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De-correlation of the mass transformation coef cients ien proposed in [ IE 93] a solution designed to accelerate the decompression phase by orthogonalization of the basis vectors b1 and b2 that span the collage vector subspace The second advantage of this approach is that we do not have to impose constraints on the scale coef cients of the fractal transformation In signal processing applications, such as data compression, the handled vectors are generally represented by a linear combination of orthogonal functions The most usual example is that of the Fourier transform where the functions are complex exponential We can resort to the Gram-Schmidt procedure to orthogonalize the decimated block b2 with respect to the constant basis vector b1 of the vectorial subspace This procedure amounts to multiply the vector b2 by the orthogonalizing matrix O = I b1 b1 T , where I is the identity matrix of dimension B 2 B 2 The orthogonalized vector b2 reads: b2 = Ob2 Practically, this amounts to removing the b2 component that belongs to the subspace spanned by vector b1 , and thus to force the mean value of block b2 to zero The new collage, now performed in the vectorial subspace spanned by the orthogonal vectors b1 and b2 , becomes: o = 2 b2 + 1 b1 r In this case, the coef cients i are independent of each other ien and Leps y show in [ IE 94b] that it is not necessary to formally orthogonalize the vectors to calculate the coef cients 1 and 2 They are directly obtained from the expressions: 1 = r, b1 1 = 2 B

2 =

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Moreover, this result is optimal, ie there exists a function of global regularity > 0 such that p (x) = l (x) for all x outside a set of zero Hausdorff dimension THEOREM 16 Let 0 < < 1 and f : [0, 1] [ , 1] be a lower limit of continuous functions Let g : [0, 1] [ , 1] be a lower semi-continuous function Assume that for all t [0, 1], f (t) g(t) Then, there exists a continuous function F : [0, 1] R such that: for all x, l (x) = g(x); for all x outside a set of zero Hausdorff dimension, p (x) = f (x) This theorem shows that, when the compatibility condition f (t) g(t) is satis ed, we can simultaneously and independently prescribe the local and pointwise regularity of a function outside a small set These two measures of irregularity are thus to some extent independent and provide complementary information 134 Signal dimension theorem Let us investigate the relationships between the dimension of a signal and its H lder exponents There is no general result concerning the Hausdorff dimension, apart from obvious upper bounds resulting from the inequalities dim( ) Dim( ) ( ) Here is a result for Dim( ) [TRI 86a] THEOREM 17 If is the graph of a continuous function f , then: 2 sup (x)

Dim( )

2 inf (x)

forward it and discard the other route requests that arrive later The resulting routes would then include the adversary, which places the adversary in an advantageous position We see that this attack is not very dif cult to launch All it requires is that the adversary be able to forward route request packets faster than can be done by valid nodes The adversary can do this by forming wormholes The adversary can also do this by choosing to ignore the delay between the receipt and forwarding of route request packets Such delays are speci ed by the routing protocols to avoid collisions of route request packets The adversary can also choose to ignore delays speci ed by the protocols to access the wireless channel Thus, in all these cases the route request packet can be rushed by the adversary A simple way to address the rushing attack is to allow for randomized selection of route request messages Thus, every node is expected to collect a threshold number of route requests Following this, the node can randomly choose to forward a route request from among the received requests Note that a timeout also needs to be associated here so that, if a node does not receive the threshold number of route requests within the timeout, it can choose from among the received route requests This mechanism does add to the latency of route discovery, though, and hence both the threshold value and the timeout value have to be chosen with care Note, though, that the mechanism is general enough and can be easily integrated with any reactive routing protocol to protect against rushing attacks 473 Sybil Attacks

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One term that needs greater explanation is product thinking Product thinking is the result of conceptualizing the fundamental product we offer another organization or market The product can be based on hardware, software, service, or a combination of these To understand the concept of product thinking, it is necessary to view it from the perspective of the customer That is why organizations marketing consumer products devote so much time and money to conducting consumer surveys, focus groups, niche penetration analysis, and so forth The ability to sell a product that measures up to

The same inequalities are true if we use the grid of the dyadic intervals: 2 sup (x)

Dim( )

2 inf (x)

.

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We do not provide the demonstration of these results, which requires an evaluation of the packing measure of the graph In the same context, we could show that if the local H lder exponents (un (x)) tend uniformly to a real , then this number is also equal to ( ) However, a much more interesting equality may be given for Minkowski-Bouligand dimension of the graph which is both simple and general THEOREM 18 Let f be a continuous function de ned on an interval J, and non-constant on J For any > 0, let us call -variation of f on J the arithmetic mean of -oscillations: 1 f, [x , x + ] J dt var (f ) = |J| J

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