Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

Journal of Applied Probability ◽

10.1017/s0021900200023780 ◽

1983 ◽

Vol 20 (03) ◽

pp. 529-536

Author(s):

W. J. R. Eplett

Keyword(s):

Gaussian Process ◽

Gaussian Processes ◽

Partial Ordering ◽

Boundary Crossing ◽

Conditional Probabilities ◽

Life Distribution ◽

Life Distributions ◽

Bounded Below ◽

Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

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An alternative approach to study irrotational periodic gravity water waves

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/s00033-021-01578-8 ◽

2021 ◽

Vol 72 (4) ◽

Author(s):

Biswajit Basu ◽

Calin I. Martin

Keyword(s):

Free Surface ◽

Water Waves ◽

Water Wave ◽

Wave Problem ◽

Water Wave Problem ◽

Stagnation Points ◽

Alternative Approach ◽

New Formulation ◽

Bounded Below ◽

Surface Dynamic

AbstractWe are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

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