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Usually, the ensemble average of an observable f is defined by f = ∫I fdμ, where μ is an invariant measure. However, in experiments, we can perform only a finite number of observations. Thus the quantity in an initial ensemble in EAMSD [Eq. ] is finite.
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Usually, the ensemble average of an observable f is defined by f = ∫I fdμ, where μ is an invariant measure. However, in experiments, we can perform only a finite number of observations. Thus the quantity in an initial ensemble in EAMSD [Eq. ] is finite.
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Roughly speaking, orbits visit an arbitrarily small neighborhood of all points on I infinitely often (see)
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Roughly speaking, orbits visit an arbitrarily small neighborhood of all points on I infinitely often (see).
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-1 E=E }.
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f (x) L+1 (μ) means that ∫I f (x) dμ <∞ and f (x) ≥0 on I.
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f (x) L+1 (μ) means that ∫I f (x) dμ <∞ and f (x) ≥0 on I.
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21
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77957189272
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1+α) k zk Γ (1+kα ), where E is the expectation. Note that "normalized" means E ( Yα ) =1.
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1+α) k zk Γ (1+kα ), where E is the expectation. Note that "normalized" means E ( Yα ) =1.
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A positive function U is regularly varying at ∞ with index α if U (tx ) /U (t) → xα, as t→∞ for every x>0.
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A positive function U is regularly varying at ∞ with index α if U (tx ) /U (t) → xα, as t→∞ for every x>0.
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This is because the induced transformation defined by S A0 (x) = T A0 τ (x) (x) for x A0 with the return time function τ (x) =min {k≥1: T A0 k (x) A0} is fully chaotic and has a finite invariant measure, where T A0 (x) is the reduced map on [-1/2,1/2 ) in the same way as T1 (x). In particular, x(-1/2,- c2 ) ( c1 , c2 ) and x (- c2 ,- c1 ) ( c2 ,1/2 ) move to the left cell and to the right cell, respectively, where c2 + (2 c2 ) z =1 (0< c2 <1/2 ).
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This is because the induced transformation defined by S A0 (x) = T A0 τ (x) (x) for x A0 with the return time function τ (x) =min {k≥1: T A0 k (x) A0} is fully chaotic and has a finite invariant measure, where T A0 (x) is the reduced map on [-1/2,1/2 ) in the same way as T1 (x). In particular, x(-1/2,- c2 ) ( c1, c2 ) and x (- c2,- c1 ) ( c2,1/2 ) move to the left cell and to the right cell, respectively, where c2 + (2 c2 ) z =1 (0< c2 <1/2 ).
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