Ordinary point

Let B be a tree with n vertices. The vertices of B are partitioned into two classes, in ordinary vertices and exceptional vertices. A point P of 5 is called an ordinary point if a branch with more than n/2 nodes originates in P a vertex is exceptional if it is not ordinary. C. Jordan has formulated the following proposition ... 

The plait point is an ordinary point on the connodal curve, and hence it is immediately evident that the specific volume and entropy in the critical state are intermediate between those of adjectent liquid and vapour phases. 

The synthesis of ideas described above was first achieved by Maxwell and is described by two equations that summarize the essential experimental observations pertaining to the reciprocally related electromagnetic effects1. In SI units and vector notation these equations, valid at every ordinary point... 

Ordinary Points of a Linear Differential Equation. We shall have occasion to discuss ordinary linear differential equations of the second order with variable coefficients whose solutions cannot he obtained in terms of Lhe elementary functions of mathematical analysis, la such cases one of the standard procedures is to derive n pair of linearly independent solutions in the form ofinfinite series and from these series to compute tables of standard solutions. With the aid of such tables the solution appropriate to any given initial conditions may then he readily found. The object of this note is to outline briefly the procedure to he followed in these instances for proofs of the theorems... 

An ordinary point x = a of the second order differential equation... 

It is obvious by inspection that the point fi= 0 is an ordinary point of the equation (17.1). Writing the equation in the form... 

If we use once more the fact that the ordinary locus is dense in Ag,d 0 Fp or every prime p and the fact that any ordinary point of Ag,d lies in one of the substacks Ag, then we get the following theorem. 

The experienced present of the second dimension includes portions of time that from our ordinary point of view are past and future. I have shown the boundaries of this second experienced present (the dotted line) tapering off toward the past and the future in an indefinite way. I suspect that they can also be moved according to how attention is focused. The overall height of this dimension is shown as quite low to represent the fact that the portions of awareness that are involved in psi phenomena are ordinarily very weak compared with ongoing brain experience in ordinary time. The still, small voice of psi is so quiet that it is generally inaudible and even when it speaks, we have to listen very carefully. 

Like in the ordinary point groups, the direct product of irreducible representations is a reducible representation and the characters for individual symmetry operations obey the relationship... 

It has to be noted that the relation between the elements of 0(3)+ (also called SO(3), the group representing proper rotations in 3D coordinate space) and SU(2) (the special unitary group in two dimensions) is not a one-to-one correspondence. Rather, each R matches two matrices u. Molecular point groups including symmetry operations for spinors therefore exhibit two times as many elements as ordinary point groups and are dubbed double groups. 

Some interactions in life have little or no elements of transference involved in them. Transference is more likely if the person or situation you are involved with has some resemblances to your parents or to unresolved situations that involved your parents. If your boss looks somewhat like your father or acts somewhat like him, transference is more likely than if there is litde or no resemblance. If you want something that seems magical from your ordinary point of view, like awakening, like being uught by a Teacher, that may also evoke transference reactions. 

Within the syntopy model, the essential algebraic structure of point symmetry groups is retained (in fact, this structure is extended), and the elements of syntopy groups are derived from ordinary point symmetry operators [252,394,395]. There are, however, alternative approaches for the generalization of symmetry, where fundamentally different algebraic structures are used. 

In Section 3.5.1, the Frobenius series method was discussed with regard to differential equations with regular singular points. In this section, a method is given that effectively deals with differential equations with ordinary points. To illustrate the importance of this method consider the relatively harmless-looking first-order initial value problem... 

Factoring of the Secular Equation. These considerations also lead to the conclusion that the secular equation of such molecules can often be factored further than would appear from the ordinary point group of the whole molecule, for any orientation of the tops. Thus in nitromethane (CII3NO2) the mo.st favorable orientation leads to a point group 6,., which would yield only two factors, of degrees 9 and 0. However, with the i])proximations used above it will be found that the secular equation ac-iually factors into throe parl.s, of degrees. 5, 5, and 4. 

Unlike trajectories of distillation at infinite reflux, which may come off the boundary elements of the concentration simplex in the saddle points S only, reversible distillation trajectories come off in ordinary points x[. ... 

Spin and spatial coordinates do not factorize in Dirac-based theories (in contrast to nonrelativistic quantum mechanics). Hence, ordinary point group symmetries are not a proper approach to understanding half-spin systems. Bethe [590] solved this problem by distinguishing between the null rotation and the rotation by 2n in spinor physics, the concepts, however, were introduced much earlier to mathematics (see the book by Altmann [591] for a historical perspective and a detailed account on double groups). 

It remains to discuss case 1, where both functions in a Kramers pair belong to the same irrep. For this case, ordinary point group symmetry does not promise any a priori blocking. We will therefore look for a unitary transformation on the Kramers pair basis that can block-diagonalize a matrix constructed over functions belonging to a case 1 irrep. The problem can be reduced to the diagonalization of a general 2x2 matrix spanned by the components of a Kramers pair. For the operator this is... 

Four YQCtOTsE, H, D and B are considered for a general description of electromagnetic field. All of these vectors are assumed to be finite throughout the entire field and are also assumed to be continuous fimctions of position and time with continuous derivatives at all ordinary points of the field. The discontinuities may occur on surfaces or the... 

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