PMC:1647278 / 75388-99059
Annnotations
{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/1647278","sourcedb":"PMC","sourceid":"1647278","source_url":"https://www.ncbi.nlm.nih.gov/pmc/1647278","text":"Appendix A\nWe give here the expression of the covariance matrix C introduced in section \"distribution of N = (N0, N1)\". The sequence Y (of length n) is generated by an homogeneous, stationary and ergodic order m Markov model of parameter π and stationary distribution μ. We want to compute the covariance of the vector N of random frequencies of size m and m + 1 words.\nFor any word w (of size hw), we introduce the following notation for hw ≤ i ≤ n\nI i ( w ) = I { w e n d i n p o s i t i o n i } = I { Y i − h w + 1 i = w } ( 45 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7BBF@\nwhere Yij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGzbqwdaqhaaWcbaGaemyAaKgabaGaemOAaOgaaaaa@30CC@ = Yi ... Yj for all i ≤ j. If hw ≥ m, we denote by\np ( w ) = μ ( w 1 m ) Π ( w 1 m , w m + 1 ) ... Π ( w h w − m h − 1 , w h ) ( 46 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5F8B@\nthe probability to see one occurrence of w at a given position in the sequence. At last, if we consider another word v (of size hv = m) and if hw = m, we denote by\nΠ δ ( v , w ) = ∑ x ∈ A δ p ( v x w ) ( 47 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaacqqHGoaudaWgaaWcbaacciGae8hTdqgabeaakiabcIcaOiabdAha2jabcYcaSiabdEha3jabcMcaPiabg2da9maaqafabaGaemiCaaNaeiikaGIaemODayNaemiEaGNaem4DaCNaeiykaKcaleaacqWG4baEcqGHiiIZimaacqGFaeFqdaahaaadbeqaaiab=r7aKbaaaSqab0GaeyyeIuoakiaaxMaacaWLjaGaeiikaGIaeGinaqJaeG4naCJaeiykaKcaaa@556F@\nthe probability to see occurrences of v and w separated by a gap of length δ.\nFor any words v an w (to simplify, we suppose that hv ≥ hw) then, for all δ ∈ ℤ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=rsiAbaa@3772@ and\nmax(hv, hw - δ) ≤ i ≤ min(n, n - δ) we have\nE MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=ri8fbaa@388C@ [Ii (v) Ii+δ (w)] = Dδ (v, w) (48)\nwhich do not depend on i.\nIt is therefore easy to show that\nE [ N ( v ) N ( w ) ] = ∑ i = h v n ∑ δ = h w − i n − i D δ ( v , w ) ( 49 ) = ∑ δ = h w − n n − h v N δ D δ ( v , w ) ( 50 ) = M ( v , w ) + O ( v , m ) ( 51 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeGabaa9fuaabmqadiaaaeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=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@9BD3@\nwhere the main part (2n - hv - hw + 2 terms) is given by\nM ( v , w ) = ∑ δ = h v n − h w N − δ D − δ ( v , w ) + ∑ δ = h w n − h v N δ D δ ( v , w ) ( 52 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6D1F@\nand the overlapping part (hv + hw - 1 terms) by\nO ( v , w ) = ∑ δ = − h v + 1 h w − 1 N δ D δ ( v , w ) ( 53 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFpbWtcqGGOaakcqWG2bGDcqGGSaalcqWG3bWDcqGGPaqkcqGH9aqpdaaeWbqaaiabd6eaonaaBaaaleaaiiGacqGF0oazaeqaaOGaemiraq0aaSbaaSqaaiab+r7aKbqabaGccqGGOaakcqWG2bGDcqGGSaalcqWG3bWDcqGGPaqkaSqaaiab+r7aKjabg2da9iabgkHiTiabdIgaOnaaBaaameaacqWG2bGDaeqaaSGaey4kaSIaeGymaedabaGaemiAaG2aaSbaaWqaaiabdEha3bqabaWccqGHsislcqaIXaqma0GaeyyeIuoakiaaxMaacaWLjaGaeiikaGIaeGynauJaeG4mamJaeiykaKcaaa@5444@\nand with\nN δ = { n − h w + 1 + δ δ ∈ [ h w − n , h w − h v [ n − h v + 1 δ ∈ [ h w − h v , 0 ] n − h v + 1 − δ δ ∈ ] 0 , n − h v ] ( 54 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7F57@\nAs we have\nC(v, w) = M (v, w) + O (v, w) - E (v) E (w) (55)\nthe problem is hence to compute M and O for all pairs of size m or m + 1 words. In order to simplify, we will just treat here the case of a pair of size m words (other cases can be derived from this special case).\nFor the main part we obtain\nM ( v , w ) = ∑ δ = m n − m N − δ μ ( w ) Π δ − m + 1 ( w , v ) + ∑ δ = m n − m N δ μ ( v ) Π δ − m + 1 ( v , w ) ( 56 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@78C7@\n(2n - 2m + 2 terms). As Pk (v, w) quickly converges toward μ(w) when k grows (convergence speed is given by λk where λ is the magnitude of the second eigenvalue of the transition matrix Π). So there exists a rank r ≥ m such as\nM ( v , w ) ≃ μ ( v ) μ ( w ) ∑ δ = r n − m ( N − δ + N δ ) + ∑ δ = m r − 1 N − δ μ ( w ) Π δ − m + 1 ( w , v ) + ∑ δ = m r − 1 N δ μ ( v ) Π δ − m + 1 ( v , w ) ( 57 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@981D@\nwhich has only 2r - 2m + 1 terms.\nAnd for the overlapping part we get\nO ( v , w ) = N 0 × μ ( v ) × I { v = w } + ∑ δ = 1 m − 1 N − δ × p ( w v m − δ + 1 m ) × I { v 1 m − δ = w 1 + δ m } + ∑ δ = 1 m − 1 N δ × p ( v w m − δ + 1 m ) × I { v 1 + δ m = w 1 m − δ } ( 58 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@BCF0@\nwhich has 2m + 1 terms.\nSo the overall complexity for the computation of one term of C is hence O(r) where the value of r is directly connected to the magnitude λ of the second eigenvalue of the transition matrix.\nIn the particular case of an order one Markov model (m = 1), we give here the complete expressions of M and O.\nFor all a, b, c, d ∈ A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFaeFqaaa@3821@, we have\nM ( a , b ) ≃ ( n − r + 1 ) ( n − r ) μ ( a ) μ ( b ) + ∑ δ = 1 r − 1 ( n − δ ) ( μ ( b ) Π δ ( b , a ) + μ ( a ) Π δ ( a , b ) ) ( 59 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaGqabiab=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@7BAA@\nO (a, b) = nμ(a) I MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=Hi8jbaa@3894@{a = b} (60)\nM ( a b , c ) Π ( a , b ) ≃ ( n − r ) ( n − r − 1 ) μ ( a ) μ ( c ) + ∑ δ = 1 r − 1 ( n − δ − 1 ) ( μ ( c ) Π δ ( c , a ) + μ ( a ) Π δ ( b , c ) ) ( 61 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8595@\nO ( a b , c ) Π ( a , b ) = ( n − 1 ) μ ( a ) ( I { a = c } + I { b = c } ) ( 62 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaGqabiab=9eapjabcIcaOiabdggaHjabdkgaIjabcYcaSiabdogaJjabcMcaPaqaaiabfc6aqjabcIcaOiabdggaHjabcYcaSiabdkgaIjabcMcaPaaacqGH9aqpcqGGOaakcqWGUbGBcqGHsislcqaIXaqmcqGGPaqkiiGacqGF8oqBcqGGOaakcqWGHbqycqGGPaqkcqGGOaaktuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab9Hi8jnaaBaaaleaacqGG7bWEcqWGHbqycqGH9aqpcqWGJbWycqGG9bqFaeqaaOGaey4kaSIae0hIWN0aaSbaaSqaaiabcUha7jabdkgaIjabg2da9iabdogaJjabc2ha9bqabaGccqGGPaqkcaWLjaGaaCzcaiabcIcaOiabiAda2iabikdaYiabcMcaPaaa@690B@\nM ( a b , c d ) Π ( a , b ) Π ( c , d ) ≃ ( n − r − 1 ) ( n − r − 2 ) μ ( a ) μ ( c ) + ∑ δ = 1 r − 1 ( n − δ − 2 ) ( μ ( c ) Π δ ( d , a ) + μ ( a ) Π δ ( b , c ) ) ( 63 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8F7D@\nO ( a b , c d ) Π ( a , b ) = ( n − 1 ) μ ( a ) I { a b = c d } + ( n − 2 ) Π ( c , d ) ( μ ( c ) I { a = d } + μ ( a ) I { b = c } ) ( 64 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8BDE@\nWith the example given in section \"validation\" we get for the expectation\nE0t MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFfbqrdaqhaaWcbaGaeGimaadabaGaemiDaqhaaaaa@3051@ = [4615.4 5384.6] (65)\nand\nE1t MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFfbqrdaqhaaWcbaGaeGymaedabaGaemiDaqhaaaaa@3053@ = [1384.5 3230.4 3230.4 2153.6] (66)\nThe magnitude of the second eigenvalue of Π is λ = 0.3, then rank r = 19 give a relative error \u003c 10 -10 and we get for the covariance\nC 0 , 0 = [ 1338.28 − 1338.28 − 1338.28 1338.28 ] ( 67 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFdbWqdaWgaaWcbaGaeGimaaJaeiilaWIaeGimaadabeaakiabg2da9maadmaabaqbaeqabiGaaaqaaiabigdaXiabiodaZiabiodaZiabiIda4iabc6caUiabikdaYiabiIda4aqaaiabgkHiTiabigdaXiabiodaZiabiodaZiabiIda4iabc6caUiabikdaYiabiIda4aqaaiabgkHiTiabigdaXiabiodaZiabiodaZiabiIda4iabc6caUiabikdaYiabiIda4aqaaiabigdaXiabiodaZiabiodaZiabiIda4iabc6caUiabikdaYiabiIda4aaaaiaawUfacaGLDbaacaWLjaGaaCzcaiabcIcaOiabiAda2iabiEda3iabcMcaPaaa@5529@\nC 1 , 0 = [ 1146.9 191.2 191.2 − 1529.2 − 1146.9 − 191.2 − 191.2 1529.2 ] ( 68 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6605@\nand\nC 1 , 1 = [ 1536.8 − 390.0 − 390.0 − 756.9 − 390.0 581.2 581.0 − 772.2 − 390.0 581.0 581.2 − 772.2 − 756.9 − 772.2 − 772.2 2301.4 ] ( 69 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFdbWqdaWgaaWcbaGaeGymaeJaeiilaWIaeGymaedabeaakiabg2da9maadmaabaqbaeqabqabaaaaaeaacqaIXaqmcqaI1aqncqaIZaWmcqaI2aGncqGGUaGlcqaI4aaoaeaacqGHsislcqaIZaWmcqaI5aqocqaIWaamcqGGUaGlcqaIWaamaeaacqGHsislcqaIZaWmcqaI5aqocqaIWaamcqGGUaGlcqaIWaamaeaacqGHsislcqaI3aWncqaI1aqncqaI2aGncqGGUaGlcqaI5aqoaeaacqGHsislcqaIZaWmcqaI5aqocqaIWaamcqGGUaGlcqaIWaamaeaacqaI1aqncqaI4aaocqaIXaqmcqGGUaGlcqaIYaGmaeaacqaI1aqncqaI4aaocqaIXaqmcqGGUaGlcqaIWaamaeaacqGHsislcqaI3aWncqaI3aWncqaIYaGmcqGGUaGlcqaIYaGmaeaacqGHsislcqaIZaWmcqaI5aqocqaIWaamcqGGUaGlcqaIWaamaeaacqaI1aqncqaI4aaocqaIXaqmcqGGUaGlcqaIWaamaeaacqaI1aqncqaI4aaocqaIXaqmcqGGUaGlcqaIYaGmaeaacqGHsislcqaI3aWncqaI3aWncqaIYaGmcqGGUaGlcqaIYaGmaeaacqGHsislcqaI3aWncqaI1aqncqaI2aGncqGGUaGlcqaI5aqoaeaacqGHsislcqaI3aWncqaI3aWncqaIYaGmcqGGUaGlcqaIYaGmaeaacqGHsislcqaI3aWncqaI3aWncqaIYaGmcqGGUaGlcqaIYaGmaeaacqaIYaGmcqaIZaWmcqaIWaamcqaIXaqmcqGGUaGlcqaI0aanaaaacaGLBbGaayzxaaGaaCzcaiaaxMaacqGGOaakcqaI2aGncqaI5aqocqGGPaqkaaa@8FB3@","divisions":[{"label":"title","span":{"begin":0,"end":10}},{"label":"p","span":{"begin":11,"end":369}},{"label":"p","span":{"begin":370,"end":449}},{"label":"p","span":{"begin":450,"end":1359}},{"label":"p","span":{"begin":1360,"end":1711}},{"label":"p","span":{"begin":1712,"end":2530}},{"label":"p","span":{"begin":2531,"end":2694}},{"label":"p","span":{"begin":2695,"end":3306}},{"label":"p","span":{"begin":3307,"end":3384}},{"label":"p","span":{"begin":3385,"end":3771}},{"label":"p","span":{"begin":3772,"end":3815}},{"label":"p","span":{"begin":3816,"end":4159}},{"label":"p","span":{"begin":4160,"end":4185}},{"label":"p","span":{"begin":4186,"end":4219}},{"label":"p","span":{"begin":4220,"end":5665}},{"label":"p","span":{"begin":5666,"end":5722}},{"label":"p","span":{"begin":5723,"end":6699}},{"label":"p","span":{"begin":6700,"end":6747}},{"label":"p","span":{"begin":6748,"end":7448}},{"label":"p","span":{"begin":7449,"end":7457}},{"label":"p","span":{"begin":7458,"end":8631}},{"label":"p","span":{"begin":8632,"end":8642}},{"label":"p","span":{"begin":8643,"end":8695}},{"label":"p","span":{"begin":8696,"end":8909}},{"label":"p","span":{"begin":8910,"end":8937}},{"label":"p","span":{"begin":8938,"end":9988}},{"label":"p","span":{"begin":9989,"end":10215}},{"label":"p","span":{"begin":10216,"end":11578}},{"label":"p","span":{"begin":11579,"end":11612}},{"label":"p","span":{"begin":11613,"end":11648}},{"label":"p","span":{"begin":11649,"end":13327}},{"label":"p","span":{"begin":13328,"end":13351}},{"label":"p","span":{"begin":13352,"end":13541}},{"label":"p","span":{"begin":13542,"end":13652}},{"label":"p","span":{"begin":13653,"end":13986}},{"label":"p","span":{"begin":13987,"end":15045}},{"label":"p","span":{"begin":15046,"end":15383}},{"label":"p","span":{"begin":15384,"end":16551}},{"label":"p","span":{"begin":16552,"end":17352}},{"label":"p","span":{"begin":17353,"end":18610}},{"label":"p","span":{"begin":18611,"end":19747}},{"label":"p","span":{"begin":19748,"end":19821}},{"label":"p","span":{"begin":19822,"end":20147}},{"label":"p","span":{"begin":20148,"end":20151}},{"label":"p","span":{"begin":20152,"end":20491}},{"label":"p","span":{"begin":20492,"end":20625}},{"label":"p","span":{"begin":20626,"end":21356}},{"label":"p","span":{"begin":21357,"end":22281}},{"label":"p","span":{"begin":22282,"end":22285}}],"tracks":[]}