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with 0 < C(t) < + For the issue of the MBM simulation, see [AYA 00], where there are some indications regarding this question We will also note the existence of the FracLab toolbox in this eld 643 Towards more irregularities: generalized multifractional Brownian motion (GMBM) and step fractional Brownian motion (SFBM) We saw in the preceding section that the MBM provides a model for locally self-similar processes with varying multifractional functions and pointwise exponents However, Hypothesis 64 essential within the strict framework of MBM is cumbersome for certain applications Let us quote two examples where we would wish for regularities which are worse than H lderian regularities First, in the rupture models which are important in image segmentation, we wish that the H lder exponent had some discontinuities Let us be clearer about this problem through a metaphor Let us suppose that we have an aerial image on which we want to distinguish the limit of a eld and a forest It is usual to model the texture of a forest by a FBM of the H lder exponent H1 In the same way, for the portion of the eld, we can think of a FBM of exponent H2 The question arises as to how to connect the two processes on the border There is a possibility of considering a MBM within the meaning of De nition 67, for which function h takes the two values H1 and H2 However, this MBM does not correspond to the image Indeed, it shows a discontinuity at the place where function h jumps However, on an image, only the regularity changes suddenly and most often, the eld remains continuous To model this type of rupture, let us recall the construction of the step fractional Brownian motion (SFBM) of [BEN 00] In addition, the H lder exponent of the MBM varies very slowly for applications with so-called developed turbulence (see [FRI 97] for an introduction to this subject) Indeed, the physics of turbulence teach us that the accessible data from measurements are not the H lder exponents of the studied quantities, but their multifractal spectrum.

The description of multifractal spectrum is beyond the scope of this chapter (see 1, 3 and 4), but it is enough to know that for a function whose H lder exponent is itself C r , this spectrum is trivial We can thus be convinced that the MBM is not a realistic model for developed turbulence In order to obtain processes whose trajectories have H lder exponents which vary abruptly, Ayache and L vy V hel have proposed a model called the generalized multifractional Brownian motion (GMBM) in [AYA 99] We present their model in the second part of this section 6431 Step fractional Brownian motion The multifractional functions associated with the SFBMs are very simple, which makes it possible to have a reasonable model for the identi cation of ruptures We limit ourselves to multifractional functions in steps:

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AODV also supports route repair using the route error (RERR) message This message is used by nodes for invalidating routes they are aware of when links supporting those routes break This could happen due to node movements, increased link error rate, and so on When a node loses connectivity to one of its neighbors, it sends a RERR message to other nodes that make use of the path through the lost neighbor The RERR message contains the list of destinations that have become unreachable The RERR may be broadcast or unicast to the affected nodes Based on the description of AODV provided so far, we see that AODV has no inherent security mechanisms As a result, malicious nodes can perform many attacks For example, a malicious node can impersonate some other node and initiate a forged RREQ message supposedly from the impersonated node A malicious node can also decrease the hop-count eld of a RREQ it is forwarding instead of increasing it This could be intended to ensure that the path through the malicious node would become the shortest path This would guarantee that all data packets will pass through the malicious node An adversarial node could also pretend to be the destination or could pretend to know a route to the destination The resulting RREP message can then be forged, ensuring that all messages to the destination will go to the malicious node A malicious node can also choose not to forward the RREQ and RREP messages, resulting in suboptimal routes In addition, a malicious node could also create forged RERR messages or choose not to forward genuine RERR messages All these potential attacks demonstrate the need to enhance the security of AODV We will describe some potential approaches for making AODV secure (at least against certain attacks)

h(t) =

(x) = lim inf un (x)

where un (x) is the cube of i=1 [ki 2 n , (ki + 1)2 n [ that contains x Other covering nets can obviously be used, but dyadic cubes are well suited for calculations 132 Theorems on set dimensions The rst theorem can be used as a basis for understanding the subsequent more technical results

Page 169 groups with separate decision criteria and expectations Our ability to tap into those needs and address them as effectively as possible is a necessary step in the process of becoming customer-based 6 Deciding on measurement parameters Once the list of key issues (critical success factors) has been developed, the next step is to determine the various parameters of actual data measurement The key parameters revolve around the effective administration of the customer satisfaction measurement instrument There are essentially three issues that required immediate attention:

THEOREM 12 Let be a nite measure such that (E) > 0 Assume that there exists a real , such that for any x E:

A secure version of AODV called Secure AODV (SAODV) has been proposed in [49, 50] SAODV provides features such as integrity, authentication, and nonrepudiation of routing data SAODV incorporates two schemes for securing AODV The rst scheme involves nodes signing the messages that they create (eg RREQ, RREP) This allows other

(x) = (x) =

Then = dim(E) = Dim(E) Proof Let > 0 Let En be the subset of E consisting of all points x such that, for k n: uk (x) +

If < 2 n , any -covering {ui } of En by dyadic cubes of rank n is such that: |ui | + for all i Therefore (En ) (ui ) and |ui |

(En ) If n is large enough then First, we deduce that H ( ) (En ) (En ) > 0 This gives dim(En ) Since En E, then dim(E) Secondly, by using the covering of En formed by dyadic cubes of rank k obtain: N2 k (En )2 k( + ) n, we

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