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9.1 INTRODUCTION The CP operation of microstrip antennas can be achieved by using either singlefeed or two-feed designs. The single-feed design has the advantage of a simple feed structure, and in addition, no external phase shifter is required. Many single-feed designs have been described in 5. However, single-feed designs usually have a narrow CP bandwidth (3-dB axial ratio) of about 1% for a microstrip antenna with a dielectric substrate. To enhance the CP bandwidth for a single-feed design, an aperture-coupled microstrip antenna using an inclined nonlinear coupling slot has been reported [1]. Details of the experimental results are given in Section 9.2. In this study, the CP bandwidth is broadened to about 2.1 times that obtained using a simple inclined coupling slot for CP operation. When a thick air or foam substrate is used, the CP bandwidth of a microstrip antenna with a single-feed design can be further enhanced. In this case, a single-feed aperture-coupled circular microstrip antenna with a slit and two stubs and having a CP bandwidth of about 7.2% has been reported [2]. For two-feed designs, single-layer, broadband CP microstrip antennas with a large CP bandwidth greater than 20% have been reported [3 5]. These broadband CP designs mainly have a thick foam or air substrate; to achieve good impedance matching, feed methods using two three-dimensional microstrip transition feed lines [3] and two aperture-coupled feeds [4, 5] have been used. Recently, two-feed designs using two gap-coupled probe feeds [6] and two capacitively coupled feeds [7] have been presented. For the former design with a thick air substrate, experimental results show that a 65% impedance bandwidth (1:2 VSWR) and a 46% 3-dB axial-ratio CP bandwidth are achieved [6]. Within the CP bandwidth, the gain bandwidth, de ned

markings of Min(Couvn (R, mc )) In the second case, it enables ring of a transition that leads to a marking greater than or equal to a marking of Min(Couvn (R, mc )) Let m be a marking of Min(Couvn (R, mc )) A marking obtained by ring of t greater than or equal to m must necessarily be greater than or equal to Sup(Post(t), m) and the marking preceding the ring is greater than or equal to Sup(Post(t), m) C(t) Summarizing: Min(Couvn+1 (R, mc )) = Min(Min(Couvn (R, mc )) t T m Min(Couvn (R,mc )) Sup(Post(t), m) C(t)) Every step multiplies by at most |T |+1 the number of minimal items The previous method gives a bound on the number of stages before stabilization So we deduce that the method is primitive recursive A priori, the shortest sequence method seems more ef cient.

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Certain adipokines may also modulate the left ventricle s response to obesity. Leptin has generated much interest but a clear picture is yet to emerge: some studies reported a positive correlation between leptin and LVM (Sader, Nian and Liu, 2003; Tritos, Manning and Danias, 2004; Paolisso et al., 2000; Paolisso, Tagliamonte and Galderisi, 2001), whereas others found a negative correlation (Barouch, Berkowitz and Harrison, 2003) or no signi cant relationship (Pladevall, Williams and Guyer, 2003; Malmqvist et al., 2002). Similarly, recent studies of the possible relationship between adiponectin

However, the interest of backward analysis will be illustrated during the presentation of extensions of Petri nets 45 The reachability problem The reachability problem consists of deciding whether, given a net R and two markings mi and mf , there is a ring sequence: mi mf The decision algorithm has been independently established by Mayr [MAY 84] and Kosaraju [KOS 82] Here we will not fully describe this proof, some features of which are very technical and to which a book is devoted [REU 89] We present the scheme of the proof and we only develop its main feature, ie the suf cient condition for reachability First, the reachability problem is solved for a more general model than that of Petri nets, called the chain of vector addition systems (CVAS) The main motivation of this generalization is intimately linked to the proof.

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One of the major limitations in the drive to improve the performance of rotating electrical machines is the temperature limitation of the materials used in the construction. In the drive to manufacture more competitive machines, effective design requires accurate computation of local temperatures. This in turn, requires accurate modelling of the thermal performance of the machine which requires detailed knowledge of the thermal boundary conditions, including

This proof can be described as follows: A size is de ned for the reachability problem This size is an item of a well-founded set (ie such that an in nite sequence of strictly decreasing sizes does not exist) We establish a suf cient condition for reachability with the help of the covering graph If the condition is not ful lled, we build a nite (possibly empty) set of problems with smaller sizes such that a solution for the initial problem exists iff a solution exists for any of the reduced problems Taking the initial problem as the root, we build a tree of problems de ning, for the sons of a problem, its reduced problems If the tree was in nite, there would be an in nite branch, which is impossible as a result of the rst point.

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