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where > 1 and 0 < H < 1 The values of the phases n are arbitrary We can directly prove [TRI 99] that, restricted to any bounded interval J, the box dimension of the graph is equal to 2 H By homogenity, Dim( ) = 2 H This con rms the fact although it is more dif cult to prove (see [BOU 00]) that the -oscillations are uniformly of the order of H In other words, there exist two constants c1 and c2 > 0 such that, for any t J and for any , with 0 < < 1: c1 H f, [x , x + ] c2 H

Figure 13 Graph of a nowhere derivable function, attractor of a system of four af ne applications, such that i = 1 and i = 1 Dimensions 4 2 and Dim are equal to 3 The Hausdorff dimension lies between 1 and 3 2 2

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Finding the value of the Hausdorff dimension of such graphs is still an open problem In the case where the n are independent random variables with same distribution, it is known that dim( ) = 2 H with probability 1 135 2-microlocal analysis In this section, we brie y present an analysis of local regularity which is much ner than the H lder exponents described above As was observed previously, neither p nor l allow us to completely describe the local irregularity of a function To obtain an exhaustive description, we need to consider an in nite number of exponents This is the purpose of 2-microlocal analysis This powerful tool was de ned in [BON 86], where it is introduced in the framework PDE in the frame of Littlewood-Paley analysis A de nition based on wavelets is developed in [JAF 96] We present here a time-domain characterization [KOL 02]

s,s DEFINITION 14 A function f : I R R belongs to the 2-microlocal space Cx0 1 if there exist 0 < < 4 and C > 0 such that, for all (x, y) satisfying |x x0 | < and |y x0 | < :

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When considering how to determine a competitive price for a project, a number of factors must be considered Figure 18 illustrates the nature of cost and price competition for a project contractor competing for business Bid price is subject to a number of diverse factors, including fixed and variable costs for the project firm, competition and market price

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Figure 14 Graph of the Weierstrass function for = 3 and H = 1 Phases n 2 are random independent variables and are identically distributed Dimensions and Dim are equal to 3 The Hausdorff dimension is almost surely close to 3 2 2

attackers to evade detection Given the disadvantages, this technique is not used widely in commercial systems Another technique for detecting malicious activity has been introduced recently although it is not commonly used yet in commercial systems This technique, called speci cation-based detection, assumes the existence of a precise protocol speci cation Malicious behavior is detected by comparing the protocol traf c with the protocol speci cation The detectors typically build precise models of expected behavior (eg by using state machines) based on the protocol speci cations and then compare the observed behavior in the network against the model The advantage of speci cation-based detection techniques is that, given accurate speci cations of normal behavior (eg protocols speci cations), malicious behavior can be detected with a high degree of certainty This eliminates the possibility that a detector will classify a normal behavior as malicious, which reduces false positives Such detectors may also spot novel attacks since these detectors do not depend on the existence of speci c attack signatures On the other hand, developing models of normal behavior for each protocol is often a fairly complex task Further, this approach requires models of normal behavior for all the protocols being used in the network in order to detect a wide range of attacks These models need to be executed for each node in the network As a result, these detection schemes require signi cant CPU resources for a large enterprise network Another limitation of speci cation-based detection is that detectors using this technique do not detect attacks that do not violate the speci cation, but exploit permitted behavior to launch an attack For example, in a ooding attack or a TCP SYN ood attack, a node s behavior is acceptable by the protocol speci cation, but the behavior is actually harmful to the operation of the enterprise and is therefore malicious Detection of attacks is typically not suf cient for protecting an enterprise because the attack may continue to cause harm to the network Therefore, intrusion detection systems are usually coupled with attack response systems Once an attack has been identi ed by the IDS, the response system is responsible for stopping the attack It could do this by isolating the malicious behavior or by cutting off the attacker from the network and if possible restoring the damage caused by the attacker Responses are of two types:.

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