Operator first kind

At the point 0 another bifurcation point (of the first kind) is encountered SU -> (SU) — U, resulting in an unstable singular point. Thus, there will be a jump from A to B and the operation will be established on the upper stable cycle. 

Stability of the first kind operator equations with respect to coefficients. 

Here we give the general formulation of the concept of stability with respect to coefficients for the difference scheme in Section 1 by having recourse to an operator equation of the first kind... 

The eigenvalue problem for the Laplace operator in the rectangle Go subject to the first kind boundary conditions... 

The general theory of iterative methods is presented in the next sections with regard to an operator equation of the first kind Au — f, where T is a self-adjoint operator in a finite-dimensional Euclidean space. The applications of such theory to elliptic grid equations began to spread to more and more branches as they took on an important place in real-life situations. 

Difference schemes for elliptic equations of general form. As we have mentioned above, the applications of ATM in the numerical solution of an operator equation of the first kind consist of several steps ... 

The operator scheme is associated with an operator equation of the first kind... 

The first kind, a localized action (often called an operation in code), is an action which a single object is requested to perform it is specified without consideration of the initiator of the action. You can recognize localized actions by their focus on a single distinguished object type ... 

Error, 4,5,6,7 Error of first kind, 14 Error of second kind, 14 Error, Measurement of, 7,8 Evolutionary operation, 64 Experimental designs, 48—63 central composite designs, 52,53,54... 

Though our investigation of p.m. of the second kind has been rather incomplete compared with that of the first kind given in former chapters, it might be safely said that p.m. gives correct results in the sense of die asymptotic expansion, provided the necessary quantity can be calculated by operations within the Hilbert space 4g. In this case, however, care must be taken to use the correct formula, e.g. (19. 6), and not the approximate one such as (19. 7). 

This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator Ay = ySlXl + Vx2x2 supplied by the first kind boundary conditions may be set up in a quite similar manner as follows it is required to find the values of the parameter A (eigenvalues) associated with nontrivial solutions of the homogeneous equation subject to the homogeneous boundary conditions... 

We begin our exposition with a discussion of examples that make it possible to draw fairly accurate outlines of the possible theory regarding these questions and with a listing of the basic results together with the development desired for them. Common practice involves the Laplace operator as the operator R in the case of difference elliptic operators A. The present section is devoted to rather complicated difference problems of the elliptic type. Here and below it is supposed that the domain of interest is a p-dimensional parallelepiped G = 0 < xa < la, a = 1,2,, p with the boundary T (a rectangle for the case p = 2), on which the boundary condition of the first kind is imposed ... 

Enantiotopic ligands and faces are not interchangeable by operation of a symmetry element of the first kind (Cn, simple axis of symmetry) but must be interchangeable by operation of a symmetry element of the second kind (cr, plane of symmetry i, center of symmetry or S , alternating axis of symmetry). (It follows that, since chiral molecules cannot contain a symmetry element of the second kind, there can be no enantiotopic ligands or faces in chiral molecules. Nor, for different reasons, can such ligands or faces occur in linear molecules, QJV or Aoh )... 

Since handedness (left-handed versus right-handed) is important in molecules the eight symmetry operations can be rethought as (C) 4 operations of the first kind (which preserve handedness) translation, identity, rotation, and screw rotation (D) 5 operations of the second kind (which reverse handedness, and produce enantiomorphs) inversion, reflection, rotoinversion, and glide planes. 

Scheme 1. Comparison of whole molecules to determine isomeric relationships. The question marks signify Superposition yes or no . The three tests are Syi —comparison by symmetry operations of the first kind (rotation, torsion) BG—comparison of bonding (connectivity) graphs vertex by vertex Syn—comparison by symmetry operations of the second kind (reflection).

The operators of discrete rotational groups, best described in terms of both proper and improper symmetry axes, have the special property that they leave one point in space unmoved hence the term point group. Proper rotations, like translation, do not affect the internal symmetry of an asymmetric motif on which they operate and are referred to as operators of the first kind. The three-dimensional operators of improper rotation are often subdivided into inversion axes, mirror planes and centres of symmetry. These operators of the second kind have the distinctive property of inverting the handedness of an asymmetric unit. This means that the equivalent units of the resulting composite object, called left and right, cannot be brought into coincidence by symmetry operations of the first kind. This inherent handedness is called chirality. 

The ion sensitive field-effect transistor (ISFET) is a special member of the family of potentiometric chemical sensors [6,7. Like the other members of this family, it transduces information from the chemical into the electrical domain. Unlike the common potentiometric sensors, however, the principle of operation of the ISFET cannot be listed on the usual table of operation principles of potentiometric sensors. These principles, e.g., the determination of the redox potential at an inert electrode, or of the electrode potential of an electrode immersed in a solution of its own ions (electrode of the first kind), all have in common that a galvanic contact exists between the electrode and the solution, allowing a faradaic current to flow, even when this is only a very small measuring current. 

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