Vectors definition

The origin of the molecular system is chosen as the instantaneous center of mass. From the vector definitions above it therefore follows that... 

C12 chains, the vectors between and the hydrogens connected to C were used for the calculation of tilt angles. The average tilt angle q>) was calculated from angles between these vector definitions and the normal of the water surface. 

Since we treat molecular configurations in generic state, we have to define rotational and vibrational vectors definitely. Rotational vectors are defined to be infinitesimal generators of the 50(3) action, which turn out to be expressed as... 

This definition implies that the direction of the vector d was taken into account and that the coefficient c(i,j,fd)sndc(j,i,f,-d) may be different. So, the symmetrical matrix consists of a two step operation ... 

There are several different fomis of work, all ultimately reducible to the basic definition of the infinitesimal work Dn =/d/ where /is the force acting to produce movement along the distance d/. Strictly speaking, both/ and d/ are vectors, so Dn is positive when the extension d/ of the system is in the same direction as the applied force if they are in opposite directions Dn is negative. Moreover, this definition assumes (as do all the equations that follow in this section) that there is a substantially equal and opposite force resisting the movement. Otiierwise the actual work done on the system or by the system on the surroundings will be less or even zero. As will be shown later, the maximum work is obtained when tlie process is essentially reversible . 

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

These two equations are a special case of the corresponding tliree-dimensional definition, connnon in XRD, with the surface nonnal n replacing the third lattice vector c. 

Thennotropic liquid crystal phases are fonned by rodlike or disclike molecules. However, in the following we consider orientational ordering of rodlike molecules for definiteness, although the same parameters can be used for discotics. In a liquid crystal phase, the anisotropic molecules tend to point along the same direction. This is known as the director, which is a unit vector denoted n. 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

The most well-known and at the same time the earliest computer model for a molecular structure representation is a wire frame model (Figure 2-123a). This model is also known under other names such as line model or Drciding model [199]. It shows the individual bonds and the angles formed between these bonds. The bonds of a molecule are represented by colored vector lines and the color is derived from the atom type definition. This simple method does not display atoms, but atom positions can be derived from the end and branching points of the wire frame model. In addition, the bond orders between two atoms can be expressed by the number of lines. 

We must now mention, that traditionally it is the custom, especially in chemo-metrics, for outliers to have a different definition, and even a different interpretation. Suppose that we have a fc-dimensional characteristic vector, i.e., k different molecular descriptors are used. If we imagine a fe-dimensional hyperspace, then the dataset objects will find different places. Some of them will tend to group together, while others will be allocated to more remote regions. One can by convention define a margin beyond which there starts the realm of strong outliers. "Moderate outliers stay near this margin. 

Anuther concept that is extremely powerful when considering lattice structures is the fi i i/imca/ lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar juoduct of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions ... 

These definitions arise from the transformation properties of vectors and can be summarized as follows If in the transfonuation of the coordinate system to another system quantities Aj, A2, ., A trans-... 

A column matrix is an ordered set of numbers therefore, it satisfies the definition of a vector. The 2x1 array... 

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... 

In this section, well-known properties of second-order positive-definite symmetric tensors and functions involving them will be cited without proof. The principal values and principal vectors (m = 1, 2, 3) of a symmetric second-order tensor A are given by... 

By definition, the vector of conjugate momenta corresponding to the vector of generalized internal coordinates q is... 

The two-dimensional Bragg condition leads to the definition of reciprocal lattice vectors at and aj which fulfil the set of equations ... 

First, we must realize that many variables exist in any structural design. We can make a list of structural variables such as sizes, lengths of objects, materials, laminae orientations, and so on. those variables all have influence just as column length, moment of inertia, and Young s modulus influence column-buckling loads. The complete list of design variables will be called the vector Xj, and that vector will have N components. That list constitutes the definition of the structural configuration. 

For a particular field s(r) we can calculate at every point r the mean and the Gaussian curvatures of a corresponding surface passing through r, the normal vector of which is n = s/s. Using the standard geometrical definitions based on h and its derivatives we obtain... 

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