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Microsoft Excel provides two ranking functions, the Rank.EQ function which assigns competition ranks ("1224") and the Rank.AVG function which assigns fractional ranks ("1 2.5 2.5 4"). The functions have the order argument, [1] which is by default is set to descending, i.e. the largest number will have a rank 1. This is generally uncommon for ...
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
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.
The family of finite-rank operators () on a Hilbert space form a two-sided *-ideal in (), the algebra of bounded operators on . In fact it is the minimal element among such ideals, that is, any two-sided *-ideal I {\displaystyle I} in L ( H ) {\displaystyle L(H)} must contain the finite-rank operators.
It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon. A similar explanation is based on atlas models, systems of exchangeable positive-valued diffusion processes with drift and variance parameters that depend only on the rank of the process. It has been shown mathematically that Zipf's ...
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, Naimark's theorem says that if {} = is a POVM acting on a Hilbert space of dimension , then there exists a PVM {} = acting on a Hilbert space ′ of dimension ′ and an isometry: ′ such that for all ,
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 parameter space is the space of all possible parameter values that define a particular mathematical model. It is also sometimes called weight space, and is often a subset of finite-dimensional Euclidean space. In statistics, parameter spaces are particularly useful for describing parametric families of probability distributions.