A lower bound for the volume of a four-dimensional closed manifold

A closed orientable four-dimensional manifoldM ⁴, whose sectional curvatures satisfy –1 K 1, is considered. If the product of the curvatures of orthogonal area elements is nonnegative and if at least at one point all the curvatures are different from zero, then the estimate vol(M ⁴) > (4/9)² is obtained for the volume ofM ⁴. A theorem on the local structure of a manifold with small volume whose curvatures at every point are of the same sign is established.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 3–8, 1991.     

Some lower bounds for the canonical volume of 3-folds of general type

Let V be a smooth projective 3-fold of general type. We show that the canonical volume K3 of V has a lower bound1/10 provided q(V) 0 and K3≥3/40 provided P4(V)≥5. We also give some clean and better form for Noether inequalities for minimal 3-folds of general type.     

Computation of lower bounds on the network cost in location problems subject to distance constraints

Methods for the computation of lower bounds on the cost of the connecting network for the continuous and discrete variants of the problem of location of interconnected objects subject to minimal or maximal distances between them are proposed. For the continuous variant, the bound is found by solving a linear programming problem. For the discrete variant, an assignment problem with a rectangular matrix containing forbidden entries is constructed. An application of the assignment problem for locating objects of various sizes is described.     

Using phase-type models to cost stroke patient care across health,social and community services

Stroke disease places a heavy burden on society, incurring long periods of time in hospital and community care, and associated costs. Also stroke is a highly complex disease with diverse outcomes and multiple strategies for therapy and care. Previously a modeling framework has been developed which clusters patients into classes with respect to their length of stay (LOS) in hospital. Phase-type models were then used to describe patient flows for each cluster. Also multiple outcomes, such as discharge to normal residence, nursing home, or death can be permitted. We here add costs to this model and obtain the Moment Generating Function for the total cost of a system consisting of multiple transient phase-type classes with multiple absorbing states. This system represents different classes of patients in different hospital and community services states. Based on stroke patients’ data from the Belfast City Hospital, various scenarios are explored with a focus on comparing the cost of thrombolysis treatment under different regimes. The overall modeling framework characterizes the behavior of stroke patient populations, with a focus on integrated system-wide costing and planning, encompassing hospital and community services. Within this general framework we have developed models which take account of patient heterogeneity and multiple care options. Such complex strategies depend crucially on developing a deep engagement with the health care professionals and underpinning the models with detailed patient-specific data.     

A sharp lower bound for the canonical volume of 3-folds of general type

Let V be a smooth projective 3-fold of general type. Denote by K ³, a rational number, the self-intersection of the canonical sheaf of any minimal model of V. One defines K ³ as the canonical volume of V. Assume p _g (V) ≥ 2. We show that , which is a sharp lower bound. Then we classify those V with small volume. We also give some new examples with p _g  = 2 which have maximal canonical stability index. Finally, we give an application to 4-folds of general type. The author is supported by the National Natural ScienceFoundation of China.