Suppose X is a Banach space. We introduce the concept of quasi-distribution semigroups on X, as a generalization of the concept of distribution semigroups introduced by Lions in [19]. In our approach, the generator A of a quasi-distribution semigroup may not be densely defined. Also introduced is a functional calculus for A in terms of the Fourier transform. For fixed r>0, k # N _ [0], define order (r, k) for a quasi-distribution semigroup and define an F r, k functional calculus. We prove that A generates a (k+1)-times integrated semigroup of exponential type r with local Lipschitz continuity if and only if A generates a quasi-distribution semigroup of order (r, k) if and only if A has an F r, k functional calculus. When A is densely defined, a quasi-distribution semigroup reduces to a distribution semigroup in the sense of Lions.1997 Academic Press

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This paper investigates the problem of protecting multicast sessions in mesh wavelength-division multiplexing (WDM) networks against single link failures, for example, a fiber cut in optical networks. First, we study the two characteristics of multicast sessions in mesh WDM networks with sparse light splitter configuration. Traditionally, a multicast tree does not contain any circles, and the first characteristic is that a multicast tree has better performance if it contains some circles. Note that a multicast tree has several branches. If a path is added between the leave nodes on different branches, the segment between them on the multicast tree is protected. Based the two characteristics, the survivable multicast sessions routing problem is formulated into an Integer Linear Programming (ILP). Then, a heuristic algorithm, named the adaptive shared segment protection (ASSP) algorithm, is proposed for multicast sessions. The ASSP algorithm need not previously identify the segments for a multicast tree. The segments are determined during the algorithm process. Comparisons are made between the ASSP and two other reported schemes, link disjoint trees (LDT) and shared disjoint paths (SDP), in terms of blocking probability and resource cost on CERNET and USNET topologies. Simulations show that the ASSP algorithm has better performance than other existing schemes.

Abstract. Motivated by a great deal of interest in operators that may not be densely defined and do not generate global integrated semigroups, we establish general perturbation theorems for local integrated semigroups and describe their applications to local complete second order abstract differential equations.1. Introduction. Let X be a complex Banach space, and consider the following local complete second order abstract differential equation on X:

Abstract. Motivated by a great deal of interest recently in operators that may not be densely defined, we deal with regularized semigroups and integrated semigroups that are either not exponentially bounded or not defined on [0, ∞) and generated by closed operators which may not be densely defined. Some characterizations and related examples are presented. Our results are extensions of the corresponding results produced by other authors.1. Introduction. Let X be a complex Banach space and let A be a closed linear operator on X. Many physical problems may be modelled as a first order abstract Cauchy problem: