Hill equation dynamics

The development of mathemafical models is described in several of the general references [Giiiochon et al., Rhee et al., Riithven, Riithven et al., Suzuki, Tien, Wankat, and Yang]. See also Finlayson [Numerical Methods for Problems with Moving Front.s, Ravenna Park, Washington, 1992 Holland and Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York, 1982 Villadsen and Michelsen, Solution of Differential Equation Models by... 

Above the canopy we adopt the solution of Raupach et al 1992 [528] for the scalar perturbations. This follows the pattern of Hunt, Liebovich and Richards, 1988 [287] by dividing the flow-field into an outer layer and a shear stress layer of depth hr In the outer layer, scalar perturbations are governed by inviscid dynamics while in the shear stress layer, changes to the scalar flux also play a role at first order. The linearised equation for the scalar perturbation induced by the hill in the outer layer is,... 

Figure 2 shows a time trace of the dynamics for N = 3, n = 6. Now the dynamics follow a stable limit cycle oscillation. This forms the basis for the synthesis of the repressilator [27]. In this case the eigenvalues of the hxed point atx] = X2 = X3 = 1/2 are = —1 — n/2 and X23 = —1 + n/4 f3nijA [34]. In this case there is a Hopf bifurcation when n = 4 so that for values of n > 4 there is a stable limit cycle oscillation corresponding to the repressilator. If the equations for the dynamics of mRNA are included, then oscillations are still found, but now oscillations can be found for smaller values of the Hill... 

Some of the applicable muscle models include the Maxwell, Voigt, Hill and Carlson models (Figure 1). In particular, the Carlson (1957) equation is used in much of this work to describe the stress-velocity relationship of cardiac muscle over the entire cardiac cycle. Min et al. (1978) found very little difference in analyzing ventricular dynamics when he alternately used Carlson s equation only during isotonic contraction and Hill s equation during isovolumic contraction. 

Streefer, V. F., ed. 1961. Handbook of Fluid Dynamics. New York McGraw-Hill. A classic handbook on fluid dynamics wifh confributions from distinguished experts. Written for engineers and scientists in the field. Deals wifh bofh fundamenfal concepts and applications. Covers fluid flow (one-dimensional, ideal, laminar, compressible, two phase, open channel, stratified), turbulence, boundary layers, sedimentation, turbomachinery, fluid transients, and magnetohydrodynamics. Includes many formulas, equations, tables, graphs, and illustrations. Each chapter has a bibliography and the volume has subject and author indexes. 

Den Hartog, J. P. 1985. Mechanical Vibrations, 4th ed. New York Dover Publications. Reprint. Originally published 1956. New York McGraw-Hill. Covers the fundamentals of mechanical vibrations. Can be used by practicing engineers as well as for classroom instruction. An elementary knowledge of dynamics and calculus is necessary, but differential equations are explained in detail. Examples have been drawn from real-life experiences of the author and his friends. Each chapter includes problems that illustrate typical practical situations with answers included at the back. 

Almost aU the biological models are nonhnear dynamic systems, including for example saturation or threshold processes. In particular, nonlinear compartmental models. Equation 9.5, are frequently found in biomedical applications. For such models the entries of K are functions of q, most commonly fcy is a function of only few components of q, often q, or qj. Examples of fcy function of q,- or qj are the Hill and... 

Barry and Hill [12-13] developed models based on computational fluid dynamics (CFD) to predict the performance of protective textile clothing. CFD software solves the governing equations for mass, momentum, and heat transfer in fluids over a 2- and 3-dimensional computational mesh. 

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