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In other words, the impulse response h of a fractional differential system breaks up r into a localized part hn of integer order n = i=1 i and a part h of purely diffusive nature We can nd in [DAUP 00, HEL 00] a great number of examples illustrating this decomposition result on some non-standard oscillators 743 Connection with the concept of long memory Finally, let us recall that such systems are said to have long memory in so far as the decrease of the impulse response (in the stable case) is not of exponential type This is determined by a generalized expansion in = 0 of distribution , which is followed by the application of the following lemma LEMMA 71 (Watson) For 1 < 1 < m < m+1 , we have:

M 1

( ) =

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developing a systematic program for customer-based project management requires a clear understanding of our customer base Creating the goals for project supply chain or value chain management are a necessary activity However, these goals are established in relation to a clearly identified customer/supplier base In other words, until we have an understanding of the key stakeholders with the potential to affect our organization s operations, it is impossible to progress very far down the path of customer-based project management This chapter examines the key elements in project stakeholder analysis, a necessary first-step to improving our ability to manage projects well in a difficult and changing environment Past organizational research and, indeed, common sense tells us that organizations and even managers within those companies cannot operate in ways that ignore the external effects of their decisions No manager makes decisions exclusive of consideration of how those decisions will affect external groups1 One way to understand the relationship of project managers and their projects vis vis the rest of the organization is through employing stakeholder

m + O( M ) (1 + m )

algorithms on them The authors in [23], however, show that this need not be the case In this paper the authors demonstrate the tractability of public key algorithms on sensor networks The authors make use of elliptic curve cryptography and show that public keys operations can be executed within 34 s While this is a tremendous improvement over earlier numbers, it is still a signi cant number A potential solution is to utilize asymmetric keys for setting up symmetric keys for subsequent communications Note though that, in a few years, the use of asymmetric techniques in sensor networks might become viable

M 1

(741) m 1 t1+ m + O(t 1 M )

h (t) =

Thus, by juxtaposing the decomposition result (739), the expression of (740) and the asymptotic analysis (740), the following characterization of stability is obtained THEOREM 72 System (734) is stable in BIBO if and only if the two following conditions are veri ed: in (739), we have e(si ) < 0, for all i; the rst exponent 1 in (740) is strictly positive It should be noted that a priori, si , although a nite number (see [BON 00]), is not known in a simple way in the general case The situation is quite different when the system is more structured, as it emphasized below 744 Particular case of fractional differential systems of commensurate orders The general result given before can then be expressed differently, by using a strong algebraic structure induced by the commensurate character of the derivation orders When saying = s , R is de ned such that H(s) = R( ) It is then enough to decompose the rational fraction R into simple elements ( ) m , to de ne

.

by inverse Laplace transform the corresponding basic elements E m ( , t) and to characterize their stability by using an asymptotic analysis similar to the preceding one The function E m ( , t) is the fundamental solution of the operator (D )m ; it belongs to the family of the Mittag-Lef er functions, which is a subset of hypergeometric special functions Using these functions, we obtain the following structure result PROPOSITION 718 We have:

.

h(t) =

rnm E m ( n , t)

with R( ) =

A re ned asymptotic analysis of the functions E m ( n , t) makes it possible to deduce the following fundamental result for BIBO stability when R = Q/P , with P, Q two coprime polynomials and 0 < < 1 THEOREM 73 We have: BIBO stability | arg | > , 2 C, P ( ) = 0 (742)

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