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The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector , the state vector . If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.
For example, for the bond of an electrical system, the flow is the current, while the effort is the voltage. By multiplying current and voltage in this example you can get the instantaneous power of the bond. A bond has two other features described briefly here, and discussed in more detail below. One is the "half-arrow" sign convention.
The configuration space of all unordered pairs of points on the circle is the Möbius strip. In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space.
The state of an isolated physical system is represented, at a fixed time , by a state vector | belonging to a Hilbert space called the state space. Separability is a mathematically convenient hypothesis, with the physical interpretation that the state is uniquely determined by countably many observations.
Non-commutative geometry takes this as inspiration for the study of non-commutative C*-algebras: If there were such a thing as a "non-commutative space X," then its () would be a non-commutative C*-algebra; if in addition the Gelfand–Naimark theorem applied to these non-existent objects, then spaces (commutative or not) would be the same as C ...
The vector space V is called the representation space of φ and its dimension (if finite) is called the dimension of the representation (sometimes degree, as in [18]). It is also common practice to refer to V itself as the representation when the homomorphism φ is clear from the context; otherwise the notation ( V , φ ) can be used to denote ...
If the size of the state space is finite, calculating the size of the state space is a combinatorial problem. [4] For example, in the Eight queens puzzle, the state space can be calculated by counting all possible ways to place 8 pieces on an 8x8 chessboard. This is the same as choosing 8 positions without replacement from a set of 64, or
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers.