Preface to the first edition

Riley, K. F.; Hobson, M.; Bence, Stephen; Riley, K. F.; Hobson, M. P.; Bence, S. J.

doi:10.1017/cbo9780511810763.003

Cited by 52 publications

(68 citation statements)

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“…We derive the approximate expression ͑44͒ for the probability i that a newly generated estimate p i e͑n͒ = N s ͑i͒ / k i for the prob- We assume that the log͓N s ͑j͒ ͔ for each interface j are independent variables ͑i.e., that the sampling at different interfaces is uncorrelated͒. Since we are adding many independent variables, we apply the central limit theorem 29 to Eq. ͑B2͒.…”

Section: Appendix B: Acceptance Probability For the Rb Methodsmentioning

confidence: 99%

“…The basis of our analysis is the fact that on firing k i trial runs from interface i, the number of successful trials N s ͑i͒ is binomially distributed, 29 E͓p i e ͔ = p i and using Eq. ͑18͒, we obtain…”

Section: Expressions For the Variancementioning

confidence: 99%

“…Since we no longer assume that all points at interface i are identical, we must now take into account the distribution of the number of times m j that point j is selected. This is, in fact, a multinomial distribution, 29,35 which has an average E͓m j ͔ = M i / N i and a variance V͓m j ͔ = M i ͓1/N i ͑1−1/N i ͔͒. Let us now do a "thought experiment" in which we first decide how many trials will be fired from each point; i.e., we fix the set of values ͕m j ͖ ͑of course, ͚ j m j = M i ͒.…”

Section: ͑C2͒mentioning

confidence: 99%

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Forward flux sampling-type schemes for simulating rare events: Efficiency analysis

Allen

Frenkel

Wolde

2006

The Journal of Chemical Physics

214

263

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We analyze the efficiency of several simulation methods which we have recently proposed for calculating rate constants for rare events in stochastic dynamical systems in or out of equilibrium. We derive analytical expressions for the computational cost of using these methods and for the statistical error in the final estimate of the rate constant for a given computational cost. These expressions can be used to determine which method to use for a given problem, to optimize the choice of parameters, and to evaluate the significance of the results obtained. We apply the expressions to the two-dimensional nonequilibrium rare event problem proposed by Maier and Stein ͓Phys. Rev. E 48, 931 ͑1993͔͒. For this problem, our analysis gives accurate quantitative predictions for the computational efficiency of the three methods.

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Section: Appendix B: Acceptance Probability For the Rb Methodsmentioning

confidence: 99%

Section: Expressions For the Variancementioning

confidence: 99%

Section: ͑C2͒mentioning

confidence: 99%

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Forward flux sampling-type schemes for simulating rare events: Efficiency analysis

Allen

Frenkel

Wolde

2006

The Journal of Chemical Physics

214

263

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“…(7) and p is some combination of x r and x i , whose value remains a constant along its characteristic curves [19]. Substituting this form of ρ into (7), we see that the characteristic curves are obtained by integrating the equation…”

Section: Probability From Velocity Fieldmentioning

confidence: 99%

Probability and complex quantum trajectories

John¹

2009

Annals of Physics

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It is shown that in the complex trajectory representation of quantum mechanics, the Born's Ψ ⋆ Ψ probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.

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“…For odd powers, the integrand is transformed by factoring out one sine and the remaining even powered sine is converted into cosine using the identity Another method used to evaluate powers of sine is by using reduction formula. A reduction formula transforms the integral into an integral of the same or similar expression with a lower integer exponent [4]. It is repeatedly applied until the power of the last term is reduced to two or one, and the final integral can be evaluated.…”

Section: Introductionmentioning

confidence: 99%

Simplified Method of Evaluating Integrals of Powers of Sine Using Reduction Formula as Mathematical Algorithm

Suello¹

2015

IJAPM

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This paper presents a simpler and shorter method of evaluating integrals of powers of sine. The reduction formula for sine is repeatedly applied to the integral of the nth power of sine until generalized formulas are derived. Since the derivation process involves recursive relations, the coefficients and exponents of the derived formulas showed certain patterns and sequences which were used as the basis for developing an easier algorithm.Key words: Integration, mathematical algorithm, powers of sine, reduction formula, trigonometric identities. IntroductionEvaluating integrals of powers of trigonometric functions is always part of the study of Integral Calculus. Integrals of powers of sine are usually evaluated using trigonometric identities and the solution depends on whether the power is odd or even. For odd powers, the integrand is transformed by factoring out one sine and the remaining even powered sine is converted into cosine using the identity Another method used to evaluate powers of sine is by using reduction formula. A reduction formula transforms the integral into an integral of the same or similar expression with a lower integer exponent [4]. It is repeatedly applied until the power of the last term is reduced to two or one, and the final integral can be evaluated. Using integration by parts, the reduction formula for sine is [5]. The methods discussed above are normally tedious and time consuming depending on the given power of sine. As shown in the study of Dampil [6], deriving generalized formulas can simplify solutions, hence, the objective of this paper is to come up with a shorter and simpler method of integrating powers of sine. Generalized formulas are derived by successive application of the reduction formula to the integral of the nth power of sine. Because of the recursive nature of the reduction formula, an algorithm is developed International Journal of Applied Physics and Mathematics 192Volume 5, Number 3, July 2015Lito E. Suello* Malayan Colleges Laguna, Pulo-Diezmo Road, Cabuyao City, Laguna, Philippines.Manuscript submitted January 25, 2015; accepted June 5, 2015.* Corresponding author. Tel.: +639497688588; email: lesuello@mcl.edu.ph based on the sequences and patterns of the coefficients and exponents of the terms of the derived formulas.

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Preface to the first edition

Cited by 52 publications

References 0 publications

Forward flux sampling-type schemes for simulating rare events: Efficiency analysis

Forward flux sampling-type schemes for simulating rare events: Efficiency analysis

Probability and complex quantum trajectories

Simplified Method of Evaluating Integrals of Powers of Sine Using Reduction Formula as Mathematical Algorithm

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