Stochastic invariances and Lamperti transformations for Stochastic Processes

Stochastic invariances and Lamperti transformations for Stochastic Processes Pierre Borgnat, Pierre-Olivier Amblard, Patrick Flandrin To cite this version: Pierre Borgnat, Pierre-Olivier Amblard, Patrick Flandrin. Stochastic invariances and Lamperti transformations for Stochastic Processes. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 25, pp.28-2. <ensl-7596> HAL Id: ensl-7596 https://hal-ens-lyon.archives-ouvertes.fr/ensl-7596 Submitted on Oct 27 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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➎ 7 34r r s P r ➒ ➋s r P 2 P r P ➈ q P r ➅❿ ➌ q8 P ❷ ➆ ➅ r ➀ 7 ➀ ➋ P P ➒ ➏➎ ➎q P P ➀ ❼ P P P ❾ q ➑➀ ➂ P P P P P P ➀ 2 8 r P r r P P 2 r P P ➀ P r s ➑ P ❹❹ ➒ ❾ q ➒ P 8 P r r P 5 P 7 r P ➄ P P 4 P P 2 ➄ 34r P ➒ ➑ P P P ➄ ➀ 4P ➀ 9 r P ❺ ❿ r ➋ 4P P P 4 P P P 2 P r ➂ P r 4 P t r s s ❷ ➂ P 2 34r ❿ t r ❶ P P P P P 2 34r P P ❸ r 6 P s ➑ r P 9 P P ➃ P P r ➇ ➀ P 834r P P ❶ r ➀ P➃ 7 r P9 P P P 8 P P ➂ r 8 P P P P ➄ ➆ P P P P P P P P P 2 ➆ 6 P ➂ ❹❺ 6 P r ➀ r P ➑ P 2➋ P 8 ➂ s P ➀ P 2➋ ➋ r 8 P 7 ➒ ➀ P ➃ r 4 r 4 P 4 7 r P P r P s 8 P ➒ r P P 4 ➆ P P 5 P P➃s ➃ r P 24➐8 P 5 P 6 r P P➃ P ➀ 4 ➋ r ➄ ❹❾ q 2 34r ➃ ❻ 9 r 9 P r P P P P ❻ ❻ 2 34r 6 P P P ➄ 9 P P 8 P ➀ s 4P 7 8 ➀ 2➋ P r P s ➂ r 8 7 r P P ➋ ➀ ❻ P ➒❻ ❾ r P 7 r P9 P ➋ r P P 234r P ➀ P P P P 8 P P 9 s 34r P P 6 r ➑ P ➑ ➀ 9 r P 8 8 P ➀ s ➆ r P9 s r P ❾ r r P 4 P P 95 P ➀ P P ➂ ❷ ➒❻ P ❻ ➌

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❾ P P ❺ ❺ 34r ❷ ❼ ❶ s ❽ P s ❷ 8 ❷ s P❼ r ➃ 5 P s P 2 6 r P 8 P s P r P s ➒ r r 8 P ➀ r 7 P s 2 8 r r 8 P P t P r r r r t ➑ ➄ ➀ r P ➇ r 2➅ 2 P P r 5 ➀ P ➎ 2 34r P P P r ➑ P P r P P r 7 r ➄ r9 P 6 P r➁ P 4 2 2➀ P 72 6 r➁ r ❷ ➄ P ➑ ➄ 4 P ➀ r 7 s P r 2 3 P r ➃ P P ➒ ➒ P s P 6 P P P ➒ r ❻ ➆❻ ❻ r s Pt s 9 P❸ P P s P ❷ P 4 P P❸ r P P P➁ 2 34r 9 ➄ ❷s r 9 P P ❶ ➓ ➐7 r 2 r ❽ P 6 s ➀ P➄ P r 7 7 ❹ ❹ q P ➄ 8 ➒ 8 4 P9 ➅ ❾q P P 4 8 ➄ ❹ ❿ ❾ ❿ ➌q r P ➂ r ➆ P P 2 ❷ r s P P r 7 r P 8 P P P 8 9 6 2 34r ➄ r P ➀ ➁ t ➄ ➀ P P P r ➃ P r P t P P ➅ ➎ ❹ q2 r P ❹❺ r P r 8 P P❷ P r P P s P r P P P ➄ P r 2 P r 4 r ➃ ❺ ❺ ❺ ❺ ❹ ❹ r r r ➆ r ➂ ❼ ❿ 2 P r P 8 ➆ P ➃ r P P ➒ r 4 r P s P P➃ 4 r P➃ P ➃ P P P ➀ P s ❾ ❾ ➎ ➎ q ➏ r➒ ➁ P r➆ ➈❺ ❺ q ❹ r P ❹ ❺ 8 8 P r P P P➑ r P P 6r P s s P ➃ P P 2❶ ➄ r P 6 r r r 4 r P P 4 ➀ P ➒❻ ❻ ❹ ❺ ➆❻ ➒❻ ❹ ➎ ❻ ❹

➌ gou fgn 3 2.5.5 2 2 4 6 time a.u. 2 3 time a.u. 2 2 fbm 2 3 time a.u. 2 3 4 5 6 pdf of increments 7 5 5 normalized by std ❻q➐ ➂➃ Ð Ò ➍➅ ❿ ➂ ❼ Ï❿ ❿ ❿❾➀ Ï ➃ ➑ ➌ ➀ ➍➆ ➏ r ➏ ❿ s ➃ ss ➃ ❿ r ➆❿ ➏ ➏Ô Ü ➂➃ ➆ ➏ Ð ➆❿ ➆ ❺è➀ s r ❿ ➅ s ➏ Ò Û s s ➃ ❿ r ➍➅ ➆➂ ❿7 Ò ❿ ➍è ➃ ❿❾➀ t ❿ ❹ ❺Ð Ò s ❿ ➅ ➍➅ ➍➆ ➂ ❿ ➎ ➃➐ ❺èÔ 9 ➂ ➅ Ù 2 9 ➍➅ Ò ➍➅ ➏ ➃❿ ➍➆ ➏s s➆➀ r ❺ ➓❺➅ ❺èÔ Ü ➎ ➏ ❿ ➍➆ ➏ ➏❿ s ❺ ❺❼ 2 9 ➀ Ñ ❿ Ï❿ s ❿ r Ô 9 ➏ ➂➃ ➃ ❼ s ➃❿ ➍➆ ➏ ➆❿ s ❼ 2 9 ➀ s s ➆r s s ❿ s Û s ➆➐ ➏➍➅ ❽ Ò ➍➅ s s s ❿ ❿ s ➂ ➃❿ ➊➂ s ➍➏ ➂ ❺èÔ r r ➓ P P 2 ➐8r P r P ❼ P s P P r P ❿ ❾ ➎ ➎q2 34r ➆ P 7 ➑ ➄ P s ➄P P P P 9 r P r 6 8 r s 4 8 ➑ 2 34r 4 8 r 9 P 8 4 r 4 r 9 ➀ P s r ➂ P P 2t34r P ➑ ➑ P r P P P r P 7 P r P r 6 r r r P 8 P r 4 7 P P P ➏➎q 2 r P9 s 6 r P P ➀ P r 7 ➑ 5 P 2 r P r P ➒ ➇ 7 P r ➃ P r P s ➇ P ➀ r 5 P t P ❸ 2 2➀ 8 P P r ➑ r ➃ P ➂

➉ Sample realization of the stationary generator.2 2 Sample realization of the Hss signal...2.3 6 7 8 9 time 2 4 6 8 2 3 time 2 Estimated variance of Hss signal (Non stationary) Increments of the Hss signal.5 5 5.5.5 2 4 6 8 time 2 3 time ➈ ❻ ➐ ➂➃ s ➏➂Û s ➃➂ ➍➆ ➏s s➆➀ï Ï ➃ ➑ ➀Ï➍➅ ➏ rð ➏ ❿ s ➃ sùs ❿ r ➆❿ ➆ ➎ ❿ ➍➅ ❿ ➍➆ ❺ ➅➐ ❾ ❺➅Ô Ü ➂➃ ➆ ➏ ➆❿ ➆ ❺è➀qs r ❿ ➅ s ➏ Ò ➃ s s ➃ ❿ r ➍➆ ➏➂ ❿ Ò ❿ ➍➅ ❿❾Ô Ü ü ➏ ➊s s ➍➆ ➏s s ➏ ➍➆ ➏ Ò ➏➍➆ ➏ ❿ ➍➆ ➏s s➆➀ ➃ ➂ ➆ ➆❿ ➃❿ ❺ ➓❺➆ ❺èÔ ❼ ➂ ➅ s ➃ ➏ ❿ ➍➆ ➍➆ ➏s s➆➀ ❺ ➀ ❿ 5 ➏ ➃ ➍➏ ➂ ➂➃r ❿ ➏➑ ❹ ➁ ➂ s ➃ ❿ s q ❽ ➅➐ ➏➍è ➏ ➍➅ ➎ ➃❿ s Ï❿ ➃❿Ð s ➂ ➃❿ Ô 9 ➃ ➏ ➃❿ ➍➆ ➏ ➆❿ s ❺ ❺➅➀ Ï ➍ ➎ ❿ s ❿ r 2s ➍➏ ❿ ❿ ❼ s s r ❿ ➅ s ➍➅ ❿ ➆❿ ➆❿ ➂ r ➍➅ ➏s s ❺èÔ Ü s ➍➅ s s s ➂ s ➓ ❿ ❻ ➂ ➆ 69 ➍ r s r s ➆ ➆ ➀ ❿ ➎ s ➆ ➏❿ s ➃ ❽ ➃❿ ➁ ❹ ➀ ❹ ❷ ➊Ô P ➐8r P ❿ ❿ q 2➇34r 9 P ➄ s P P ❷ 5 P ➀ r ❶ ➂ P r P P s 2 34r P r P❸ P P➁ 7 r r❸ 4 P 2 P r P P❷ s P Pt r ➃ ➀ r P ➀ P 2 ➋ ❹ r ➃ P r 8 s P r 8 P P P 2❶ ➋ t ➑ r r r s r P ➃ P P P r 9 8 r P P P P r P P 8 r ➄ 6 ➀ r 4 P 2 ➄ r ➃ ➃ P ➀ ➃ P ➃ t r r P s ➑ P P r P Pt P P P 2 P r ➑ P r P r ➄ r ➃ r P r ➑ r ➄ P P ➀ P ❸s s P 2 r❽ P❽ ❼ P❼ P P P 8 P ➃ ❷ ➃ r ➂ P P r ➃ ❾q 2 P ❸ ➂ 9 r ➀ s r 9❻ 9 P 5 P s

Stationary generator Self similar process, H=.8.3.2.2...2.3.. lin scale lin scale.4 2 3 4 5 time.5..2.3.4.5 Mellin spectr..5 Fourier spectr...2.3.4.5 Reduced frequency log scale in db.2 2 3 4 5 time 2 2 4 6 8 Fourier spectr. model FSpectr. Mellin spectr. model MSpectr...2.3.4.5 Reduced frequency 3 ➆➂ ➂ ➃❿ s ➆➂➃ ➊s ➂ ➍➅ s sï Ï ➃ ➑ Ô Ü Ù➂➃ ➆ ➏ 9 s ➃ ❿ r ➆❿ ➆ ❺ s5➍➆ ➏ s5 ❿ Õ 4 35Õ ➆➂ ➆ ❶ ➀à❹ ❺è➀ s r ❿ ➆ s Ò ß ➂➃ ➆ ❿ Ö ➎ Ï ➃ Ð❿ s r ❿ s Ò ➏ Ù Ò ❿ ➍è ❿ Õ 4 35Õ➓Ô Ü ➃ ➊s s ➍➆ ➏s sü ➏ ➍➅ Ò ➍➅ ➃❿ ➍➆ ➏s s➆➀ ➂➃ ➏ ➆❿ ❿ 7❿ ➆ ➍➆ ➂ ➂ r ❺ 7 ➓❺➆ ❺➅Ôq ❼ ý 2➂➃ ➆ 9 ❿ ❺➅ s ➃ ➏ s ➍è ❿ ➃ s ➃❿ ➆ s ➆ ❿ s Ò Ó ❿ ➓ ➂ s ❿ ➏ ➍➆ ➂➃➂ r ❺ ➂ ➅ s ➃❿ ➂ ❿ ➏ s ➍➏ ➂ ➀ ➃❿➎➂➃ s ➍➆ ➂➃ Ô Ü ➒ ➃ ➏ Ïs ➏➍➅ s ❿➎ s ➃ ➃ ❿ ❾ 3 ➏➂➃➂ ❿❽s ➍è ➀ ➍➆➍➆ ❿ t ❿❽ 2❷ ❺èÔ ➆ 9 Ïs ➍è Ò Ï ➍ s ❿ ➆ ➂ r ➅ß ❿ ➏ ➍➆ s ❿ ➆ ➃ ➏ ❿ ➃ s ➃❿ ➍➅ ➆ ➆❿ s ➓s ➃ ❿ r ❺ ➏ s ➍➅ ➂➃ ➆➐❽ ➏➍➏ s ➓ ➃ s ➆ Ï➂ ❿ ❿ ❽ ❿ r s ➂ ➂❾ ➏s ❿ ❿ ➍➅ s9 ❿ ❽ s ❿ ➂ r ❿ ❿ ➍➆ ❿ ❿ r ➏ ➃❿ ➍➅➂ s ➏➂ 2s Ï❿ ❿ ❺➅Ô ❿Ù ➆ ➏s ➏ 3 ➆➂ ➂➃ ❿s ➏➍➅ Ò s s ➂qs ➍è ➓❺ r5 ➆➂ ➂ ➏ ❼ ➏➂➃➂ Ï ➃ ➅ ➂ s➏➀ ➆ ❿ Õ 4 Ó❶❺ Ï s ➍➆➍➆ ➅ ➆ s r Ò ❿ r ➆❿ Ûß ➍➏ ➃ ❿ ❿ s r s ➆ Ós ➏ ➆➂➈ ❿❽ ➍➆ ❿❽ ➃ ❺èÔ P r t P P r 2 Pt r r r P ➆ P P ➂ ➃ r P➇ r s s ➓ ➐7 r ➄ P9 r 7 9 r P 9 8 s ➇ ➀ P P 9 P ➄s 2 6PtP P ➃ ➒ P ❷s P 8 P P❸ ❹ ❹ q2 P P 7 r s ➀ ➋ P ➄ P9 P r s P ➋ ➋3 8 P ➈➉ q2 34r ➂ P P P ➂ P r ➄ P P r s ➀ P ➄ 8 ➃ 8 P s P ➄ ➂ P P P P 6 r 2 P r 8 P P r 6 6 r ➆ P 7 r P P ➀ 9 ➋ r ➆ P 7 r r

❹ r P P P ➂ s P t P P r r s P t ➀ r 9 P r P P 2➇ ➑ P P r 9P P P P s r 6 P9❻ s 4 ➑ P P P P r ➈❺ r ➀ r r r P P ➂ P P P 9 ❷ P r P r r P P ➀ 2 r 9 2834r r ➀ s P ➒ r ➃ P P P s 4 P ➀ r P P P r 2 8 ➑ P r ➆ 8 P 8 7 5 P 8 P 7 r ➃ P r r 7 r P P P P P ➀ r P 5 ➀ P P P ➀ r r P r P P 2➑34r r P 4 r s r P ➆ P P P P 2➋➐8 P 5 P r P 8 ➂ P P ➀ P P ➒ ➆ P P➄ P P q P➄ r ➀ P ➂ ❿ ➎ q P P ➃ P ➈ q2❼ P❸ 4 4 r P 7 r P ➑ 2 ➋ P r r P r 4 P 4 P P ➀ ➑ s s P 5 P P 2➋34r r ➄s P s r9 P 8 P P ➋ P 4P P P P P ➃ P ➀ P ❿q2 ❾ ➓ q ❼ ➒ q ÐP q ❼ 34r 5 ➀ P ➎ 8 P ➄ P ❽ P P P 8 ➒ r ➆ P P s P P P 2 P r ❸ ➆ P P 9 ❸ P s ➀ P❸ P P❸ ➃ s 4 P r r P ➀ P P 4 P P P P 2 34r 9 s ➀ 8 ➂ P ➇ r ➑ P ➎ 2 ➋ P ➑ P➇ ➑ P 6 r ➄ P 6s ➂ P 2 P P ➑ P P➃ P P r P r P ➋ 4 r P 6 6s t P P 8 9 s P❷ r P ➂ P P r P P9 4 P P 2 34r P P ➀ ➄ P P P P P ➄ P P P ➀ 4 ➀ s 6 2 P r s ➀ ❽ t ❽ r ➂ P P P P ➀ ➁s 6 P P P q 2 P ➀ 9 P ➁ P P P P P P P r ➇ P r 6 6 ➀ P P r ➄ ➀ 8 P P 9 r r r ➂ P s P r r 28 6 P ➋ P 4 r r P r r ➃r r P ➀ P s r P9 8 r ➃ r 2 ➂ P r P 5 P ➋ r 4 P P 6 r P P➃ P ➎ 2 ➋ P ➀ P r P ❽ P P P P P

❹ ❹ P ➑ ➄ r P ➑ 4 r❷ P tr r P P ➀ ❶ P❸ r ➇ P P❸ ➇ ➈➎ q P t ➏➎ ❾ q2 P 9 r r ➃ r r r P 5 P t r ❷ P P P P P 2 ➄ ➃ P r P P P P s r P P 6 P P 8 P P s r P ➒ ➈ q 2 ➆ P ➒P P P r s P P P r ❷ P P r ➃ P P P 2 9 P❶ ➂ 8 r P P P ❻ ➅ ➎ q6 r r ➇ 4 P ❻ r 5 P P9s r P ➆ r P P P P 6 ➆ r ❻ ❺ ❺ P 4 s ➄ ➃ P P P ❻ ❺ ❹ ➎ ❻ ❹ ❿ 34r 7 P 5 P ➌ r r r P r 7 r P r 8 P P P P P P 5 P r P s ➃ ➑ P P 2 34r r P P P ❻ ➅ q P ➇❻ r r P➑ r 8 P P 6 ➃ P r r➑ ➆ 8 ➑ s 8 P P ➒ P P 4 ➃s 2 P➁ P 9 ❻ P ❻ P❶ r ➇ 9 r 9 9 P P 6 ➑ r P r P ❻ ❻ ❻ ❹❾ P P ➀ r P P r r P P P ➀ ❻ ❻ 2 6 6 r 8 r P P P ➋ r r 4 ➄ P P 7 P P ➋s ➃ P P ➀ ➀ 4P P P P ➃P ❻ ❹ ➌ r r s ➇ P P ❻ 2 34r P P ➀ 6 r ➄ P➇ P rt P P r ➀ 6 r ➑ P P P r 6 P ➇ P r P P➄ 6 P P 2 ➆ P P P P s P P P P P r P P 6 s r ➑ ➀ P r ➑ P➃ r ➄ 2 ➓➋➂ r❷ P➇ P➇ ➄ Pt P r ➄P ➂ r 2 ❻

❹ ➆ ➀ P P P P 7 ➀ 7 P P P P P P 7 r r 2 ❼ 8 ➂ 6 r r P ➆ ➑ P ➅❿ q r ➑ ➄ ➀ r P P P P P P P 2➁34r P ❸ P P❶ P❶ P P P r ➊ r r P ❺ s P 7 r ➑ P P 2 ➆ ➋ P 8P P P P ➑ P r r r P P ❺ 34r P t s P ➄ P 2 P r ❿ ➄ r ➀ P P t ➃ P r r P ➇ P r P P P P P ➂ ➃ ❻ ➒❻ P 9s r r P P s r P P P ❶ r s r ➀ P❶s r P ❸ ➃ 4 4 2 P r ➂ 8 6 P P r P r P r P r r s P r ➆ P s P P➃ r 6 P P q P ➄ ➀ r P s P P ➃ 6 ❹ q P r P 4 ➀ s ❾ ❹❹ q2 ❹➉ ➈ Ï ❼ P P➑ r ➒ P P r ➀ 4 P P P r ➑ P P 2 2➀ P P 4 P ➀ 2➄ ➑ r r ➄ ➐7 P ➆ P r r P r r ➂ ➒ ❻ P ➀ ➂ r r r9 r ➑ P P P P 4 ➐7 P P ➂ ➒ ➒❺❻ P9 r 2 34r 8 8 ➃ P P P r 8 P ➇ ➀ ➋ ➏➎ ➌ ➆❿ ❹ ➆❿ q2 P P r s P Pt r P ➄ P ➀ r P r r ➂ P P❷ r ❹ q2 ➑ ➑ P P 7 ➋ P ➂ P P ➇ t➀ 2 P r ts r 8 ➇ s r r ➒ r P 8 tr 2 34r 9 ❸ 9 P tr P P r ➀ ➄ P P P 6 8 ➅ ❹ q 2 34r ➒ P P 6 r r r ➒ P P ➄ P ➒ P P t 7 2 34r P❷ ➂ P P r ➋ r ➆ P ➃ 2

❹ 34r r P s r ➅ ➉q2 P P r P❸ r ➇ r t P ➀ 7 P P r P 4 r 6 s P P➃ r P r r r 4 7 ➀ r r P ➃ Pt P P s P ➄ ➃ ➂ P P r P P P r 2 ❻ ❻ ❹ 34r 5 P❽ s ➀ r 9 P❽ 5 P 2 6P ➆ P ➑s 7 r P P➄ r ➂ P P 2 P➃ P 8 r 7 r P P➃ P ➀ 5 P r r r r s P P P P r P 26➓ P ➋ r P r 6 2 26s P r P ➒ r P r 2 5 P ❹ ➇ r 9 P r ➃ P❷ P P ❻ ❻ ❺ P 3 P ➃ r P r ➄ r ➄❻ r P r r r 6 P➃ ➀ P r ➃ ❻ ❺ 6 ❻ ❻ ❻ ❹ ❻ ❻ r P t ➁ r P r P r P P r P r P 6 P P P 2734r 6 r P r 4 P ➃ P ❺ P 2 P ➀ P r ➂ 9 P P 5 P r P ❹❹ q r 9 ➄ 4s P r 2 P➃ ➂ P9 ➎ P P 7 r ➀ r❷ r P P r r 2 P 7 P P ➇ r r s P P ➒

❹ ➎ 5 Sample realization of EC DSI Estimated variance of the EC DSI with time 5 5 5 4 3 2 2 4 6 8 time 2 4 6 8 time ❻ ➐ ➂➃ ➅ ➍ s r s ➆ ý Ï Û 5Ý s ➍➅ ➅ ➍➆ ➂➃ ➉ ❿ ❿ ➍➅ Ô Ü ➏➂ s❽ s ❿ ü ➅ ➊ r ❿ Ðs ➍➏ ➂ Ù ❿ ❿ ß ➂Û ➆ ü Ò ❽❻ ➂➃ ➏ 69 ➍ r s r s ➏ ❺è➀ Ï ➃ ➓ ➏ s ➏❿ ➃ ❿ s ❺ ❺ ❺ ❺èÔ ❺ s Ï Û ➅ ➊❿ s Ï Û ➅ ➊ r ❿ ❿ ➍➅ ❺ ❺ ❿ 9 s ü ❺5 ❿ ❺ sù ➆ ➏ ❿ s ➍ Ý ➉ Ï ➃ ➅➐ ❿ ➆❿ ➑ ➀q Ô Ô➃➀ ❺ ❺➅Ô Ü s ➆➂ ➀qÑ ❿ Ï❿ s Ý ➉➓❻q ➂➃ ➏ 69 ➍ r s r s ➆ Ï ➆ ➀ s s ➃ ü ➃❿ ➅ ➂ s➓ ➃❿ ➀ ➂➃ ❿ 5 Ï ➃ ➃ s ➆ s ➍➅ ➅ s ➃ ❿ü ➃❿ü ➃ Ô ❾ ➅ s ➂➃ ➎ ➍➆ ➏s s Ï ➃ ý➑ ❹ ❿ ➑❶ ❼ ß ❿ Ï s ➍➅ ➂➃➂ ➃ ❿ s ➍➅ ➍➅ ➆❿ s s➎s Ï❿ Û Ô 4Ï 5 ➏s ❿ ➍➅ ➍➆ ➏s s➆➀ ❿ ➆ ➍➏ ➂ ➂➃r5 Ï ➃ ➑❹ ➁❽ ➏ ➂ s ❿ s➏ô Ü ➂➃ ➊ ➏ ➍ s ➍➆ ➃➂ ➂ ❿ s ❼ ❿ ➍➅ s ➃ ❿ Ý ➉❼ Ò sï ➏➂ÓÔ P r 4 ➋ P 7 P r 4 s P P 5 P 9 r ➀ r 7 ➃ r ➀➍ P r ➃ s ➄ P 8 ➃ P ➈ ➎ q2 34r ➃ 8 s r➃ r ➃ P P P 2434r 6 P P P P r P s P P P ➅➌ ❿ q ❹ r ➑ r 2 2 24 P s P P❷ 24 P r 8 P P ➆ P ➋ P s ➋ P ➀ r r 2 P r r P s P P P P P P s 2 P➑ ❿ 7 r ➒ s r P 6 P r 6 P P r ➑ P P 9 ➃ P➃ r 8 P P 2 P P P P r P P 6 P ➃ ❹ 34r ➂ P ➀ P s r 7 Pt r s ➄ P r 2 P 7 r r P P 7 4P P P s P ➀ P r 6 P P ➄ P P P P 2 34r ➎ ❿

❹ ❿ Weierstrass function λ =.4; H=.6.5.5 log(fourier spectrum) Analysis of a Weiertrass function 6 Fourier spectrum 4 model in f 2H 2 2 4.5 6.5..5.2.25.3.35.4.45.5 Reduced frequency 2.5 2 2.5 Mellin Cyclic spectrum.5.5 3..2.3.4.5.6.7.8.9 time 2 3 4 5 6 7 8 Mellin scale ❾ ❻ ➐ ➂ ❿ Ú ➆ ➆ s s s 35 ❿ ➆➂ Ò ❿ ➍è ➃ ❿❾➀ s ➆ß ❿ ❿ ➏t ➃ ❿ ❹ ➁ ❺ ❿ ➂ ➅ ➀ Ï Û Ö➑ ➁ ➀ ❿ ➁ Ô Ü s ➃ ➂ ❿❽ sï Ï ➃ 5 ➂➃ Û ➏ ❿ ➏ ➏s➏➀ ❿ ➏s9 ❾ ➆ 9 Ò t ➆❿ ➍➆r ➏ ➃❿ Ö❶ ➑ Ô ❿ ➃ 9 s Ï❿❽ ➆➂ ➆ ➏❿ s ❿ ➂ r s s r ➏ ❿ sï ❺➅ ❺➅Ô9 ➆ ➏➀ ❿❽ s ➃ ➒ ➆ s ➏➍➅ ❺➅ ❺➅➀❾s s Ï Û Ù➂ ❿ ❿ ➏ ➏❿ ❿ ➍➅ ➓ ➏➂ Ò s ➏➍➅ ➃❿ s❼s ➍è Øs s ❿ ➓s ➃Ñ ❿ Ò ❾ ➅ Ò ➏ ➏ s ➍➆ ➂➃ ➃ ➂➃ 9 ➆ 9 ➏ ❿➎ s ➃ ➍➅r ➍➆➂➃ ➍ s ➏➍➅ Ï ❺9 sï ➆ß ❿ ➏ ➎ ❿ ❹ ➌ ❺q s ➏ ➏s ➆❿ ➈Ô Ü ➏s ❿ s ß s r ➃❿ ❿ ➆ ➍➆ ➂ ➂ rù ❿ ❿ ➆ s ➏ ❿ s Ò ➀ ❿ ➆❿ 5 ➂ ➒ ➃ ➏ ❿ s Ò s➓ Ù ➏s ➎t ❿ ➃ r ➃❿ ❿ Ó❹ ❹❺èÔ9➒ ❿ ➂➃➂ r➎ s s ❿❽ Ò t ➆❿ ➍➆r Ô9Õ ➆ sï ➂ s r❽➍➅ ❿ s ➆ ➃ sq ❿➎s ➆ ➆ ➂ ➂ ➍ Ñ s❼ ❿ ➏ ❿ ➏s ➂➃ ❼ ➏ q s ➂ ➍ Ñ s➆ô Ü Ñ s s ➆ Ù ❿ü3 ➆➂ ➂ ➃❿ s ➍➆ ➂➃ r ➂➃ Ø❹ ➋ ❸ Ô ➒ ➓ ❿ ➆➐ ➍è ➂➃rÙs ➆➂➃ s ➃➂ ➍➅ s s➏➀ ❿ 9 ➂ 5ß ❿ ❿ ➂ rù ❽ ➏ Ñ ❿ ➑ s ➆rÙ ➏ ➆➂ ➆ ❿ s Ý ➉èÔ P P s 6 ➑ P ➀ P r 6 r r P P P 9 2 34r 8 P s 9 8 P ➋ r P P P r P P ❹ ➋ P ➀ ➂ P 7 P r P P Pt P P P s ➄ 2 P t ➄ P t P r r P ➂ ➆ P r P P s r 2➒34r ➒ ❿q r 6 P r P s P P P r s ➀ P 4 4s 5 P r 2 ➆ ❷ ❾ ➒ q ❶ q P q 9 r 6 r ➒ P P s 6 r ➄ r P P P s P ❸ r P 5 P ➇ ➎ 8 P 2 P P P r ➃ s P P P P r 8 r P ➆ 8P P P r P r 8 ➑ P P ➋ r 7 r ➒ r ➋ P➄ P P P P➇ 2 r 6 P tr s P➇ ts ➇ 2 P r r ➋ ➇ ➈➎ q P ➃ r s ➃ 2 s P 82 P P➃ P r P ➂ ❹ ➌ ❹➉ q2 34r P9 6 8s P s r ➂ P s P 7 P

❹❾ P 8 r 2 ➑ P r P P r 4 9 P P P P r ➀ r P ➃ P P ➄ ➀ r P ➀ s P P 2❼34r ➇ P P s ➇ P P ❷ r P ❸ ➒ P r t ➑ P 8 ➀ ➇ P P P P P P ➎ 2 2 6 P ❷ r 9 P P P P ➀ P P r P➑ r ➋ s 4 P ➒ r 7 s ➀ P P P 2 ➀ ➋ ➑ ➑ ➀ ❷ 34r r P s r P P r 5 P ➀ 234r r P P 6 ❶ P ➃ r P P r ➄ ➑ s P t P P ➄ 6 ➀ r➇ 4 r r P P ➄ ➀ ➑ ❹ ➎q 2 r P r 7 r r r ➒s P 7 6 7 2➋ 8 r P r r r Pt P s ➄ r r r r P P ➄ P P P P ➆s r➇ 6 P P ➇ P P r rt P r 4 P P 2 34r r r 4 9 2 s r ➃ 9 r 8 P P ❶ s r ➃ ➀ P 2 P 7 r 6 4 ➋ r 8 r 6 ➂ r ❷ r r ❸ 27 6P r P P 6 P P 6 ➋ P P P 8 r ➃2 ➁ ➒r r 4 P r 6 7 r 2 P P P P 8 ➄ r P P P q P ➄ P 72➋ ❸ P P P P r P s 2 8 r r r ➂ ❽ P➄ P r P r 6 7 r 8 2 ➆ s 6 r r ➃2 r 6 r r r ➇ P 6 ➂ P r s P 4 ➒ P 4 P r 7 ➃ P 2 P9P ➁ P P P P r 6 P P9 r P ➃ P r ➃ P ➇ 8 P r P ➇ ➀ P 9 r ➀ P 2➑ s ➄ P 8 P 2 r P P r P P P P r P ➂ s ➀ r 4 P 6 ❾ r r P P P P 2 34r P P P s t P P P P➇ r ➄ r P ➋2 3➒ ➑ r ➄P ➑ r P 7 P s P 6 P 7 8 s P P ts P P 2 r r P r r P 9 P 8 8 ➄ P P r 2 r r P P ➑ P P r P

❹ ➌ s r 2 P r s r r ➂ P P r 2 P r ➑P P 6 P s r r ➑ P P P ➀ ➌ 34r ➑ 5 P❷ ➃ P P ❹ 2 6 r ➄ P r 5 9 P P P ➆ 6 s P r ➑ r ➄ P P 6 P P P ❷ ➀ r r ➃ P 2 P 4 P P P P 62 s 4 r P P 2 34r P P ➃ r r 6 r ➂ P r P 4 7 ➀ 34r ➂ P r 8 ➑ P r r ➂ ➄ r ➂ P P P r ➑ ➄ ➀ P P r 6 P r ➑ P 2 P r ➑ P r r r P ➀ 2 r r P 6 P P P P P 2 34r 7 ➂ P ❹ 7 r P P r ➂ P 7 r r 2 ❷r r P P❽ r ➇ P r ❷ P t ➄ r ➆ P P 9 P P ➑ ➃ P r ➑s P P 2t34r 9 P P s r P ❷ r r ➑ P P r ➄ 5 P r r P ❸ ➎ 2❼34r ➇ ➃ P❸s P P 2 r P P P P P P P ➂ s r➇ r ➄ r P ➃ P P ➃ 2 34r ➑ s P s P P P❷ P r ➀ r s 5 P r r P 2❷34r ➂ P r P ➆ P P P P 6 ➄ r P s P ➉ ❹ P ➋ 5 P P 34r r 4 P ➆ P P➄ P P P ➀ P s P➃ ➄ P P r ➒ ➒ P P 2 P r ➎

❹➉ 34r P r r P ➃ P P P s P ➀ 7 P P 7 4 P s s P s P ➃ ➀ 8 P ➆ r ➃ 2 6 r r ➃ r P ❶ P P P 2 ➃ P P P r P P s P r s P r ➅ ❽ P ❷ ➂ 2 P r 8 9 ❸ r 8 P9s 8 r ➄ ➀ ➀ ➈➎ q ❹ ➎q 6 P P P➇ ➄ P ➀ P ➄ ➀ 2➑34r P ➃s P r ➄ r P ➄ ➀ P 4r 8 s ➂ 2 Pt P r P s P r r P ➇s 2 P r r ➃ ➏➎ q P r ts s P P ❹➉ ➒❹➌ q27 P r ➀ ➃ r r ❽ ➀ P s 7 7 r P➄ r 4 2 7r r 7s r ➀ r P P r P 2 P P P ➑ P r r ➀ r r P ❼ r P9 P P ➋ ➀ 4s r P 27 P r P 9 9 P P P ➇ 2 s r r ➒ P ➀ P P P r 2 6 r 7 P P 8 ➄ P P P P P r P P s 2 r 6 P P P ➀ 8 s 6 r P r r P ➑ P r 7 P s P P r P r P ➀ P 2 r r P P➄ r 8 7 P P 4 P r P s r 2 34r ➀ P➑ s s 8 P P r 7 r 4 ➋ P P t P ❽s s s 8 P ❶ P➁ r P r P 7 s s s P 8 ➒ P P r P P s P P r ➄ P r P P ➀ 6 r 2 7 8 9 4 P P s 6 s P r 4 P➃ r r ➇ s r ➇ P P t ➀ r P 2 34r r ➂ r s ➑ P r P ➄ P P P P P 2 4 P P ➀ P r P P q 72 34r P r P P 7 ➂ ❹ ❿

❹ 34r r P➃s r P s ➂ ➂ ❹ ➀ r 34r 7 P r 9 r P P 7 4 r ❼ r ➃ P 7 s 6s P q 2➒34r r ➀➍ 9 P P P r P r 4 r ➀ r P r 4 s P r 5 P s 8 P r ➄ P r ➀ ➑ ➀ r ➃2834r r ➇ P P t s r ➄ s 2 34r P➇ P 9 ➋ s 6 P ➑ ➃ ❷ r P ➄ P ➂ ❶s P 2 s r P P P r ➃ r ➃ P ➄ r P s ➃ r ➀ ➀ P ➀ 2❶3➒ P P P r ➀ P r r ➀ ➄ r ➄P ➀ ➑ r ➀ 2 6 P r ➄ P r ➑ ➄ P 6 P P 2 Pt r r r P t 8 ➈ r P➆ P r 2 9 P P s 8 P❷q 2 q P 2 P r 8 r ➄ ➒ r 8 r P s P P❶ ➅ q P r P P 6 P➃s ➂ ❹ ➀ ➂ ➀ 34r P ➒ r r 4 P➃s ➈ r P 34r 6P ➋ Pts 8 P r ➄ ➀ P t r ➄P r r P P P P P 6 ➑ 2 34r P➄ 8 7 r P s r r r 6 ➀ ➀ r➃ r ➄ ➀ P 2 34r P r r r P ➀ 6 P ➃ s r 6 r 6 P ➑ r P 7 P➄ r P r P9 ➒ P P 2 8 8 r 5 8 9 P9 ➄ ➀ P P 7 9 ➂ r 7 P s r➃ ➀ P P 2 34r P P➄ r 4 P P P P r P P r 8 P P P 6r P s 4 r ❶ P P P ➀ P s ❹ P s ❽ ❶ 8 ➀ r

P P P r P ➀ 2 P r ➂ 8 r r P P ➃ r P Pt s P r P P P P 8 P P❷ P r P ➆ P r 2 7 P9 ❾➄ ➂ 7 P P P9 r P ➀ P ➒ r s P P➃ P r 6 P➃ 7 P P➃ 2 P 2 fractional Brownian motion, H=/3 2 fssi fbm,a ± =±.5 2 X = 2,X + =5.5 3.5 4 2 4 6 8 2 4 6 8 fssi fbm, a =37(au),a + =+ fssi fbm, a =37(au),a + =+.5.5 X ± =± X = 2,X + =5.5.5 2 2 4 6 2 2 4 6 X ± =± 2.5.5.5.5 2 fssi fbm, a =37(au),a + =45(au) 2.5 2 4 6 time, a.u. X = 2,X + =5 2.5.5.5.5 2 fssi fbm, a =37(au),a + =45(au) 2.5 2 4 6 ❻q➐ ➂ Ò ➍➅ ❿ ➂❾ 9 Ï❿ ❿➎ ❿ sï Ï Û ➓ß ❿ Û ➓s ➃Þ➏ s ➍➆ ➂➃ ❿ ❿ ➍➅ ➏ rô Ü ü➂➃ ➆ ➏ 2 9 Ï Û ➌ ➀ ➂ ➆ ➅➐ ❿ ➏❿ Ò ➓ ➆➂ ➍➅ ➃ r ❿ü Ò ➂ ➂➃r ➆ ➆➂ ➏ ➂ ➆❿ ➍➆ Ô➓3 ➂ ➂➃ ➆ ➏ s 2 9 ➆ ❿ ➀ s ❿ ➏ ➏➂➃ Ô 9 ➂ ➅ s 2 9 9ß ❿ Û ❼s ➃Þ➏ q ❿ ➆ ➂ Ô Ü ❾s q 2 ❼ 9 ➏ ➃❿ ➂ Û 9 ➆ ❼ ➏ ➆❿ ❹ ❿ ❶ Ô 3 ➂ ❼ ➃ ➏ s ❼ 2 ❼ ➏ ❿ ➂➃ ➃ 9 ❿ ➅ 9 ➆ ➏❿ ❹ ❿ ❶ ➀ ❿ ➓ ❿ ➓ ➆➂ ü ❿ Ô 9 ➃ ➏ ❾s 9 s ➂ ➂ ❿ ➎ ❿Ù ß ❿ ➃ ➃❿ ➏ ➂ q Ô Ü Ù➂➃ ➆ ❿ ➆➂ s➍➅ ➏s ❿ sï ➎ s ➂ s ➏➂Û s ➃➂ ➍➅ ➏s s ➏s➏➀ Ï ➆ ➏ s❾ 9ß ➏ s ➆ ➍➅ s ❿ s s ➍➆ ➏s s s➈ s s ➏s s s Ñ ❿ ❽ ß ❿ Û s Þ➆ s ➍➆ ➂➃ ➆ ❿ ❿ ➍➅ 2s s ❺èÔ

❹ 5 P❶ ➎ r ➄ P P P P s s P ➃ r ➑ P P r P 9 r ➄ P ➃ P 2 34r r ➀ P P r 4 P r 6 6 s P 9s P r 7 P P P 8 P r P P r P ➀ r s P P r r r ➑ r ➂ P 4 8 9 2t 7 r 9 P r r P P P r P s 6 7 P 85 P P 2 3 6 r P ➒ r ➃ P r 4 r ➀ 8 s P P➇ ➄ P P s P P ➀ ➑ t 9 ➋ r P P P P 2 P➄ r r 4 s P P r 4 2 34r 7 r 5 ➀ ➍ 2 r P❷ r 9 P 7 P P r P r P P P 6 P➃ 2 6 P 4 r 8 r❸ P P P ❽ ❷ P r P ➇ s P r ➑ 8 r s 8 P r 8 r P P P 2 5 P❸ ❹ ❹ ➆ r P P s r 2 P r ❸ P ❶ r r ➑ ➆ ➇ P❷ ❹ ➉ ❹➌ q➒ 6 P P ➀ 2 r P r ❽ r s P r r ❼ 4 8 2 ➑ s P r P ➅❿q➒ r➇ ➀ ➑ r 8 ➀ P s 9 r r ➋2 r P➁ 8 P ❷ r ➀ P r s s 9 ➇ P❸s s P s 7 ➑ ❼ P 8 2 8 ➂ s P P r➇ P P 2 P P P ➀ P P 7 s s P ➀ 9 P P 4P P P 2 34r r P P s s 9 P P P ❺ ❺ 2 34r r P P s s ➀ P P P ❹ ➋ ➌ r ❸ P P 2 34r r ❹ ➌ q s r ➃ r s s ❷ 2 6 9 P P 8 P ➌ r P P ❹ ❺ P r 8 r P ➄ r r P P P P 2 34r P P P P P P r r r P9 P r P ➀ r P 7 P r r 8 2 34r 6 7 P r s ➆ r 2 ❾ ➈ ➒ 9r r 9 9 ➊ ➂ P ➇ r 9 ➆ P P P r P t P ➑ 8 s ➑ ➃ r P

x x non linear amplitude warping, X =-2 and X =5 X =-2 and X =5 - + - + 5 2 4 3 2 - -2 - -8-6 -4-2 2 4 6 8 y p.d.f..5.5 - -8-6 -4-2 2 4 6 8 x non linear amplitude warping, X - =-2 and X + =2 X - =-2 and X + =2 2.8.6 p.d.f..4 -.2-2 - -8-6 -4-2 2 4 6 8 y -.2-4 -3-2 - 2 3 4 x ❾ Ü ß ❿ ➂➃ ➆ ➎ ➏ ➍➅ s 5s ➍ ❿ ❿ ➂ ➃❿ ➏ ➃ r s ➏ ü ➂ Û s ➃ ❿ ➂ s➏➀ Ò 9 ➍➆ s s➆ô Ü Ïß Ï ❿ q ➃ ➏s ➍➅ ➏s ❿ ❿ ➃➂ Û r ➆❿ s Û r Ò ❿ ➍➅ ❿ sï ➃❿ ➏ ➈➀ Ï ➏❿❽ ❿ ➃ ➂ s ➃ ❿ ➂ s s s ❿❾Ô r ➂ P ❷ P ❼ P P ➀ r s P ➃ P ➀ 2 P❷ P P ➀ r ➑ P 8 r 8 t r ➑ ➂ P ➀ ➃ P P ➑ P t ➋ P ➄ ➄ ➑ P P 8 7 P ➆ P ➒ ➆ r ➒ r P ❼ 4 r❽ s r ❹ 6❿ q➑ P ❽ P ➇r P s P P P r 7 r ➏➎ ❹➌ q2 P 8 8 r r P 9 P P P 2 r r 7 8 ➄s ➄ r P r P ➑ P ➑ P P P P P t Pt ➑ ➇ P 5 P 2➄ r P P 9r s P s P ➁ P ➁ r r r r ts P t P P P P r 2 34r 4 r ➋ 8 ➄ ➋ ➄ ➀ 4 r P P r s ➇ P r ➄ r P ➀ r r ➆ ➒ P P P ➆ P ➆ r P r s P➄ P P P s r 2 Ï ❼ 2➅ 2 P r ➋ r ➃ r 3 s P 8 P r ➒ r 4 8 2 ❾ Ô Õ r➀ Ô ➒ ➂ ❿ ❿❾➀ 3üÔ Ü t t ❾➀ ❿ Ý➓Ô ➆ ➃ ➍ ❾Ô Ú ➆➂ ➅ s Ò ü ❿ ➂ r s s➆➀ ➏s ➃ ❿❾➀ ❿ s r ❿ ➏s s s ➍➆ ➂➃ ❿ Ô ➉ ❿ Ô❾ Ñ5 ❿ Ú Ô❾Ú ➂➃➂ ❿ ➏ ➏➀ ➏ ➃ s➏➀ Ô Ú ➃➂ ➆r➀ ❹ ➑➑ ➑ Ô ❹ Ô ÔÕ ➓ ➂ ➈➀ Ô 9 ❿ ➏➀ ❿ Ô ➒ ➂ ❿ ❿❾Ô ➍ s ➍Ï ➍➅ ➏s s ➏s Ï ➃ ß ❿ Û s ➃Þ➏ s ➍➏ ➂ Ï ➃❿ ❿ ➍➅ Ô ➉ ❿ è➀ ➂ ❶ ➏➁ ➀❾ s ➏❷ ❸ ❹ ➀➈ ❿ ➒ ➀ Õ ➀ ❿ ❹ ➑➑ ➌ Ô

➏❷ ➀35 r ➌ ❹ ➌ ➀35 ➍ ❸ ❸ ➌ Ô ÔÕ ➓ ➂ ❿ Ô3 ➍ ➏➂ÓÔ ➆ s s s ➃ ❿ s ❿ ➏s ➓ ➂➃ ➂➃ ❿ ➄ s➆ô ➉ ❿ ➀ ➏s ➋ ❹❹ ➀ ➏➋➋ ➋ Ô ➁ 4 Ô ❿ ➃ Ñ ❿ Ý Ô ❿ ➏s➏Ô Ú ➏ ➎ ➆➂ ➅ ❼ s ➏s➏ ❿ ➃ r t ➃ ➂ ➆❿ ➍➅ ❿ ❽s ❿ ➂ ➍➅ s s ❿ Ô ➉ ❿ ➀ ➂ ➉ ➉ ➉è➀ sï➌❹ ➑ ➏➋➋ ➁ Ô ❶ ➓ Ô ➏❿ ➂ ➏Ô èôù69 ➀ 69 ➓ ➀ ➋ ➋ ❷ Ô ❷ Ô 9 ➆ ❿❾Ô èô56❼ ❿ ➂ ➂Ó➀ 8 ➆ q Ñ ➀ ➏➋➋ ➁ Ô à❸ 3üÔ 9 ➆ r ❿ Ô ❾ ➆ Ï s➏ô ❿❽ Ú ➆ ➆ s s s ➊3Ù ❿ ➏➂➃ ❼ Ò ➍➅ ➂❾ Ò ❿ ➍è ➃ ❿❾Ô ➀ ➌ ❸ ➑ ➁ ❶ ➋ ➁ ➁ ➀ ➏➋ ➑ Ô Ô 9 ➆ ❿ ➈➀ Ô 9 ➆ ❿ ➈➀ ❿ Ô Ô ➂ ➆Þ Ô Ý s ➍➅ ➅ 3 ➆➂ ➂➃ ❿ ❿ s Ò Ò qs ❿ ➂ ❿ ➂➃r s s➏ô➉ ❿ ➀ s ➏❷➑ ➌ ➏❷➑ ❷ ➀ ➏➋ ➋➑ Ô ➋ Ô 9 ❿ Ô r ➄ ➆ ➃ s ➍ s ➏❿ s ➃ ❿ ❿ ➆➂ s ➆ ➄ ➏❿ ➄ ➏ ➆ s s ➃ ❿ ❿ ➏s➏Ô ➉ ❿ ➓Ô ➀ ➏ ➃ ➏➀ ➆➀ ➏s ❹ ➌ ➌➑ ➀ ❹ ➑ ➑ ➁ Ô ➏➑ Ô➈ 9 ❿ ➏➀➈ Ô➈➒ ➂ ❿ ➃❿❾➀ ❿ Ô Ô➈Õ ➓ ➂ ➈Ô ➍ s ➍ s ➍➅ ➅ s ➍➆ ➂➃ ❿ ❿ ➍➆ Ô ➀ ➋ 2❷❺➅ ➏ ➁ ➀ ❿ ➓❹ ➑ ➑ ❹ Ô ➃ ➏ 9 ❿ ➏Ô ➆Ô q Ý ➏s s➆➀ ❻❼➍➅ ➂➃ ❿ ➂ s ➄ ➄ ➆ ➃ ➏ r ❿ ➀ 8 ➏ ➓ ➆ ❹ ➑➑❹ Ô ❹ 6 Ô 9 ➍➏➍➅ ❿ ❿ Ô ➓ ➆➂ ➂ Ô➒ ➊ ❿ ❿ ß ➂Û ➆ ➃❿ Ü ➆ r➓ ❿ ➂➃ ➍➆ ❿ s➏ô ➀ ➌ ➁ Ó❷❺è ➏❷ ❹ ➏❷ ❹❸ ➀ Ý ➍➅ ➆ ➏ ➏➋ ❷ Ô ➏➌ Ô ❼ ❿ ➍ Ñ Ó➀ 3üÔ 35 ➃ ➀ ❿ Õ Ô Ú ➆ ❿❾Ô Ü ➆ ❿ s Ò ❿ Ò 9s ➆➂➃ ➊s ➂ ➍➅ s s ➏s➏Ô ➀ ➁ ➁ ❹❶ ➁ ❹ ➀ ➏➋ ➋ ❸ Ô ➆➁ Ô 6❼ ❿ s➏➀ 4 Ô 4Ï ➃ ➀ ❿ ➎ Ô Õ r Ô ➍➏ ➂ ❿ ❿ ❼ ❿ ß ❿ Û ➆➂ r s ➃ ➂ ➍➆ s ➍➆ ➏s➏Ô ➉ ❿ ➅➀ ❿ r ❹ ➑ ➑➌ Ô ❶ ➉èÔ Ý ➏➍ ➃ s➆ô èô ➉ Õ 3ü➀ ➏➋ ➋ ❹ Ô ➏❷ Ô Ý ❾Ô èô Ú ➃➂ ➆r➀ ➏➋❷ ❸ Ô ❸ Ô Ý ➂➃➂ Ô ➒ ❿ Û s ➍➏ ➂ s Þ➆ ❿ ❿ ➍➆ Ô ➀ ➆➁ à❸ ❶❸ ➀ ❹ ➑ ➑➑ Ô ➏ Ô Ý ➂➃➂ ❽ ❿ ➒9Ôq ❿ ➏ ➏Ô s s ➃ ➂ s s ➍➏s s ➍➆ ➂ ❿ ❿ s r s ➏ s➏ô ➆➀ ❷ ❶ ❺è á❸ ➋ ❸ ➋ ➋❷ Ô ➏➋ Ô Ý ➂ ➂➃ ➀➈➒❼Ô➈ ❿ ➏ ➏➀ ❿ Ý Ô❾ ❿ ➆ ➀ ➃ s➏ô Ô❾❻❼Ý ➍➅ ➆❿ ➍➆ ➏s ➃❿ ➆ ➀ ➏➋➋ ❸ Ô ❹ ➑ Ô ❻ ➓ ➏➍ s ❿ ➎3 Ô 35 ➃ Ô Ô ❿ ➍➅ ➆ ❿❽ ❿ ➃ ➆ s ➃ r q ➏s s➏➀ ❹ ➑ ➑ ❹ Ô ❹ Ô ➒ ➂ ➍➅ ❿ ➆ Ô Ô❽Ú ➂ ➆r ➀ ➏➋ ➋➑ Ô ❹ ❹ 3üÔ9➒ s ➆ ÔØ ➍➆ ➂➃ ❿ ➀9 ❿ ➃ ➆ s ➂➃ ➃ r ❿ ➏❿ ➂ ➃Þ ➃ ❿ ➆ rô ➉ ❿ ➒❼Ô Ô Ú Ô9 ❿ ➀Ï Û ➀ ➀❼ ➂➃ ➏ ❷ è➀9 ➏s ➌ ➋ Ô ❿ ➏ ➆ ➂ ➀ ➏➋ ➌ Ô ❹ ➌ Ô ➒ ➂ ❿ ❿❾Ô ➍➆ ➂ ➅ ➊ ❿ ❿ ❼Ú ❿ ➆ s ➍è ❿ ❽s ➆➂➃ ➊s ➃ ➂ ➃ r Ô➎➉ ❿ qô Ü s ➀ ➃ ➀ è➀ ➏s ➆➁ ➋ ❶ ❹ Ô ❻ ➂ s ➆ ➆ ➀ ➏➋ ➋➑ Ô ❹ ➁ Ô ➒ ➂ ❿ ❿❾Ô P èô Õ ➍➆ ➆ ➍ q ➏s s➆➀ ➋ ➋ ➋ Ô ❹ ❶ Ô ➒ ➂ ❿ ❿ ❿ ü Ô 9 ❿ ➏Ô ❿ ❽➍ ü ➍➅ s Û ➃ ❿ü ÏÚ ➆ ➆ s s s 35 ❿ ➆➂ Ò ❿ ➍è ➃ ❿ s➏➀ ❿ ➆ ➅ Ò t ➆❿ ➍➆r ❿ ➆ ➅ ❿❾Ô ➀ ❶ ➌ ➁ ➆➁ ❷ ➀ ➆ ➆ ➏ ❹ ➑➑ ➌ Ô ❹ ❷ Ô9➒ ➂ ❿ ❿❾➀ Ô9 ❼ ❿ ➀Ï ❿ Ô Ô9Õ ➂ ❾Ô ➒ s ➃ ❿ ➃ rü Ðs ➏➂Û s ➃➂ ➃ r ➀Ï ❿ ➍ Ñ q ❿ s9 ❿➎ ➏ ➈ ❿ s Ò ➃ ❿❾Ô ➉ ❿5 Ô 4 ❿ ❿➎ ❿ 3üÔ Ý ❿ ➀ ➃ s➏➀ è➀ ➂➃ ❷ ❹ è➀ ➏s ❸ Ô ❿ ➏ ➆ ➂ ➀ ❿ ➓❹ ➑➑ ➌ Ô ❹❸ ➓Ô ➒ s ➍ ❾Ô ➅Ô569 ➀ 69 ➓ ➀ ➏➋ ➋ ❶ Ô ❹ Ú Ô ❿ ➆ ÔÖÜ ➎s ➏➍➅ ➂ ➍➆ ➏➂ ❿ ➏ r Ï➍➆r ➍➅➂ s ➃ ❿ r ➃ ➅ s ➆ ➃ s➆ô ➀ ➌ ➌❷ ➀ ➏➋ ❷ Ô ❹ ➋ Ú Ô➈ ❿ ➏ ❿ qô ➒ ❿ Ñ s➏ô6❼ ➍è ➏ Þ➏ ❿ ➍➆r ➍➅➂ s ➃ ❿ r ❿ s ❿ ➂ ➍➅ s s ➏s➏Ô ➀ ➉ Ü9 ❹ ❺➅ ➁ ➏➁ ➀ ❿ r ➋ ❸ ❶ Ô ➌ ➑ Ô ➃❿ ❿ 3üÔ 4 ➍ Ô ➍è ➂ ➏❿ s Û r s ➃ ➃ ❿ Ò ❿ s ➂ ➃❿ Ò 9 ➓ ➂Û ➃ ➂ ➍➆ ➃

➎ s ➃ ❿ r ➍➆ ➏s s s➆ô è➀ ➁❷ ➏➑➑ ➋ ➏➑ ❹ ➀ ❹ ➑ ➑➌ Ô ➌ ❻ Ô ➂ r s ➏ Ô❽ ➏ ➍➏ ➂ ➂➃r ➍➆ ➏➂ ➏ ❿ s t ➆❿ ➍➆ ➏s➏Ô ➀ ❹ ➌ ❶ ➌ ➀ ➏➋ ❷ Ô ➌❹ ➓Ô Ôq rü ❿ 8 Ôq➒9Ô ❿ Ô ❿Ð ➍➆➂ s s ❿ ❿ s ❿ r ➍➆ ➏s s s➆ô ➀ ➋ Ó❹❺è ➏➌➌ ❶ ➁ ➀ ➏➋ Ô ➌ ➌ ❻ Ô ❿ ❿ ❿❾Ô ü ➆ ➏s ➆❿ ❿ s ❿ ü ➂ ➃ ü ➃➂ ➃ r Ô➉ ❿ Ô ❿ Ó➀ Û ➀ ➀ ➂ ➌ Ô 3 ➆ ➆❿ 6❼ Ô➃➀ ➋ ❷ ❶ Ô ➌ ➁ Ô ❿ ➀ Õ➓Ô ❿ s ➆❿❾➀ 3üÔ Ú Ô ❾ ➏ ➀ Ô ➂➃ ➏ ➏➀ ❿ üý➓ô ❿ ➆ ÔÖÕ ➃ 2 ➍➅ ➂ ➂ ➆ ➃ ➍➅ ➃ rù ❿ ß ❿ ➃ ➆ ➊s ➃Þ➏ 5 4Ï ➆➂ ➆ ❿ ➍➅ Ò ➏ t Ñ ➏ s ➍ Ñ s➆ô ➀ ➏➑ ❶ à❹ ❶ ➁ ❶ ❹ ❶ ➁ ❸ ➀ ❹ ➑ ➑➑ Ô ➌❶ Ô ➆ Ô ➆ ➃ ➊s ➂➃ s ➍ s ➍ ➍➅ ➏s s ➏s➏Ô ➀ ➏➑ ➁ ❷❹ ❸ ➀ ➏➋❷❹ Ô ➌ ❷ ➓ Ôá6 Ô ❾ ❿ Ö➈Ô Ô 35 ❿ ❿ rô ➆❿ ➏ ➂➃ Þ➆ ❿ s ➏ ➃❿ ➊ ➂➃ ➏❿ ➍ Ñ ➍➅ s s ➏s➎ ❿ s s ➍➅ ➏ s ➆➂➃ s ➃➂ ➍➅ s s ➏s➏Ô á➀ ➌❷ ➌ ➋ ❷ ➌ ➋ ❹ ➀❹ ➑ ➑➌ Ô ➌ ❸ 3üÔ ➏ Ô Ô5Ý➓Ô ❿ 8 s ❿ ❽6❼ ❿ r ➀ ❿ ➍➆ ➅ ❿❾➀ ➏➋❷❹ Ô ➌ ➓ Ô 35 ➂ ➂ ➏Ô Ô Õ ➍➆ ➆ ➍ q ➏s s➆➀ ➋ ➋ ➋ Ô ➌ ➋ Ô 35 ❿ ➆➂ Ô ➆Ô➎➒ ➏ ➆ ❿❾➀ ❿ ➒ ❿ ➍➆ s ➍➅ ➀ ➏➋ ❹ Ô ➁➑ Ô 35 ❿ ➆➂ ➏ÔÏÕ ➓ ➂➃ ➃ Ò ➍è ➂ 9 ➂➃Ñ Ï❿ 9 ➂➃➂ s ➏ ➅ Ô ➀ ❹ ➑ Ó❹❺è á❸ ➑ ❸ ➌ ➀➆➒ ➏ r ➏➋➋ ➋ Ô ➁ Ô 35 ❿ ➆➂ 9 ❿ Ô q ❿ 8 s s➏ô ➒ ➍è ➃ ❿ ➂ 9 Ï❿ ❿ ❿ s➏➀ Ò ➍➅ ❿ ➂ ❿ s ➏s9 ❿ ➂➃ ➍➆ ❿ s➏ô ➀ ➑ 2➁ ❺è ➁ ❹❹ ➁➌ ❸ ➀ ➋ ❷ Ô ➁ ❹ ❻ Ô 8 ➅ ❿ 3üÔ ➂ ➃ ➏➂➃ Ô➎3 9➄ ➏➂➃ s ❿ ➅ s r ❿ ❾ ➏s ➈➇ ❿ ➍➆➂ s s s ❿ ➐ s ➃➂ ➏s❼ ➅ ➄ ➆ ➂➃ ❿ Ô➉ ❿ ➀ ➏s➌ ➑ ➌ ❶ Ô ➉ 8 4Ï➉ Õ ➀ ➏➋➋ ➋ Ô ➁➌ Ô 8 ➂➃ Ô èô5ú ➂ ➍➅ ➆❿ ➃ß ➍ ➀ ❿ ➀ ➏➋ ➋➌ Ô ➁ ➁ 6 Ô 8 Þ➆ ❿ ❿ ➓Ô Ô ➃❿ ➏ ➏s ➃ ❿➓ s ➏➂Û s ➃➂ ➍➅ ➏s s ➏s❾ ➆ ➇ s ❿ s Ò ➃ ❿❾Ô ➀ ➌ ❸ Ó❹ ❺➅ ➁ ❹ ➋ ➁ ❶❹ ➀ ❿ ❹ ➑ ➑➑ Ô ➁ ❶ Ô Ñ ❿ ➓Ú Ô Ú ➂ ➂➃ ❿ ➏ ➏➀ ➏ ➃ s➏ô ÔÚ ➃➂ ➆r➀ ❹ ➑➑ ➑ Ô ➁❷ Ô ➏ ❿ ❿ 4 Ô ➆❿ s s Ô 4Ï ➏ ➍➅ ❿ ❿ ❿ s ❿ r➓s ➍ s ➍ ➍➆ ➏s s s s ❿ Û r r ➅ Ò ❿❾Ô è➀ ➁➌ ➌ ➋➌ ➌ ➋ ❸ ➀ ➏➋➋ ➋ Ô ➁ ❸ ➓Ô ➂➃ ➏ ❿ Ý➓Ô ❿ ➅ Ô 6❼ ➂ ➅➐ ➅➐ ❿ ➆❿ sq ❿ ➂ ➆ ➍ ➍➅ ➏➍➅ ❿ s ❿ Ò s s r s ➆ s➏ô ➅➀ ❷ ➌❹❸ ➌ ❶ ❶ ➀ ➏➋➋ ❷ Ô ➁ ➓ Ô ❿ ➃ s Ñ r ❿ 3üÔÜ t t ❾Ô ➅ÔÙ6❼ ❿ ➂ ➂Ó➀ ➏➋➋ ➁ Ô ➁➋ Ô s ❿ ❿ Ô ❾Ô 8 ❿ ➅ ➍ ➏s ➃ ❿ ➎❿ ❿ s ❿ r s ➂ ➍➆ ❿ ➍➅ s ➍è Ô ➀ ❸ ➏➑ ➏➋ ➀ ➏➋ ➋ ❹ Ô ❶ ➑ 4 Ô➈ ➂➃ ➆ ❿❾ÔÙ ➍➆ ➂ ➂ r❽s ❿ r➎ ❿ ý ➍➆ ➏s s s➆ô ➀➈➌ ➏ ❹ ➏ ❸ ➀ ➏➋ ❶❸ Ô ❶ Ý Ô ❿ ➅ ÔÖÝ s ➍➆ ➆ s ➍➆ ➂➃ ❿ ❿ ➍➅ ❿ ü➍➅ ➂ ➅➐ ➏❿ s ➃ ❿ s➆ô è➀q❹ ➋ ❸ à❹ ➌➋ ❹ ❸ ➑ ➀ ➏➋➋ Ô ❶ ❹ Ý Ô ❿ ➆ Ô Ô5 ❿ ➏ ➆ ➂ ➀ ❹ ➑➑ ➑ Ô ❶ ➌ Ú Ô ➏ ➏Ô ➏ ➃ s ❼ ➆❿ ➏ ➂ s ➆➂➃ ➊s ➂ ➍➆ ➏s s s➆ô ➀ ❶ ❹ ➋ ➋ ❹ ❷ ➀ ➋ ❸ Ô ❶ ➁ Õ➓Ô Ï➄ ➍➆s ❿ Ô ÔØ35 7 ➏s ❿ 5 ➆ ➏ s 2 9 ➍ Ï ➃ ➆ ➆ ➍➏ ➂ s ➂➃ ❿ Ô❽➉ ❿ Ô ❿ ❿ ➀ 8 ➆ ➏ ➏➋➋ ➋ Ô ❶ ❶ Õ➓Ô Ü Ô Õ Ô Ú 5 ❿ Ù Ô❾6❼ ❿❾Ôü ➓ ➂ ➃ ❿ s ❿ r➎ ➍➆ ➏s s ❿ ➑ Ô ➀ ➌ ➁➑➋ ➁ ➌❹ ➀ ➏➋➋ ➁ Ô ❶ ❷ ➓ Ô Ú ❿ ➏➂➃➂ Ô Ô5 q ➆❿ ➍➅ ➂➃➂ ➀ ➋ ➋ ❷ Ô ❶❸ Ô Þ➆ ➍➅ ❿ 4 Ô qô s r ❾Ô Õ ➍➅➂ s s➓ s ➍➅ ❿ ➊ ➆ ➓s ❿ rs ➆➂➃ ➊s ➃ ➂ ➍➅ s s ➏s Ò ➏❿ ➆❿ Ô ➀ ➁ ❶ Ó❹ ❺➅ ➌➋ ❷ ➁ ➑ ➀ ➏➋ ➋ ❸ Ô