Inner and Outer Product

Vectors can be multiplied in two ways one can calculate the inner product, which is a scalar, and one can take the outer product, the result being a matrix. 

The inner product of two vectors is the sum of the products of the corresponding elements. For example  

The length of a vector or the 2-norm of a vector, denoted by o, also uses the inner product. The length of vector a is defined as  

The outer product of two vectors is obtained by multiplying the first element of a column vector by a row vector, thus forming the first row of a matrix. Subsequently, the second element of the coluirm vector is multiplied by the row vector, and so forth. In the example of Eq. (20.7) this would lead to  

Both vectors do not necesarily have to be of the same length. 

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The inner and outer products of two vectors produce a scalar and a matrix, respectively. Two 3-vectors may also be multiplied together to generate a new 3-vector, a procedure called a vector or cross product, with the result given by eq. (16.15). 

Since there is no capillary moisture conduction to the product surface in the frozen product, no first drying stage exists. It starts with the second drying stage. The ice or sublimation front continuously moves inside the product. The required heat for sublimation is transferred to the product by conduction or radiation, or with a combination of both. Moisture vapor diffusion in the dried product follows the Knudsen molecular flow model (see Table 5-2). In the third drying stage residual liquid moisture desorbs to the inner and outer product surface from its bonded state. Since desorption occurs only after complete sublimation of ice, the heat transfer has to be reduced to avoid heating the product over the permissible limit. 

It is also possible to define various mixed products, which are combinations of inner and outer products. Of particular interest is the inner product of a vector a with a bivector b A c, which is defined as the antisymmetric part of their geometric product. In this case, a direct calculation yields the vector... 

Using the distributive properties of inner and outer products, any products of linear combinations can be expanded into linear combinations of products. Thus any scalar-valued expression can be expanded into a polynomial in the inner products of pairs of vectors, bivectors, or trivectors, together with scalars and triple products. Inner products of bivectors and trivectors can be further expanded into polynomials in the inner products of vectors only, using the equivalence to Grami-ans derived in the previous section. Moreover, if two triple products occur in any term, we can likewise expand them as a Gramian into a polynomial in the vector inner products, since... 

Figure 8.49 STEM-BF images of hydrated samples, (a) Inner (IP) and outer product (OP) regions of a paste of alite hydrated for 90 days at 20°C (w/c = 0.4) (b) inner and outer product region of a CEM I cement hydrated for 90 days at 20°C (w/c = 0.4).

Labeling Regulations. The Food, Dmg and Cosmetics Act requires that the cosmetic product be safe under conditions of use and that labeling is not false or misleading. Under this Act, the labeling of a cosmetic product must contain the name and address of the manufacturer, packer, or distributor the net contents and any appropriate warnings. This information must appear on the label of the product, both inner and outer containers. 

Zinc sheet is produced by the pack-rolling process, in which 2—40 rough-roUed sheets are stacked together in packs and roUed simultaneously. Packs must be spUt frequentiy, interchanging inner and outer sheets to equalize temperatures and reductions. With care, good quaUty sheets can be produced by this technique but considerable variations in properties can occur. If a bright ductile product is desired, roUs are held at 120—150°C the reduction on the last pass is 20—40%. 

In this expression A and Q are distance dispersion resulting from electron-vibrational coupling, and frequency tensor (assumed identical in reactant and product states), respectively (work of formation of precursor and successor states is omitted). If we assume that the frequency tensor is diagonal, then we have simply a sum of independent terms for all inner and outer contributing modes. At sufficiently high temperature, the hyperbolic tangents become unity and we obtain the usual (in this approximation) high-temperature expression ... 

MerimetsanAlchemy took place at the Me-rimetsa rehabilitation centre in Tallinn, Estonia in May 2006. As a participatory fashion and social therapy project it aimed at intersecting value production from fashion with manifold hands-on therapy work, replacing some of the sweatshop like production processes at the centre. The endeavor was a reflection of both inner and outer change and the process took form in the shape of garments and photographs. 

Mitochondria are found in all cells. Found in the cell cytoplasm, these structures are surrounded by both an inner and outer membrane. The outer membrane is fairly smooth. The inner membrane has deep folds called cristae, which contain electron-transport compounds. It is on the cristae that most ATP is produced in the final stage of cellular respiration. Mitochondria are called the powerhouses of the cell because they are the site of ATP production, and ATP provides the energy for most cell activities. 

Note that asymmetry between the inner and outer interfaces was observed also for large unilamellar vesicles [146]. However, the ratios of the rates of 3Chl quenching and the yields of radical-ion products on the inner and outer membrane surface for large unilamellar vesicles are of the opposite character compared with small unilamellar vesicles. 

Thus, N represents the fraction of the disk-electrode-formed product (i r). Couples that are used for the determination of the collection efficiency include Fen(CN)6 /Fem(CN)6, Br /Br, hydroquinone/quinone, and Fen(Cp)2/ Feni(Cp)2, where HCp represents cyclopentadiene. The collection efficiency N depends only on the electrode geometry (the disk radius and the inner and outer ring radii). 

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