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Generalization for arbitrary triangles, green area = blue area Construction for proof of parallelogram generalization. Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure ...
Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore rumor spreading, notably through the use of social network analysis software. Under the umbrella of social networks are many different types of graphs. [ 17 ]
The descriptions of these events have elements of truth, and the "big lie" generalizations merge and eventually supplant the public's accurate perception of the underlying events. After World War I the German stab in the back explanation of the cause of their defeat became a justification for Nazi re-militarization and revanchism.
[1] Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry).
The list or nondeterminism monad maps a set X to the set of finite sequences (i.e., lists) with elements from X. The unit maps an element x in X to the singleton list [x]. The multiplication concatenates a list of lists into a single list. In functional programming, the list monad is used to model nondeterministic computations.
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 ...
A theory does not change into a scientific law with the accumulation of new or better evidence. A theory will always remain a theory; a law will always remain a law. [33] [36] [37] Both theories and laws could potentially be falsified by countervailing evidence. [38] Theories and laws are also distinct from hypotheses.
Let be a vector space over a field. [6] For instance, suppose is or , the standard n-dimensional space of column vectors over the real or complex numbers, respectively.In this case, the idea of representation theory is to do abstract algebra concretely by using matrices of real or complex numbers.