Function two-dimensional

Complex numbers owe their origin to the quest for the square root of a negative number. Thus the so-called imaginary number i = is a fundamental element of complex numbers, written as z = X + iy, in which x is the real part and y is the imaginary part. Although real numbers quantify physical quantities, complex numbers provide very convenient representations of many physical phenomena. In quantum mechanics, the wave function is a complex function. Two-dimensional, incompressible, irrotational flows are represented by a complex flow potential, w = 9 h- t /, with 9, the velocity potential, as the real part, and /, the stream function, as the imaginary part. 

S. Jiang, J. Chen, J. Tang, E. Jin, L. Kong, W. Zhang, and C. Wang, An nanoparticles-functionalized two-dimensional patterned conducting PANI nanobowl monolayer for gas sensor. Sens. Actuators B, 140, 520-524 (2009). 

Echocardiography is a widely applied noninvasive modality that provides data on cardiac structure and function. Two-dimensional (2D) echocardiography with Doppler flow assessments can characterize hemodynamics and systolic/diastolic function and display cardiac structures and measurements of cardiac chamber sizes. Three-dimensional (3D) echocardiography offers the potential to enhance assessments of cardiac function, structure, and geometry. 

Grinceviciute N, Snopok B, Snitka V. Functional two-dimensional nanoarchitectures based on chemically converted graphene oxide and hematoporphyrin under the sulfuric acid treatment. Chem Eng J 2014 255 577-84. 

Correlation function Two-dimensional System Weakly coupled 2-D system liquid crystal... 

In the case of the adiabatic flash, application of a two-dimensional Newton-Raphson iteration to the objective functions represented by Equations (7-13) and (7-14), with Q/F = 0, is used to provide new estimates of a and T simultaneously. The derivatives with respect to a in the Jacobian matrix are found analytically while those with respect to T are found by finite-difference approximation... 

A step-limited Newton-Raphson iteration, applied to the Rachford-Rice objective function, is used to solve for A, the vapor to feed mole ratio, for an isothermal flash. For an adiabatic flash, an enthalpy balance is included in a two-dimensional Newton-Raphson iteration to yield both A and T. Details are given in Chapter 7. 

Some discontinuities may be identified by a conventional two-dimensional ultrasonic technique, from which the well-known C-scan image is the most popular. The C-scan technique is relatively easy to implement and the results from several NDE studies have been very encouraging [1]. In the case of cylindrical specimens, a circular C-scan image is convenient to show discontinuity information. The circular C-scan image shows the peak amplitude of a back-scattered pulse received in the circular array. The axial scan direction is shown as a function of transducer position in the circular array. The circular C-scan image serves also as an initial step for choosing circular B-scan profiles. The latter provides a mapping between distance to the discontinuity and transducer position in the circular array. 

The alternative approach is to treat the film as a nonideal two-dimensional gas. One may use an appropriate equation of state, such as Eq. Ill-104. Alternatively, the formalism has been developed for calculating film activity coefficients as a function of film pressure [192]. 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

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