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In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a ...
The group (Z,+) of integers is free of rank 1; a generating set is S = {1}.The integers are also a free abelian group, although all free groups of rank are non-abelian. A free group on a two-element set S occurs in the proof of the Banach–Tarski paradox and is described there.
The Math Images Project is a wiki collaboration between Swarthmore College, the Math Forum at Drexel University, and the National Science Digital Library. The project aims to introduce the public to mathematics through beautiful and intriguing images found throughout the fields of math. The Math Images Project runs on MediaWiki software, as ...
For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products.
The hydrophobic effect can be quantified by measuring the partition coefficients of non-polar molecules between water and non-polar solvents. The partition coefficients can be transformed to free energy of transfer which includes enthalpic and entropic components, ΔG = ΔH - TΔS. These components are experimentally determined by calorimetry ...
A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is the scalar triple product, which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where the latter is a ...
A polar molecule has a net dipole as a result of the opposing charges (i.e. having partial positive and partial negative charges) from polar bonds arranged asymmetrically. Water (H 2 O) is an example of a polar molecule since it has a slight positive charge on one side and a slight negative charge on the other. The dipoles do not cancel out ...
In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects. [1] The biproduct is a generalization of finite direct sums of modules.