Minors of determinant

The cofactor A , of the element is the signed minor of determined by the rule A = (—The value of Al is obtained by forming any of the equivalent expressions 2- =i OijAij, Z =i /A , where the elements must be taken from a single row or a single column of A. 

That is, a determinant of order n is obviously defined in terms of determinants of order n-1. In Equation (A.11), the detenminant is called the minor of element A-n, and the quantity is called the... 

If B— [bij] is an N xN matrix in which bu equals the degree of vertex i, bij = —1 if vertices i and j are adjacent and bij = 0 otherwise, then the number of spanning tree of G is equal to the determinant of any principal minor of B [hararybO]. The extremes occur for totally disconnected graphs that have no spanning trees and thus a complexity of zero, and for complete graphs of order N that contain the maximum possible number of distinct trees on N vertices. ... 

The contractile apparatus may be thought of as the sum of those intracellular components which constitute the machinery of chemomechanical transduction. It is the set of proteins which convert the chemical energy of the terminal phosphate ester bond of ATP into mechanical work. The structure of the contractile apparatus is determined by the connections between the various protein molecules via specific binding sites or, in a minority of cases, via labile covalent linkages. The kinetics of the contractile machinery are determined by the regulation of changes in these connections. 

The coefficient G,y can be determined from the minors of the determinant A, as shown by Rescigno and Segre [1] ... 

The problem of determining the mathematical form of the rate expression for a chemical reaction is one that involves a combination of careful experimental work and sound judgment in the analysis of the data obtained thereby. In many cases the analytical techniques discussed in Section 3.3 are directly applicable to studies of reversible reactions. In other cases only minor modifications are necessary. 

Most organic reactions occur in multi-step reactions, with only a small minority of organic reactions proceeding with a single step. We find, experimentally, that it is extremely unlikely for any two steps to proceed with the same rate constant, which means that we can only follow one reaction at a time. And the reaction that can be followed is always the slowest reaction step, which we call the rate-determining step - a term we often abbreviate to RDS. 

The properties of the minors of the secular determinant of an alternant hydrocarbon may again be used to show that the integrals for which the index is even in (44) and odd in (45) and (46) are zero. It follows that the finite change Aq is an odd function, of Sa, while AFg and Apgt are even. Any inequalities between values of any index for two different positions u), as defined in equations (31) to (34) which arise as first terms of the corresponding infinite series in (44) to (46), persist term-by-term in the expression for the exact finite changes (Baba, 1957). In consequence, the broad agreement with experiment found earlier in the description of ionic and radical reactions by the approximate method carries over to the exact form. 

Does not identify those patients who react to a minor antigenic determinant (i.e., anaphylaxis) does not reliably predict the occurrence of late reactions patients with a negative skin test may still have allergic reactions to therapeutic penicillin... 

Antimicrobial agents are frequently used before the pathogen responsible for a particular illness or the susceptibility to a particular antimicrobial agent is known. This use of antimicrobial agents is called empiric (or presumptive) therapy and is based on experience with a particular clinical entity. The usual justification for empiric therapy is the hope that early intervention will improve the outcome in the best cases, this has been established by placebo-controlled, double-blind prospective clinical trials. For example, treatment of febrile episodes in neutropenic cancer patients with empiric antimicrobial therapy has been demonstrated to have impressive morbidity and mortality benefits even though the specific bacterial agent responsible for fever is determined for only a minority of such episodes. 

A second method of determining major and minor element abundances in meteorites is X-ray fluorescence analysis. In this method, a sample of meteorite is ground into a fine powder and irradiated by a monochromatic X-ray beam. The atoms in the sample absorb some of the X-rays, causing electrons to be ejected. The ejection of an electron causes other electrons to drop to lower energy levels to fill the vacancy, and characteristic X-rays are emitted. The X-ray energies are unique to each element and permit the chemical composition to be determined. X-ray fluorescence was used to determine the compositions of meteorites in the late 1950s and 1960s. If done correctly the compositions have the same accuracy as wet chemical analyses, but some of the measurements in the literature have problems with alkali elements. 

It is apparent from Table IV that trace elements determined by the x-ray fluorescence method are limited to those occurring in whole coals at concentrations of at least a few parts per million. Elements such as selenium, mercury, and antimony, which are generally present in whole coal at levels below 1 ppm, cannot be determined by this method. The major elements in coal, hydrogen, carbon, oxygen, and nitrogen, cannot be determined by x-ray fluorescence, but this should not inhibit the use of the method for trace and minor element determinations. 

Shipe, W. F. and Senyk, G. F. 1980. Evaluation of Milko-Tester Minor for determining fat in milk. J. Assoc. Offic. Anal Chem. 63, 716-719. 

---