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If is an integer, the answer is , but the precise – or even asymptotic – amount of unfilled space for an arbitrary non-integer is an open question. [ 1 ] 5 unit squares in a square of side length 2 + 1 / 2 ≈ 2.707 {\displaystyle 2+1/{\sqrt {2}}\approx 2.707}
An important class of functions when considering limits are continuous functions. These are precisely those functions which preserve limits , in the sense that if f {\displaystyle f} is a continuous function, then whenever a n → a {\displaystyle a_{n}\rightarrow a} in the domain of f {\displaystyle f} , then the limit f ( a n ) {\displaystyle ...
Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models.These and other types of models can overlap, with a given model involving a variety of abstract structures.
The above concept of relation [a] has been generalized to admit relations between members of two different sets (heterogeneous relation, like "lies on" between the set of all points and that of all lines in geometry), relations between three or more sets (finitary relation, like "person x lives in town y at time z "), and relations between ...
Flowchart of using successive subtractions to find the greatest common divisor of number r and s. In mathematics and computer science, an algorithm (/ ˈ æ l ɡ ə r ɪ ð əm / ⓘ) is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. [1]
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; [1] "degeneracy" is the condition of being a degenerate case. [2] The definitions of many classes of composite or structured objects often implicitly include inequalities.
A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique ...
That is to say, () for any small (that is, indexed by a set , as opposed to a class) diagram of objects in . A continuity space is a generalization of metric spaces and posets, [ 21 ] [ 22 ] which uses the concept of quantales , and that can be used to unify the notions of metric spaces and domains .