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In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric.
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. [1] Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model ).
Inductive reasoning is any of various methods of reasoning in which broad generalizations or principles are derived from a body of observations. [1] [2] This article is concerned with the inductive reasoning other than deductive reasoning (such as mathematical induction), where the conclusion of a deductive argument is certain, given the premises are correct; in contrast, the truth of the ...
Automorphic forms are a generalization of modular forms to more general analytic functions, perhaps of several complex variables, with similar transformation properties. [44] The generalization involves replacing the modular group PSL 2 (R) and a chosen congruence subgroup by a semisimple Lie group G and a discrete subgroup Γ.
The general theory for Lie groups deals with semidirect products of the two types, by means of general results called Mackey theory, which is a generalization of Wigner's classification methods. Representation theory also depends heavily on the type of vector space on which the group acts. One distinguishes between finite-dimensional ...
[60] [73] An intimately connected factor is that the sample is random and representative. This means that it includes a fair and balanced selection of individuals with different key characteristics. For example, when making a generalization about human beings, the sample should include members of different races, genders, and age groups.
Some of President-elect Donald Trump's cabinet picks are facing hurdle after hurdle on the road to confirmation but not those on his economic team (at least so far).
Aikin and Casey expanded on this model in 2010, introducing a third form. Referring to the "representative form" as the classic straw man, and the "selection form" as the weak man, the third form is called the hollow man. A hollow man argument is one that is a complete fabrication, where both the viewpoint and the opponent expressing it do not ...