Geometrical point

From a geometric point of view PCA can be described as follows ... 

From a geometrical point of view, it is evident that all molecules having an aU-trans conformation in the alkyl chain close to the chiral centre (see mark in... 

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... 

From a geometric point of view, the autonomous fixed point is the organizing center for the hierarchy of invariant manifolds. From a technical point of view, it is also the expansion center around which all Taylor series expansions are carried out. If the TS trajectory is to take over the role of the fixed point, this observation suggests that it be used as a time-dependent coordinate origin. We therefore introduce the relative coordinates... 

Remark 1.3.3. As every ideal of colength n in R contains m , we can regard it as an ideal in R/m". Thus the Hilbert scheme Hilb"(fZ/mn) also parametrizes the ideals of colength n in R. We also see that the reduced schemes (Hilbn(ii/m ))re,j are naturally isomorphic for k > n. We will therefore denote these schemes also by Hilb"(R)rei. Hilbn(R)red is the closed subscheme with the reduced induced structure of the Grassmannian Grass(n, R jmn) of n dimensional quotients of R/mn whose geometric points are the ideals of colength n of k[[xi,..., z[Pg.10]

The geometry of this problem is shown in Figure 8.11. The linear equality constraint is a straight line, and the contours of constant objective function values are circles centered at the origin. From a geometric point of view, the problem is to find the point on the line that is closest to the origin at x = 0, y = 0. The solution to the problem is at x = 2, y = 2, where the objective function value is 8. 

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... 

From a geometric point of view, clays can be packed rather closely. Muds containing clays, however, have a higher porosity than sand. The higher porosity of the clays is caused in part by the high water content (swelling), which in turn is related to the ion-exchange properties. 

From a geometrical point of view only, this structure could be compared with that of CsCl, with 1 Ca in place of Cs, and the centre of a 6 B octahedron in place of the Cl atom (in the centre of the cell with its axes parallel to the cell axes). Ca is surrounded by 24 B in a regular truncated cube (octahedra and truncated cubes fill space). A number of hexaborides (of Ca, Sr, Ba, Y and several lanthanides and Th, Np, Pu, Am) have been described as pertaining to this structural type. 

In this chapter we return to the question of the geometrical interpretation of the algebraic approach. Specifically, we need to make contact with the concept of the potential function which is central to the geometrical point of view. For example, in three dimensions, one has the Schrodinger equation (1.2)... 

Geometry-based approach from a geometrical point of view, a cavity is a concave empty space that can be described using 2D (surface) or 3D shape descriptors (19-21). We consider three regions in the protein environment the protein bulk, the bulk solvent and the cavity space. The protein bulk is the space filled by the protein atoms. The bulk solvent is the space outside the protein which differentiates from the space inside the protein which defines the cavity where the drug-like molecule is supposed to bind. The identification of protein pockets by numerical methods suppose the capacity to discriminate first the protein bulk from the rest... 

Our conclusions based on the above argument is that the appearance of new facets does not occur simultaneously with the sharp increase in surface solute concentration, but rather a few degrees below the transition. At the transition temperature, the composition of the "geometric points" which represent the orientations at which the compositional change has occurred on the rounded parts of the equilibrium form are indeed enriched in solute. However, the areas associated with the compositional change are too small to be detected by means of the 100 nm analytic probe of the SAM used in this study. The compositional changes can only be detected once the new facet has appeared, and has reached dimensions ofthe order ofthe probe size, or larger. 

The mutual orthogonality of the character vectors is reminiscent of the axes of a Cartesian coordinate system, and suggests the valuable idea that the character vectors of a group form a basis for the symmetry. Any vector can be resolved into components of different symmetry types. The projection of any vector onto any symmetry species is calculable. So we have returned to the geometrical point of view ... 

Suppose k is an algebraically closed field of characteristic p > 0 and suppose x Spec(fc) — Ag>5 is a geometric point given by the polarized abelian variety (X0, A0) over Spec (A ). In [Cr] the complete local ring of Agtd at x is computed. We will describe the result. 

Therefore, we conclude that the sequences S = (, ... j. ) associated to the geometric points of 2 are locally constant on 2. In other words 2 is a disjoint union... 

A complete discrete valuation ring R of characteristic p. Put S — Spec(i ), let rj be the generic point of 5, rj a geometric point lying over tj and let s be the special point of S. 

Finally let us give a list of the isomorphism types of the complete local rings at geometric points x Spec(fc) — X in characteristic p ... 

Using the above what can we say about complete local rings of < (2,p) at geometric points are of the simple types listed above. In particular <5(2, p) -+ Spec(Z) is a locally complete intersection morphism of relative dimension th ree. 

Let x Spec(Fp) — A9td be a geometric point defined by the polarized abelian variety (Xo, Ao) over Fp. We would like to answer the following question If 6 = 6(Xo,Ao), does it follow that x lies in the closure of Ag, Fp Unfortunately, we do not know the answer for p< 3. 

Let x Spec(fc) — Z(pm,S) be a geometric point given by (Xo,Ao) over Spec(Ar). Let ( OiMo) be a principally polarized abelian variety over Spec(fc) such that there exists a diagram... 

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