Non-linear mathematics

A non-linear mathematical model, which is a set of ordinary differential equations, for the process in the SPBER was developed.19 The model accounts for the heterogeneous electrochemical reaction and homogeneous reaction in the bulk solution. The lateral distributions of potential, current density and concentration in the reactor were studied. The potential distribution in the lateral dimension, x, of the packed bed was described by a one dimensional Poisson equation as ... 

Even if the quasi-steady-state approximation is valid, all of the forms of the reaction rate equation (5.3) given above involve the concentrations of at least some of the intermediate species. If we are to be concerned with the intermediate species, and particularly when we wish to employ the methods of non-linear mathematics (or even direct numerical computation) then we must find ways of including such quantities. The appropriate rate equations for the concentration of H, O and OH are readily constructed... 

The non-linear mathematics approach begins with the set of governing differential equations but puts great emphasis in the initial stages of analysis on the true steady-states of such systems, i.e., the combinations of concentrations for which all the rate equations, including those for the reactants and products, become zero. This is only appropriate in so-called... 

Intellectual movements which have led to the development of catastrophe theory have their origin in philosophy, mathematics and the experimental sciences. The evolution of philosophical views, the progress in the area of non-linear mathematical methods and the development of experimental methods related to the investigation of stability of processes have created favourable conditions for the appearance of catastrophe theory. 

Thus, we have derived the non-linear mathematic model of the electrochemical process with low-conductive surface film that can decompose with an exponential temperature-dependent rate. Essentially, this is an example of a thermokinetic model. 

The first step in the application of optimization was in stmctural design in the 1960s when Schmit [12] proposed a rather general new approach, which served as the conceptual foundation for the development of many modern structural optimization methods. It introduced the idea and indicated the feasibility of coupling finite element stmctural analysis and non-linear mathematical programming to create automated optimum design capabilities for a rather broad class of stmctural design problems. 

Sobol, I.M. 1993. Sensitivity analysis for non-linear mathematical models. Mathematical Modelling and Computational Experiment 1 407-414 Translated from Russian I.M. SoboT. 1990. Sensitivity estimates for nonlinear mathematical models, Matematicheskoe Modelirovanie 2 112-118. 

Certainly, a more complete and accurate description of a generalized transition would be associated with analyzing its microscopic measures, which solution associates strong non-linearity mathematically exploitable with respect... 

Sobol , I.M. Sensitivity analysis for non-linear mathematical models. Math. Model. Comp. Exp. 

H. W. Kuhn and A. W. Tucker, Non-linear Programming, in J. Neyman, ed., Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951 Thomas L. Saaty, Mathematical Methods of Operations Research, McGraw-Hill Book Co., New York, 1959. 

Data given in Tables 1-6 clearly show a significant dependence of P2 and p4 on amine concentration, that is, at least one of the apparent rate constants kj contains a concentration factor. Thus, according to the mathematical considerations outlined in the Analysis of Data Paragraph, both p2, P4 exponents and the derived variables -(P2 + p)4> P2 P4 ind Z (see Eqns. 8-12) are the combinations of the apparent rate constants (kj). To characterize these dependences, derived variables -(p2+p)4, P2 P4 and Z (Eqns. 8,11 and 12) were correlated with the amine concentration using a non-linear regression program to find the best fit. Computation resulted in a linear dependence for -(p2 + p)4 and Z, that is... 

By comparing equations 1-4 and 9-11, one can see a one-to-one correspondence between terms in the mathematical description and the numerical program. The flexibility of TOPAZ derives from the fact that the parameters of equations 9-11 (p, Cp, kx, ky, qg, P, y) can be linear or non-linear functions of spatial coordinates, time, and temperature. 

Finite-element techniques can cope with large, highly non-linear deformations, making it possible to model soft tissues such as skin. When relatively large areas of skin are replaced during plastic surgery, there is a problem that excessive distortion of the apphed skin will prevent adequate adhesion. Finite-element models can be used to determine, either by rapid trial-and-error modelhng or by mathematical optimisation, the best way of... 

See also the theoretical description of a micro reactor for optical photocatalytic dissociation of non-linear molecules in [140]. Here, a mathematical model for a novel type of micro reactor is given. Rotating non-linear molecules at excitation of valent vibrations are considered, having a magnetic moment. Resonance decay of molecules can be utilized with comparatively weak external energy sources only. 

Basically two search procedures for non-linear parameter estimation applications apply. (Nash and Walker-Smith, 1987). The first of these is derived from Newton s gradient method and numerous improvements on this method have been developed. The second method uses direct search techniques, one of which, the Nelder-Mead search algorithm, is derived from a simplex-like approach. Many of these methods are part of important mathematical computer-based program packages (e.g., IMSL, BMDP, MATLAB) or are available through other important mathematical program packages (e.g., IMSL). 

Non-linear parameter estimation is far from a trivial task, even though it is greatly simplified by the availability of user-friendly program packages such as a) SIMUSOLV (Steiner et al., 1986), b) ESL, c) a set of BASIC programs (supplied with the book of Nash and Walker-Smith, 1987) or d) by mathematical software (MATLAB). ISIM itself does not supply these advanced features, but ISIM programs can easily be translated into other more powerful languages. 

In the last decades not only thousands of chemical descriptors but also many advanced, powerful modeling algorithms have been made available, The older QSAR models were linear equations with one or a few parameters. Then, other tools have been introduced, such as artificial neural network, fuzzy logic, and data mining algorithms, making possible non linear models and automatic generation of mathematical solutions. 

In a self-reproducing, catalytic hypercycle (second order, because of its double function of protein and RNA synthesis) the polynucleotides Ni contained not only the information necessary for their own autocatalytic self-replication but also that required for the synthesis of the proteins Ei. The hypercycle is closed only when the last enzyme in the cycle catalyses the formation of the first polynucleotide. Hypercycles can be described mathematically by a system of non-linear differential equations. In spite of all its scientific elegance and general acceptance (with certain limitations), the hypercycle does not seem to be relevant for the question of the origin of life, since there is no answer to the question how did the first hypercycle emerge in the first place (Lahav, 1999). 

J.E.Dennis, R.B. Schnabel Numerical methods for unconstrained optimization and non-linear equations, Prentice Hall Series in computational mathematics, New York, (NY, USA), 1983... 

In a transported PDF simulation, the chemical source term, (6.249), is integrated over and over again with each new set of initial conditions. For fixed inlet flow conditions, it is often the case that, for most of the time, the initial conditions that occur in a particular simulation occupy only a small sub-volume of composition space. This is especially true with fast chemical kinetics, where many of the reactions attain a quasi-steady state within the small time step At. Since solving the stiff ODE system is computationally expensive, this observation suggests that it would be more efficient first to solve the chemical source term for a set of representative initial conditions in composition space,156 and then to store the results in a pre-computed chemical lookup table. This operation can be described mathematically by a non-linear reaction map ... 

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