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Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous. [6] A chain complex is said to be exact if the image of the (n+1)th map is always equal to the kernel of the nth map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
A homologue (also spelled as homolog) is a compound belonging to a homologous series. [1] Compounds within a homologous series typically have a fixed set of functional groups that gives them similar chemical and physical properties. (For example, the series of primary straight-chained alcohols has a hydroxyl at the end of the carbon chain ...
For example, if X is a topological space then the singular chains C n (X) are formal linear combinations of continuous maps from the standard n-simplex into X; if K is a simplicial complex then the simplicial chains C n (K) are formal linear combinations of the n-simplices of K; if A = F/R is a presentation of an abelian group A by generators ...
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
Homogeneity and heterogeneity; only ' b ' is homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image.A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, income, disease, temperature, radioactivity, architectural design, etc.); one that is heterogeneous ...
In organic chemistry, a homologation reaction, also known as homologization, is any chemical reaction that converts the reactant into the next member of the homologous series. A homologous series is a group of compounds that differ by a constant unit, generally a methylene (−CH 2 −) group.
In the above example graph, we can add a two-dimensional "cell" enclosed between the edges c and d; let's call it A and assume that it is oriented clockwise. Define C 2 as the free abelian group generated by the set of two-dimensional cells, which in this case is a singleton {A}. Each element of C 2 is called a 2-dimensional chain.