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In mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
Superposed order of the Colosseum. Superposed order (also superimposed) [1] is one where successive storeys of a building have different orders. [2] The most famous ancient example of such an order is the Colosseum at Rome, which had no less than four storeys of superposed orders. [3]
In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition if and only if it satisfies the other definition. In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object.
For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency-domain linear transform methods such as ...
is the linear combination of vectors and such that = +. In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).
Classical elements such as superimposed orders, which refers to the architectural system of using different styles of columns for each storey of a building, was introduced and often used for decorative functions in classical architecture. [4] One of the most popular examples of superimposed orders was on the classical façade of the Colosseum. [19]
Another column can now be observed up close in the St. Peter's Treasury Museum. Other columns from this set of twelve have been lost over the course of time. If these columns really were from one of the Temples in Jerusalem, the spiral pattern may have represented the oak tree which was the first Ark of the Covenant, mentioned in Joshua 24:26. [3]
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution [1]) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object of this category.