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Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and; the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of ...
The circuit rank of a hypergraph can be derived by its Levi graph, with the same circuit rank but reduced to a simple graph. = (+) + where g is the degree sum, e is the number of edges in the given graph, v is the number of vertices, and c is the number of connected components.
An example provided in Slater's original paper is for the iron atom which has nuclear charge 26 and electronic configuration 1s 2 2s 2 2p 6 3s 2 3p 6 3d 6 4s 2.The screening constant, and subsequently the shielded (or effective) nuclear charge for each electron is deduced as: [1]
Trouton’s rule can be explained by using Boltzmann's definition of entropy to the relative change in free volume (that is, space available for movement) between the liquid and vapour phases. [ 2 ] [ 3 ] It is valid for many liquids; for instance, the entropy of vaporization of toluene is 87.30 J/(K·mol), that of benzene is 89.45 J/(K·mol ...
For typical ionic solids, the cations are smaller than the anions, and each cation is surrounded by coordinated anions which form a polyhedron.The sum of the ionic radii determines the cation-anion distance, while the cation-anion radius ratio + / (or /) determines the coordination number (C.N.) of the cation, as well as the shape of the coordinated polyhedron of anions.
There are two possible structures for hydrogen cyanide, HCN and CNH, differing only as to the position of the hydrogen atom. The structure with hydrogen attached to nitrogen, CNH, leads to formal charges of -1 on carbon and +1 on nitrogen, which would be partially compensated for by the electronegativity of nitrogen and Pauling calculated the net charges on H, N and C as -0.79, +0.75 and +0.04 ...
In organic chemistry, the Le Bel–Van 't Hoff rule states that the number of stereoisomers of an organic compound containing no internal planes of symmetry is 2 n, where n represents the number of asymmetric carbon atoms.
The nullity of M is given by m − n + c, where, c is the number of components of the graph and n − c is the rank of the oriented incidence matrix. This name is rarely used; the number is more commonly known as the cycle rank, cyclomatic number, or circuit rank of the graph. It is equal to the rank of the cographic matroid of the graph.