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better at the sprint/track disciplines may obtain an advantage in the decathlon. KEYWORDS: athletics, cluster analysis, personal best, classical scaling Introduction The decathlon takes place over two days and consists of a combination of ten track and field disciplines the order of which is invariant: day 1: 100 m race (100m), long jump (LJ), shot put (SP), high jump (HJ) and 400 m race (400m); day 2: 110 m hurdles (110mH), discus (DT), pole vault (PV), javelin (JT) and 1500 m race (1500m). Actual performance results in terms of time in the track disciplines and

scalable with respect to the number of sub- domains, is quasi-optimal polylogarithmic with respect to subdomainmesh size, and is independent of coef- cient discontinuities and ratio of mesh sizes across subdomain interfaces. Numerical experiments support the theory and show that the deluxe scaling improves signicantly the performance over classical scaling. Keywords: Interior Penalty Discretization, Discontinuous Galerkin, Elliptic Problems with Discontinuous Coecients, Finite Element Method, FETI-DP Algorithms, Preconditioners, Deluxe MSC 2010: 65F10, 65N20, 65N30 DOI

Abstract

Approximate analytical solutions concerning lifetime of soluble solid particles in an unbounded stagnant medium have been developed by simple application of fractional half-time derivative in the Riemann-Liouville sense to express the relationship between the net surface mass flux and the concentration at the interface. The solutions start with the initial formulation of Rice and Do on the time-depletion of the radius of a spherical particle expressed through terms including the solubility parameter as the only key parameter controlling the process of dissolution. The two approximate developed solutions use different scaling and dimensionless variables: The 1st solution is developed by an introduction of a similarity variable [xxx] while the 2nd solution applies the classical scaling using the initial sphere radius as a length scale that leads to dimensionless radius r = R/R0 and time τ = Dt/R0 2. Both solutions provide approximate relationships close to that of Rice and Do.

References [1] O. Arizmendi, T. Hasebe, Classical scale mixtures of Boolean stable laws, Trans. Amer. Math. Soc. 368 (2016), 4873-4905. [2] O. E. Barndorff-Nielsen, S. Thorbjornsen, Self-decomposability and Lévy processes in free probability, Bernoulli 8(3) (2002), 323-366. [3] O. E. Barndorff-Nielsen, S. Thorbjornsen, A connection between free and classical infinite divisibility, Infin. Dimens. Anal. Quantuum Probab. Relat. Top. 7(4) (2004), 573-590. [4] O. E. Barndorff-Nielsen, S. Thorbjornsen, Classical and free infinite divisibility and Lévy processes

means of a classical scaling procedure if necessary [3, 5] . The unknown rate coefficients were adjusted appropriately until all the calculated and observed ion intensities converged. In this manner, the rate coefficients kl, k3, k4 and the ratio A-jb/̂ i were obtained from the experimental results at low pressure. Subsequently, the high pressure data reported previously were evaluated similarly on the basis of the reaction mechanism (1) — (4), employing the values for k^/k^ derived from the low pressure data. This evaluation provided again rate

). A set of data must be fixed by the user concerning the optimization problem definition before the implementation of the design methodology (see Dietz et al., 2005). For instance, the annual demand for each product is presented in Table 1. Table 1. Product demands Product Production (kg/year) Insulin 1500 Vaccine 1000 Chymosin 3000 Protease 6000 Three sizes are available for each equipment item: large (L), medium (M) and small (S). The classical expressions used for computing the investment cost of the equipment items follows a classical scaling law. Of

multiphase reactor with RPT”, AIChE J., Vol. 41, No. 2, 439-443 (1995). Leuenberger, H, “Scale-up in the 4th dimension in the field of granulation and drying or how to avoid classical scale- up”, Powder Technology, Vol. 130, No. 1-3, 225-230 (2003). Matsen, J.M., “Scale up of fluidized bed processes: principle and practice”, Powder Technology, Vol. 88, No. 3, 237-244 (1996). Mostoufi, N., Chaouki, J., “Local solid mixing in gas-solid fluidized beds”, Powder Technology, Vol. 114, No. 1-3, 23-31 (2001). Pugsley, T., Tanfara, H., Makus, S., Heping, C., Chaouki, J

low throughputs, mostly in the μl/min range. Consequently, the productivity is very low, rarely exceeding the synthesis of a few hundred milligrams of product, which is needed for product characterization. In order to consider microfluidic techniques as an alternative production technology, their productivity must be significantly enhanced. The usual approach in microreaction technology for increasing throughput is the so-called numbering-up or equaling-up approach. Classical scale-up is not applicable here as the use of microfluidic structures is compulsory for the

/dt. are defined , which are applied to the position vector χ to obtain the two "classical" velocities, (5) To recover local reversibility of the time differential element, the two derivatives are combined in terms of a complex derivative operator, dt (6) which, when it is applied to the position vector, gives a complex velocity, . v. - v_ dt Now, the total derivative with respect to / of a function f(x, t) contains finite terms up to the highest order. For a fractal dimension Z>/.· = 2, it writes - _.. i//^a/ dx } ι a2/ dt dt dt 2dx.dx, dt (8) The "classical" scale

=1.0721, indicating a worse clustering. 3.2 A classical scaling approach Usually, in MDS applications the objects under study have many characteristics, thus they correspond to points in a high-dimensional space. The main problem of MDS consists of mapping the given data points P i into points Q i in a space of “small” dimension, conserving the distances as far as possible. Through this reduction of dimensionality the complexity of the problem can be considerably simplified. In this section we reduce the three-dimensional space used for the illustration of text