다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Fractional Differential Equations

https://www.chester.ac.uk/sites/files/chester/rep377.pdf

Analysis of Fractional Differential Equations

We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional CTRW This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding Turning our attention to spatially 1-D CTRWs with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation Finally, we treat the `time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process

Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects

In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. It is known that whereas the diffusion equation describes a process where the disturbance spreads infinitely fast, the propagation velocity of the disturbance is a constant for the wave equation. By replacing the time derivatives in the above standard equations with pseudo-differential operators interpreted as derivatives of non integer order (nowadays misnamed as of fractional order) we are lead to generalized processes of diffusion that may be interpreted as slow diffusion and interpolating between diffusion and wave propagation. In mathematical physics, we may refer these interpolating processes to as fractional diffusion-wave phenomena. The use of the Laplace transform in the analysis of the Cauchy and Signalling problems leads to special functions of the Wright type. In this work we analyze and simulate both the situations in which the input function is a Dirac delta generalized function and a box function, restricting ourselves to the Cauchy problem. In the first case we get the fundamental solutions (or Green functions) of the problem whereas in the latter case the solutions are obtained by a space convolution of the Green function with the input function. In order to clarify the matter for the non-specialist readers, we briefly recall the basic and essential notions of the fractional calculus (the mathematical theory that regards the integration and differentiation of non-integer order) with a look at the history of this discipline.

On the evolution of fractional diffusive waves

The fractional Poisson process and the Wright process (as discretization of the stable subordinator) along with their diffusion limits play eminent roles in theory and simulation of fractional diffusion processes. Here we have analyzed these two processes, concretely the corresponding counting number and Erlang processes, the latter being the processes inverse to the former. Furthermore we have obtained the diffusion limits of all these processes by well-scaled refinement of waiting times and jumps

/pdf/on-the-fractional-poisson-process-and-the-discretized-stable-u0m56l2hym.pdf

On the fractional Poisson process and the discretized stable subordinator

Fractional calculus and the Schrödinger equation

We revisit the Mittag-Leer functions of a real variable t, with one, two and three order-parameters f;; g, as far as their Laplace transform pairs and complete monotonicty properties are concerned. These functions, subjected to the requirement to be completely monotone for t > 0, are shown to be suitable models for non{ Debye relaxation phenomena in dielectrics including as particular cases the classical models referred to as Cole{Cole, Davidson{Cole and Havriliak{Negami. We show 3D plots of the response functions and of the corresponding spectral distributions, keeping xed one of the three order-parameters.

Francesco Mainardi

Papers

Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects

On the evolution of fractional diffusive waves

On the fractional Poisson process and the discretized stable subordinator

Fractional calculus and the Schrödinger equation

for anomalous relaxation in dielectrics