Experimental Mathematics

With the availability of computers, it has been recognized that not only can some problems (which up to that time were beyond the possibility of being numerically solved) be reexamined and solved, but also new problems can emerge, be discovered, considered, and even solved, which were unknown in pre-computer time. Let us briefly mention two such problems that have some novelty, both of which have some connection with chemistry (i) the proof or almost a proof of the Kepler conjecture about dense packing of spheres and (ii) the problan relating to Ulam s spiral. 

We will end with the outline of a new problem of experimental mathematics that arose in the process of writing this book and relates to isospectral graphs. 

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Contents Basic Physical Concepts. - Hyperfine Interactions. - Experimental. - Mathematical Evaluation of Mossbauer Spectra. - Interpretation of Mossbauer Parameters of Iron Compounds. - Mossbauer-Active Transition Metals Other than Iron. - Some Special Applications. 

Monte Carlo methods comprise that branch of experimental mathematics which is concerned with experiments on random numbers (Addnl Ref N)... 

W. P. Reinhardt, Complex Scaling in Atomic Physics A Staging Ground for Experimental Mathematics and for Extracting Physics from Otherwise Impossible Computations, In Spectral Theory and Mathematical Physics A Fest Schrift in Honor of Barry Simon s 60th... 

Figure 6.8 shows the plot of Mt vs. t for Equation (6.36). As time approaches infinite, the graph gives a straight line, which is a representation of Equation (6.37) for the time lag. The time intercept is calculated from Equation (6.38). This intercept and the slope [ = 4n ab(b - a)KCj ], which is the steady-state release rate, are used to determine the diffusion and partition coefficients experimentally. Mathematical expressions, which include the amount of the drug diffused from freshly prepared and stored geometrical membranes (sheet and cylinder) at steady state are presented in Table 6.2. 

Because D+, D, and have been determined, both M and N are known experimentally. Mathematically, the following four solutions result from the possible choices of sign in Eq. B.8 and B.9 ... 

In addition to in vivo and in vitro experimentation, mathematical models and quantitative structure-permeability relationship (QSAR) methods have been used to predict skin absorption in humans. These models use the physico-chemical properties of the test compound (e.g. volatility, ionization, molecular weight, water/lipid partition, etc.) to predict skin absorption in humans (Moss et al 2002). The models are particularly attractive because of the low cost and rapidity. However, because of the above-mentioned factors influencing dermal absorption, mathematical models are of limited use for risk assessment purposes. Since these models are currently not accepted by regulatory agencies involved in pesticide evaluations, they will not be further discussed in this chapter. 

Crystalline arrays of atoms diffract X-ray beams directed at them in a pattern characteristic of the arrangement of atoms in the crystal lattice. Diffracted X-ray beams interfere with each other in a constructive or destructive manner to create a diffraction pattern that contains within it the structural information of the diffracting crystalline array. Experimental, mathematical, and empirical techniques are used to obtain the three-dimensional structure from the diffraction pattern. 

L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. In Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics computational issues in nonlinear science, pages 259-268, Amsterdam, The Netherlands, The Netherlands, 1992. Elsevier North-Holland, Inc. 

Jacobson, A. and Gordon, R. (1976) Changes in the shape of the developing vertebrate nervous system analyzed experimentally, mathematically and by computer simulation. J. Exp. Zool. 197,191-246. 

Experimental Mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns. 

We will introduce a few problems and a few solutions relating to experimental mathematics in the section following the story of isospectral graphs. 

The search for pairs of graphs having the same molecular ID number or molecular prime number ID can be viewed as an illnstration of experimental mathematics or, if you wish, experimental chemical graph theory. The concept of molecular ID numbers has been extended later to cover rings [81] and alternative ID numbers were considered [82]. The examination of the discrimination power of molecular identification numbers has been very recently revisited [83-86], clearly indicating that this topic has not yet been completely explored. New indices have outperformed earlier ones and show limits of classical measnres, such as the Balaban index /. 

Feigebaum Mitchell Jay (1945-) US. phys., famous for discovering the constant 4.6692... for ith bifurcation (limi , = di/di+l) named after him (Feigebaum numbers), his disclosures spanned new field of theoretical and experimental mathematics... 

The discussions in the following sections will focus on the thermal-hydraulic properties and characteristics of flows in parallel channels and NCLs. The hterature on general aspects of the analytical, experimental, mathematical modeling, numerical solution methods, and computational aspects of these flows will be briefly reviewed. These aspects when associated with specific Gen IV systems will also be discussed. 

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