A probabilistic interpretation of the Dzherbashyan fractional integral

The evaluation of water bodies “at risk” of not achieving the Water Framework Directive's (WFD) goal of “good status” begs the question of how big a risk is acceptable before a programme of measures should be implemented. Documentation of expert judgement and statistical uncertainty in pollution budgets and water quality modelling, combined with Monte Carlo simulation and Bayesian belief networks, make it possible to give a probabilistic interpretation of “at risk”. Combined with information on abatement costs, a cost-effective ranking of measures based on expected costs and effect can be undertaken. Combined with economic valuation of water quality, the definition of “disproportionate cost” of abatement measures compared to benefits of achieving “good status” can also be given a probabilistic interpretation. Explicit modelling of uncertainty helps visualize where research and consulting efforts are most critical for reducing uncertainty. Based on data from the Morsa catchment in South-Eastern Norway, this paper discusses the relative merits of using Bayesian belief networks when integrating biophysical modelling results in the benefit-cost analysis of derogations and cost-effectiveness ranking of abatement measures under the WFD.

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A note on fractional integral operator associated with multiindex Mittag-Leffler functions

Filomat ◽

10.2298/fil1607931c ◽

2016 ◽

Vol 30 (7) ◽

pp. 1931-1939 ◽

Cited By ~ 19

Author(s):

Junesang Choi ◽

Praveen Agarwal

Keyword(s):

Integral Operator ◽

Fractional Calculus ◽

Fractional Integral ◽

Fractional Integration ◽

Fractional Integral Operator ◽

Integration Formula ◽

Special Cases

Recently Kiryakova and several other ones have investigated so-called multiindex Mittag-Leffler functions associated with fractional calculus. Here, in this paper, we aim at establishing a new fractional integration formula (of pathway type) involving the generalized multiindex Mittag-Leffler function E?,k[(?j,?j)m;z]. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.

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Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

Journal of Applied Probability ◽

10.1017/s0021900200039103 ◽

1990 ◽

Vol 27 (03) ◽

pp. 545-556 ◽

Cited By ~ 2

Author(s):

S. Kalpazidou

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Asymptotic Behaviour ◽

Probabilistic Interpretation ◽

Stationary Markov Chain

The asymptotic behaviour of the sequence (𝒞 n (ω), wc,n (ω)/n), is studied where 𝒞 n (ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n ) until time n and wc,n (ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω c ) which represents uniquely the chain (ξ n ) as a circuit chain, and ω c is given a probabilistic interpretation.

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