In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left[Y\right]\hookrightarrow k\left[X\right] between their coordinate rings, such that k\left[X\right] is integral over k\left[Y\right]. This definition can be extended to the quasi-projective varieties, such that a regular map f\colon X\to Y between quasiprojective varieties is finite if any point like y\in Y has an affine neighbourhood V such that U=f^{-1}(V) is affine and f\colon U\to V is a finite map (in view of the previous definition, because it is between affine varieties).

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.

Finite State Machine

The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.

Finite State Machine

Finite State Machine

Finite State Machine

<mathematics, algorithm, theory> (FSM or "Finite State Automaton", "transducer") An abstract machine consisting of a set of states (including the initial state), a set of input events, a set of output events, and a state transition function. The function takes the current state and an input event and returns the new set of output events and the next state. Some states may be designated as "terminal states". The state machine can also be viewed as a function which maps an ordered sequence of input events into a corresponding sequence of (sets of) output events. A deterministic FSM (DFA) is one where the next state is uniquely determinied by a single input event. The next state of a nondeterministic FSM (NFA) depends not only on the current input event, but also on an arbitrary number of subsequent input events. Until these subsequent events occur it is not possible to determine which state the machine is in. It is possible to automatically translate any nondeterministic FSM into a deterministic one which will produce the same output given the same input. Each state in the DFA represents the set of states the NFA might be in at a given time. In a probabilistic FSM [proper name?], there is a predetermined probability of each next state given the current state and input (compare Markov chain). The terms "acceptor" and "transducer" are used particularly in language theory where automata are often considered as abstract machines capable of recognising a language (certain sequences of input events). An acceptor has a single Boolean output and accepts or rejects the input sequence by outputting true or false respectively, whereas a transducer translates the input into a sequence of output events. FSMs are used in computability theory and in some practical applications such as regular expressions and digital logic design. See also state transition diagram, Turing Machine. [J.H. Conway, "regular algebra and finite machines", 1971, Eds Chapman & Hall]. [S.C. Kleene, "Representation of events in nerve nets and finite automata", 1956, Automata Studies. Princeton]. [Hopcroft & Ullman, 1979, "Introduction to automata theory, languages and computations", Addison-Wesley]. [M. Crochemore "tranducters and repetitions", Theoritical. Comp. Sc. 46, 1986]. (2001-09-22)

finite state machine