous to the Darboux-Jouanolou-Ghys integrability theorem for the existence of rational first integrals for one-dimensional foliations.     

Minimal distortion morphs generated by time-dependent vector fields

A morph between two Riemannian n-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian n-manifolds and, with respect to these classes, prove the existence of minimal distortion morphs and diffeomorphisms. In particular, we consider the class of time-dependent vector fields (on an open subset Ω of Rn+1 in which the manifolds are embedded) that generate morphs between two manifolds M and N via an evolution equation, define the bending and the morphing distortion energies for these morphs, and prove the existence of minimizers of the corresponding functionals in the set of time-dependent vector fields that generate morphs between M and N and are L² functions from [0,1] to the Sobolev space .     

Sharp constant for a 2D anisotropic Sobolev inequality with critical nonlinearity

For the 2-dimensional anisotropic Sobolev inequality of the form

     

Remarks on Hard Lefschetz conjectures on Chow groups Dedicated in celebration of SCIENCE CHINA on the occasion of its 60th birthday

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we show that they are equivalent to the well-known conjectures of Beauville and Murre.