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Valuations and Unambiguity of Languages, with Applications to .. - Fernau, Staiger (Correct)
....of the point OE F (y) The such defined map OE F : Sigma n ; ae fi F ) X; ae E ) is Lipschitz continuous. The set AF = OE F ( Sigma n ) is called attractor of F . We denote the s dimensional outer Hausdorff measure on (X; ae E ) by H , and the corresponding Hausdorff dimension by dimH [28]. For IFS, Moran s open set condition (OSC) is well known as an assumption alleviating the determination of the Hausdorff dimension of AF [22] Provided there is an open bounded nonempty test set V X such that F (i) V ) V for any i 2 Sigma n , and that, furthermore, for any i; j 2 Sigma n , ....
C. A. Rogers. Hausdorff Measures. Cambridge at the University Press, 1970.
Risk Management with Extreme Value Theory - Klüppelberg (2002) (Correct)
....es the usual conditions, which guarantee that (X t ) t 0 is ergodic with stationary density (3.3) s(r) s(l) 1 and jmj 1 : 3. 4) For proofs of the above relations and further results on di usions we refer to the monographs Karatzas and Shreve [52] Revuz and Yor [76] Rogers and Williams [78], or any other advanced textbook on stochastic processes. The following formulation can be found in Davis [25] Proposition 3.1. Let (X t ) t 0 satisfy the usual conditions (3.4) Then for any initial value X 0 = y 2 (l; r) and any u t r, jP (M t u t ) F (u t )j = 0 ; where F is a ....
....all t 0, Y t = Z t a:s: for some stochastic process ( t ) t 0 . The random time t has a representation via the local time of the process (Y t ) t 0 . This is a consequence of the Dambis Dubins Schwarz Theorem (Revuz and Yor [76] Theorem 1.6, p. 170) Theorem 47.1 of Rogers and Williams [78], p. 277 and Exercise 2.28 of [76] p. 230. For z 2 (l; r) denote by L t (z) the local time of (Y s ) 0 s t in z. Then by the occupation time formula (cf. Revuz and Yor [76] p. 209) t = L t (z)dm ou (s ou (z) ds ; t 0 : Notice also that t is continuous and strictly ....
Rogers, L.C.G. and Williams, D. (2000) Di usions, Markov Processes, and Martingales, Vol. 2., 2nd Ed. Cambridge University Press, Cambridge.
In Search of Sensitivity in Network Optimization - Chen, Pandit, Meyn (2003) (Correct)
....is given by H = # 1 . Proof A proof of V uniform ergodicity is included in [30] The remaining results may be found in [11, Theorem 6.2] To bound the error between W and X on [T 1 , T 2 ] we apply the following lemma. The proof follows from consideration of an exponential martingale ([48], Section I 16) Lemma A.3 Let N be a Brownian motion with zero drift and infinitesimal variance # and N(0) 0. Then for any T 1 , T 2 with 0 T 1 T 2 and any D 0 we have t#[T 1 ,T 2 ] N(t) D exp 2# D Proof of Theorem 4.6 Part (ii) follows easily from Part (i) and ....
L.C.G. Rogers and D. Williams. Di#usions, Markov processes, and Martingales. Volume I: Foundations. Cambridge University Press, Cambridge, Second edition, 2000. 47
The Hausdorff dimension of the visible sets of connected compact.. - O'Neil (2003) (Correct)
.... 0 and (B(x,r) r s for x E X, 0 r 1 . Here B(x, r) denotes the closed ball with centre x and radius r. If is a Radon measure on the plane and s ] then Is( denotes the s energy of given by The Hausdorff dimension of a set is defined in the usual way via Hausdorff measures, see [1, 2, 6, 10]. The following theorem summarises some useful equivalent ways of finding the Hausdorff dimension of a set. 3 Theorem 2.1 Let A be an analytic subset of a Euclidean space, R n. Then dimH(A) sup s c ] JMS(A) 0 sup s c ] There is I JM( with (A) 0 and Is(l) o sup dimH(K) K C A and K ....
C. A. Rogers, Hausdorff Measures, Cambridge University Press (1998).
Orlicz Capacities And Applications To Some Existence.. - Fiorenza, Prignet (Correct)
....4.8. Let us now extend the results to a general set E, not necessarily compact. There exists E countable intersection of open sets such that E E and h (E) and Cap ;A (E) Cap ;A (E ) Cap ;A is an outer capacity according to [4] and h is an outer measure according to [23]) Using the fact that h satis es the assumptions of Choquet s theorem (see [23] Chapter 2.7) one has h (K) K compact; K E Moreover, using Theorem 9 of [4] one has also Cap ;A (E Cap ;A (K) K compact; K E Hence the results obtained for compact sets can be extend ....
.... exists E countable intersection of open sets such that E E and h (E) and Cap ;A (E) Cap ;A (E ) Cap ;A is an outer capacity according to [4] and h is an outer measure according to [23] Using the fact that h satis es the assumptions of Choquet s theorem (see [23] Chapter 2.7) one has h (K) K compact; K E Moreover, using Theorem 9 of [4] one has also Cap ;A (E Cap ;A (K) K compact; K E Hence the results obtained for compact sets can be extend to general sets. Remark 4.14 The proof of Theorem 4.13 follows the ideas used to ....
C.A. Rogers Hausdor Measures, Cambridge University Press, 1970.
Absolutely Continuous Spectrum of Stark Operators - Christ, Kiselev (Correct)
....= d(E) for every E 2 I in the support of : Similarly, we also obtain that for a.e. the solution u (x; E) satis es the bound (5. 21) for the energies in the support of : By the choice of and well known relation between D derivatives and dimensional properties of Radon measures (see [30]) it follows that for a.e. the restriction of the spectral measure to I is supported on a set of Hausdor dimension d(E) and gives zero weight to any set of Hausdor dimension d(E) To show the dynamical bound, notice that Theorem 1.2 of [20] says that if the spectral measure ....
C.A. Rogers, Hausdor Measures, Cambridge University Press, London, 1970
In Search of Sensitivity in Network Optimization - Chen, Pandit, Meyn (2001) (Correct)
....H = xp(x) dx is given by H = # . Proof A proof of V uniform ergodicity is included in [24] The remaining results may be found in [7] ut To bound the error between W and X on [T 1 ; T 2 ] we apply the following lemma. The proof follows from consideration of an exponential martingale ([41], Section I 16) Lemma A.2 Let N be a Brownian motion with zero drift and in nitesimal variance . Then for any T 1 ; T 2 with 0 T 1 T 2 and any ; L we have t2[T 1 ;T2 ] N(t) L exp 2 L 2 Setting the hitting time as = infft : c W 1 (t; w; max( 1 ....
L.C.G. Rogers and D. Williams. Di usions, Markov processes, and Martingales. Volume I: Foundations. Cambridge University Press, Cambridge, Second edition, 2000.
Correspondence Principles for Effective Dimensions - Hitchcock (2001) (1 citation) (Correct)
....X, the computable entropy rate of X, and the constructive entropy rate of X, respectively. Classical Hausdor dimension may be characterized in terms of entropy rates. Theorem 5.2. For any X C, dimH (X) inf H(X) A proof of Theorem 5. 2 can be found in [5] it also follows from Theorem 32 of [4]. Staiger proved the following relationship between computable entropy rates and Hausdor dimension. Theorem 5.3. Staiger [6] For any X 2 2 , dimH (X) inf HDEC (X) Putting Theorems 4.4, 5.2, and 5.3 together, for any X 2 2 we have dimH (X) cdim(X) dim comp (X) inf H(X) ....
C. A. Rogers. Hausdor Measures. Cambridge University Press, 1998.
Porosities And Dimensions Of Measures - Jean-Pierre Eckmann Esa Self-citation (Measures) (Correct)
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C. A. Rogers, Hausdorff Measures, Cambridge University Press, Cambridge, 1970.
An Optimal Skorokhod Embedding for Di usions A.M.G. Cox - And Hobson Department Self-citation (Usions) (Correct)
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L. C. G. Rogers and D. Williams. Di usions, Markov processes, and martingales. Vol. 2: It^o calculus. Cambridge University Press, Cambridge, 2000.
Lower bounds on wave packet propagation - Packing Dimensions Of Self-citation (Measures) (Correct)
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C. A. Rogers, Hausdorff measures, (Cambridge University Press, Cambridge, 1970).
An Optimal Skorokhod Embedding for Diffusions - Cox, Hobson (2002) Self-citation (Usions) (Correct)
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L. C. G. Rogers and D. Williams. Di usions, Markov processes, and martingales. Vol. 2: It^o calculus. Cambridge University Press, Cambridge, 2000.
Fractal Measures on Infinite Dimensional Sets - McClure Self-citation (Measures) (Correct)
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C. A. Rogers, Hausdorff Measures.Cambridge University Press, 1970.
An Optimal Skorokhod Embedding for Diffusions - Cox, Hobson (2002) Self-citation (Usions) (Correct)
....case where n ( from which it follows that j ( j : using the representation (16) As before we can also show (19) holds by using the correspondence 7 . 3 Applications to di usions We now work with the class of regular (time homogeneous) di usions (see [8], V. 45) Y t ) t 0 on an interval I R, with absorbing or inaccessible endpoints, and vanishing at zero. Consider the problem of determining when and how we may embed a distribution on I in the di usion. Since the di usion is regular, there exists a continuous, strictly increasing scale ....
L. C. G. Rogers and D. Williams. Di usions, Markov processes, and martingales. Vol. 2: It^o calculus. Cambridge University Press, Cambridge, 2000.
A unifying class of Skorokhod embeddings: connecting - The Azema-Yor And (Correct)
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Rogers, L. C. G., Williams, D., 2000. Di#usions, Markov processes, and martingales. Vol. 2. Cambridge University Press, Cambridge, ito calculus, Reprint of the second (1994) edition.
Optimal stopping of the maximum process: a converse to the.. - David Hobson April (Correct)
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Rogers, L. C. G. and Williams, D.; Di#usions, Markov processes, and martingales. Vol. 2. Cambridge University Press, Cambridge, 2000. 17
Risk Management with Extreme Value Theory - Klüppelberg (2002) (Correct)
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Rogers, L.C.G. and Williams, D. (2000) Di usions, Markov Processes, and Martingales, Vol. 2., 2nd Ed. Cambridge University Press, Cambridge.
Scaled Dimension and Nonuniform Complexity - Hitchcock, Lutz, Mayordomo (2003) (Correct)
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C. A. Rogers, Hausdor# Measures, Cambridge University Press, 1998.
Universal Duality in Conic Convex Optimization - Schurr, al. (2004) (Correct)
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C. Ambrose Rogers. Hausdor measures. Cambridge University Press, London, 1970.
Effective Fractal Dimensions - Lutz (Correct)
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C. A. Rogers. Hausdor Measures. Cambridge University Press, 1998.
Stationarity and Second Order Behaviour of Discrete.. - Klüppelberg, Lindner, .. (Correct)
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Rogers, L.C.G. and Williams, D. (2000) Di usions, Markov Processes, and Martingales, Volume 2. It^o Calculus, (2nd Ed.). Cambridge University Press. Cambridge.
Jeffreys' prior is the Hausdorff measure for the Hellinger and.. - Doster (1998) (Correct)
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C.A. Rogers. Hausdor Measures. Cambridge University Press, 1970.
The Hausdorff Dimension Of The Hyperspace Of Compact Sets - McClure (Correct)
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C. A. Rogers, Hausdorff Measures.Cambridge University Press, 1970.
Finite gap potentials and WKB asymptotics for.. - Kriecherbauer, Remling (2001) (4 citations) (Correct)
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C.A. Rogers, Hausdor Measures, Cambridge University Press, Cambridge, 1970.
Dimension Theory of Smooth Dynamical Systems - Schmeling (Correct)
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C. Rogers, Hausdor measures, Cambridge University Press, London{New York, 1970
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