Orthogonality principle

Use the nonhomogeneous boundary condition and the orthogonality principle to obtain the expansion coefficient. 

Use the nonhomogeneous boundary condition And orthogonality principle of eigenfunctions To obtain coefficient accompanying expansion Hence temperature T has a final expression. 

ISCAD The IMS/SAW chemical agent detector (ISCAD) is an NRL program outside the Joint CBD Program. It strives for ultralow false-alarm rates by integrating two CW point detection technologies (IMS and SAW) that have orthogonal principles of operation. The objective is a handheld chemical detector with subsecond equilibrated chemical detection at threat levels for survey mode and vehicular applications. The ISCAD incorporates NRL s pCAD, which dramatically improves SAW signal kinetics and tolerance to environmental effects. 

Anderson, A.G., et al. (2008) DMS-IMS2, GC-DMS, DMS-MS DMS hybrid devices combining orthogonal principles of separation for challenging applications. In Proceedings of SPIE, 6954. 

The rules for constructing the benzene molecular orbitals are straightforward, but require symmetry and orthogonality principles that have not been presented in this book. 

The slit-ultramicroscope of Siedentopf and Zsigmondy follows the orthogonal principle (see Fig. 22) the light of a self-regulating arc lamp d is concentrated by... 

While samples such as these have obviously been the focus for much GC X GC work in the past, the technology still remains to be demonstrated for many other sample types. It is likely that in the near future, as many more applications are studied, a general theory-or at least a guide to column selection for GC X GC applications-will reveal a logical approach to selection of phases that embodies the principles of orthogonality of separation. 

Applications of Newton s Second Law. Problems involving no unbalanced couples can often be solved with the second law and the principles of kinematics. As in statics, it is appropriate to start with a free-body diagram showing all forces, decompose the forces into their components along a convenient set of orthogonal coordinate axes, and then solve a set of algebraic equations in each coordinate direction. If the accelerations are known, the solution will be for an unknown force or forces, and if the forces are known the solution will be for an unknown acceleration or accelerations. 

Let us now consider the possibilities for deriving an eigenfunction for a particular excited state. The straightforward application of the variation principle (Eq. II.7) is complicated by the additional requirement that the wave function Wk for the state k must be orthogonal to the exact eigenfunctions W0, Wv for all the lower states although these are not usually known. One must therefore try to proceed by way of the secular equation (Eq. III.21). A well-known theorem15 25 says that, if a truncated... 

The Linear Algebraic Problem.—Familiarity with the basic theory of finite vectors and matrices—the notions of rank and linear dependence, the Cayley-Hamilton theorem, the Jordan normal form, orthogonality, and related principles—will be presupposed. In this section and the next, matrices will generally be represented by capital letters, column vectors by lower case English letters, scalars, except for indices and dimensions, by lower case Greek letters. The vectors a,b,x,y,..., will have elements au f it gt, r) . .. the matrices A, B,...,... 

Although it would be possible in principle to choose any set of orthogonal axes in the structural unit to define Ox,x2x3, it is implicit in the discussion of orientation in polymers that the structural unit also has at least orthorhombic symmetry (Point group D2) with regard to the development of orientation. As can be appreciated this can lead to an element of awkwardness in dealing with the results of orientation measurements, because the molecular situation is often more complicated. 

Helical undulators build on this principle by using two orthogonal magnetic field arrays [82, 83]. These permit transverse excursions in perpendicular x and y directions. If the arrays have a relative longitudinal shift, this introduces a phase to the induced perpendicular excursions and when the phase is 90° the electron trajectory can follow left- or right-handed corkscrew paths The emitted radiation is correspondingly right- or left-handed CPL. 

We have applied FCS to the measurement of local temperature in a small area in solution under laser trapping conditions. The translational diffusion coefficient of a solute molecule is dependent on the temperature of the solution. The diffusion coefficient determined by FCS can provide the temperature in the small area. This method needs no contact of the solution and the extremely dilute concentration of dye does not disturb the sample. In addition, the FCS optical set-up allows spatial resolution less than 400 nm in a plane orthogonal to the optical axis. In the following, we will present the experimental set-up, principle of the measurement, and one of the applications of this method to the quantitative evaluation of temperature elevation accompanying optical tweezers. 

Because HPLC and HPCE are based on different physico-chemical principles, HPCE may be expected to address areas in which HPLC has shortcomings [884]. One such area is time of separation. In terms of speed of analysis, selectivity, quantitation, methods to control separation mechanism, orthogonality, CE performs better than conventional electrophoresis and varies from HPLC (Table 4.49). CE has very high efficiency compared to HPLC (up to two orders of magnitude) or GC. For typical capillary dimensions 105—106 theoretical plates are common in CE compared to 20 000 for a conventional HPLC column and... 

The principle of FA and PCA consists in an orthogonal decomposition of the original n x m data matrix X into a product of two matrixes, F (nxk matrix of factor scores, common factors) and L (kxm matrix of factor loadings)... 

In principle, in the absence of noise, the PLS factor should completely reject the nonlinear data by rotating the first factor into orthogonality with the dimensions of the x-data space which are spawned by the nonlinearity. The PLS algorithm is supposed to find the (first) factor which maximizes the linear relationship between the x-block scores and the y-block scores. So clearly, in the absence of noise, a good implementation of PLS should completely reject all of the nonlinearity and return a factor which is exactly linearly related to the y-block variances. (Richard Kramer)... 

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