N-dimensional space

The state variables are the smallest number of states that are required to describe the dynamic nature of the system, and it is not a necessary constraint that they are measurable. The manner in which the state variables change as a function of time may be thought of as a trajectory in n dimensional space, called the state-space. Two-dimensional state-space is sometimes referred to as the phase-plane when one state is the derivative of the other. 

The simplest and fastest techniques for grouping molecules are partitioning methods. Every molecule is represented by a point in an n-dimensional space, the axes of which are defined by the n components of the descriptor vector. The range of values for each component is then subdivided into a set of subranges (or bins). As a result, the entire multidimensional space is partitioned into a number of hypercubes (or cells) of fixed size, and every molecule (represented as a point in this space) falls into one of these cells [57]. 

In the illustration of Fig. 29.4 we regard the matrix X as either built up from n horizontal rows of dimension p, or as built up from p vertical columns x,.of dimension n. This exemplifies the duality of the interpretation of a matrix [9]. From a geometrical point of view, and according to the concept of duality, we can interpret a matrix with n rows and p columns either as a pattern of n points in a p-dimensional space, or as a pattern of p points in an n-dimensional space. The former defines a row-pattern P" in column-space 5, while the latter defines a column-pattern P in row-space S". The two patterns and spaces are called dual (or conjugate). The term dual space also possesses a specific meaning in another... 

As indicated above, if the components of a vector X in n-dimensional space are real, the vector can be written as a column matrix with n rows. Similarly, a second vector Y in the same space can be written as a column matrix of the... 

To determine whether sample patterns in a database are similar to one another, the values of each of the n parameters that define the sample must be compared. Because it is not possible to check whether points form clusters by actually looking into the n-dimensional space—unless n is very small— some mathematical procedure is needed to identify clustered points. Points that form clusters are, by definition, close to one another, therefore, provided that we can pin down what we mean by "close to one another," it should be possible to spot mathematically any clusters and identify the points that comprise them. 

In an n-dimensional space L, the linear operators of the representation can be described by their matrix representatives. This procedure produces a homomorphic mapping of the group G on a group of n x n matrices D(G), i.e., a matrix representation of the group G. From equations (6) it follows that the matrices are non-singular, and that... 

By changing the basis in the n-dimensional space L, the matrices D(R) will be replaced by their transforms by some matrix C. The matrices D (R) = CD R)C l also provide a representation of the group G, which is equivalent to the representation D(R). It should be clear that equivalent representations have the same structure, even though the matrices look different. What is needed to avoid any possible ambiguity are appropriate aspects of D(R) which remain invariant under a change of coordinate axes. One such invariant is easily defined in terms of the diagonal elements of the matrix, as... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

A simplex is a multidimensional geometrical object with n+1 vertices in an n dimensional space. In 2 dimensions the simplex is a triangle, in 3 dimensions it is a tetrahedron, etc. The simplex algorithm can be used for function minimisation as well as maximisation. We formulate the process for minimisation. At the beginning of the process, the functional values at all corners of the simplex have to be determined. Next the corner with the highest function value is determined. Then, this vertex is deleted and a new simplex is constructed by reflecting the old simplex at the face opposite the deleted comer. Importantly, only one new value has to be determined on the new simplex. The new simplex is treated in the same way the highest vertex is determined and the simplex reflected, etc. 

Consider a trajectory in the R" n-dimensional space of x t) and a nearby trajectory x t) + 6x t), where the symbol 6 means an infinitesimal variation, i.e. an arbitrary infinitesimal change not tangent to the initial trajectory. Eq.(55) can be linearized throughout the trajectory to obtain... 

Each spectrum Is regarded as a point In an N dimensional space. The coordinates of each point are the absorbance values at each wavenumber Interval. Several metrics are widely used.(19) The first Is the N dimensional cartesian distance between points 1 and j. 

Simulation of network formation in n-dimensional space using lattice or off lattice computer simulation. 

Clustering is a branch of exploratory analysis able to provide answers about the presence of groupings among objects or variables, by means of a similarity measurement (Vandeginste et al., 1998). The similarity among two objects is defined as an inverse fimction of their distance the more two objects are distant, the less they are similar. Several metrics may be used to evaluate the distance D between two objects i and j in a n-dimensional space. The most common are... 

Geometrically, the factors may be considered as orthogonal axes in n-dimensional space (n = number of variables). The variables are points in that space and the coordinates of the points are the loadings of the variables on the factors. 

The possibility of describing chemical structures numerically with the aid of physico-chemical parameters and indicator variables puts us in the position to determine similarity or dissimilarity of chemical compounds more objectively. Chemical compounds can be represented as points in an n-dimensional space whose coordinates are formed by the parameters which are used to characterize the compounds. This space is therefore called parameter space. The distance of two... 

A simplex is a geometrical figure defined by + 1 vertices in n dimensional space (e.g. in two dimensions this would be a triangle). The procedure is to calculate the energy at n + 1 points and then to replace the worst point (the point of maximum energy) by another P which is the other side of the hyperplane defined by the remaining n points. If P is the centroid of these n points, then the new point is obtained by invertingthrough that is... 

If an n dimensional space is characterized by the n orthonormal basis functions /i, /s, / , then, by definition, the scalar product is... 

With the restriction of uniform mass densities the diffusion equation in n-dimensional space becomes... 

These n conditions define a point in n-dimensional space. We now move away from the stationary point in a controlled manner by relaxing only one of these conditions. For example, we may no longer require the second component of the gradient to be zero. We are then left with n-1 conditions, which define a line in n-dimensional space passing through the stationary point Eq. (6.3) ... 

As remarked in Section II.E an essential feature of the asynchronous treatment is the fact that certain Boolean states have two or more possible next states. Note that this has no philosophical pretentions at the level of the problem of determinism a Boolean state may have more than one possible next state depending on the time delays, essentially in the same way as, in the continuous description, different values of the parameters may result in different trajectories. But, in addition, it must be realized that a Boolean state is not punctual it would correspond, in the continuous description, to a whole domain in the space of variables. More specifically, in a continuous description, one can cut the n-dimensional space of variables into 2" boxes, each of which corresponds to a state of the Boolean description. In the continuous space of variables a state is entirely defined by its coordinates in the Boolean description, it is defined in a broad way by a Boolean number and in more detail by data about the carryover of the time delays. Finally, the simplicity of the Boolean treatment permits to include in an especially easy way the possibility that the values of the time delays fluctuate with time (of course, this has more to do with the problem of determinism). 

Due to the special structure of MATLAB, readers should be familiar with the mathematical concepts pertaining to matrices, such as systems of linear equations, Gaussian elimination, size and rank of a matrix, matrix eigenvalues, basis change in n-dimensional space, matrix transpose, etc. For those who need a refresher on these topics there is a concise Appendix on linear algebra and matrices at the end of the book. 

Since, for the time when c runs through a multitude of positive vectors, Inc moves across the whole of the linear n-dimensional space (In projects a positive real semi-axis to the total axis), the only limitation on if resulting from the existence of a PDE is... 

Figure 15 Schematic illustrating the concept of a simplex. A simplex is a geometric shape formed from N +1 vertices in an N-dimensional space. Hence, for one, two- and three-dimensional spaces, the simplex points comprise the vertices of a line, triangle, and tetrahedron, respectively. The simplices can move through their respective spaces by undergoing repeated reflections and/or changes of shape (see main text).

The ultimate dream for an excellent evaluation would be to delineate the dimensional space (16e) of catalytic reactions. The three principal axes in such a space could be (1) the d" configuration of the metal, (2) the coordination number, and (3) the element group (i.e., in the new ACS nomenclature 3,4,... 12). In terms of the dimensional space many of the transformations considered would occupy a limited volume. In a more elaborate n-dimensional space one could add ligand type and other considerations to better represent the whole set of transformations. 

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