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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 ...
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.
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}.
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.
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 ...
An exact sequence (or exact complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. A short exact sequence is a bounded exact sequence in which only the groups A k, A k+1, A k+2 may be nonzero.