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The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis , a domain is a non-empty connected open subset of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n ...
In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function.
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. [1] ( Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor).
There are algorithms to solve all the problems addressed in this article over the integers. In other words, linear algebra is effective over the integers; see Linear Diophantine system for details. More generally, linear algebra is effective on a principal ideal domain if there are algorithms for addition, subtraction and multiplication, and
Assume in some area we want to find solution () of the equation: = (), = (,, …,) with boundary conditions: = (), The basic idea of fictitious domains method is to substitute a given problem posed on a domain , with a new problem posed on a simple shaped domain containing ().
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Boundary value problems are similar to initial value problems.A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).
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