Orthogonality definition

Orthogonal transformations preserve the lengths of vectors. If the same orthogonal transformation is applied to two vectors, the angle between them is preserved as well. Because of these restrictions, we can think of orthogonal transfomiations as rotations in a plane (although the formal definition is a little more complicated). 

The unit vectors Ci, whose exact definition, meaning and interpretation depend on the particular application at hand, are called basis vectors and form the elements of a basis. They are particularly simple to work with because they are orthogonal. This means that... 

Since the deformation tensor F is nonsingular, it may be decomposed uniquely into a proper orthogonal tensor R and a positive-definite symmetric tensor U by the polar decomposition theorem... 

In conclusion, we observe that many writers in the modern literature seem to agree about the convenience of the definition (Eq. 11.67), but that there has also been a great deal of confusion. For comparison we would like to refer to Slater, and Arai (1957). Almost the only exception seems to be Green et al. (1953, 1954), where the exact wave function is expanded as a superposition of orthogonal contributions with the HF determinant as its first term ... 

S u P space. If the determinants j> are built on orthogonal orbitals, equation (6) is automatically fulfilled which ensures that equation (5) is also valid due to the definition of H°. The matrix elements of H° are then easily calculated ... 

Generally, two vectors that are orthogonal in S will be oblique in 5 , unless the vectors are parallel to the coordinate axes. This is illustrated in Fig. 32.4. Furthermore, if X and y are orthogonal vectors in S, then the vectors W x and W" y are orthogonal in 5 ,. This follows from the definition of orthogonality in the metric W (eq. 32.12) ... 

The above definitions allow the property of orthogonality of the characters to be Stated in the form... 

In [case A] and [case E] mentioned above, the best wave function thus obtained is of particular practical importance. The set of N orbitals appearing in these functions is in general definitely determined, except for an arbitrary numerical factor of which the absolute value is unity, as being mutually orthogonal and having a definite "orbital energy [cf. 

This chapter examines another measure of the space used by 2D separations subject to correlation. Some researchers use the words, peak capacity, to express the maximum number of zones separable under specific experimental conditions, regardless of what fraction of the space is used. By definition, however, the peak capacity is the maximum number of separable zones in the entire space. No substantive reason exists to change this definition. The ability to use the space, however, depends on correlation. In deference to previous researchers (Liu et al., 1995 Gilar et al., 2005b), the author adopts the term, practical peak capacity, to describe the used space. The practical peak capacity is the peak capacity, when the separation mechanisms are orthogonal, but is less than the peak capacity when they are not. The subsequent discussion is based on practical peak capacity. 

The author anticipates that many readers will find the results reported here to be commonplace. If so, then why do we so often report the individual peak capacities of the two dimensions and their product as the 2D peak capacity One answer—the conservative one—is that the latter is indeed the maximum number of peaks that can be separated, in agreement with the definition. A more realistic answer is that it is easy to do and appears more impressive than it really is—especially to those who fund our work. In fact, as a practical metric it is often nonsense. Because orthogonality is so difficult to achieve, especially in 2DLC, the peak capacity is a measure of only instrumental potential, not of separation potential, and consideration of... 

The general definition of a projection has been given on p. 23 in Eq. (2.37). For the purpose of illustration let us write down an example. If s = (Si,Sj,Sk) is a representation of the scattering vector in orthogonal Cartesian coordinates, then the aforementioned ID projection is... 

Operator definitions, phase properties, 206-207 Optical phases, properties, 206-207 Orbital overlap mechanism, phase-change rule, chemical reactions, 450-453 Orthogonal transformation matrix ... 

The vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

The purpose of this Chapter is not to present an exhaustive theory of linear algebra that would take more than a volume by itself to be presented adequately. It is rather to introduce some fundamental aspects of vectors, matrices and orthogonal functions together with the most common difficulties that the reader most probably has encountered in scientific readings, and to provide some simple definitions and examples with geochemical connotations. Many excellent textbooks exist which can complement this introductory chapter, in particular that of Strang (1976). 

The covariance matrix is factored using the diagonal matrix A and the eigenvector matrix U as U UT. Since 5 is symmetric and positive-definite, the eigenvalues are positive and the eigenvectors orthogonal. The inverse S 1 of S can be expanded as UA 1UT and the transformation... 

Alternative definitions lack the (— l)m term. P,(x) is therefore a concise expression for P,°(x). Note that, because of the derivative term in equation (2.6.14), if m > l, P,m(x) = 0. Advanced calculus would show orthogonality properties with... 

Substitution of this into (17), summation over t, followed by k and l, leads, with the help of the orthogonality relations, the representation property of the D, and the definition (10), to the result... 

A supportive method that is orthogonal to the candidate method is also selected on these terms. The elution order of the supportive method is by definition significantly different from the candidate method. Ideally the method screening experiments will provide two sets of conditions, as shown in Figure 4. The utilization of the supportive method maximizes the probability that new unknown related compounds, which possibly co-elute in the candidate method, will be detected and taken into account when evaluating results for subsequent batches of DS or new DP formulations. 

The process of orthogonal testing highlights one of the major technical challenges in solution-phase processes the separation of individual products from each other. In SPS, of course, products are, by definition, always attached to some kind of solid support, such as resin beads or pins. In solution-phase experiments, products are thoroughly mixed with each other. Even if one can identify which among all possible products are desirable, some way must he found to separate those products from those of no interest to the experimenter and from excess reagents that may he present in the reaction mixture. 

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