Search results
Results From The WOW.Com Content Network
The infinite words, or ω-words, can likewise be viewed as functions from to Σ. The set of all infinite words over Σ is denoted Σ ω. The set of all finite and infinite words over Σ is sometimes written Σ ∞ or Σ ≤ω. Thus an ω-language L over Σ is a subset of Σ ω.
Square-free words do not have adjacent repeated factors. [1] To clarify, "dining" is not square-free since "in" is repeated consecutively, while "servers" is square-free, its two "er" factors not being adjacent. Thue proves his conjecture on the existence of infinite square-free words by using substitutions. A substitution is a way to take a ...
A possible solution to this problem is that, instead of defining a numeric utility for each infinite outcome sequence, we just define the preference relation between two infinite sequences. We say that agent i {\displaystyle i} (strictly) prefers the sequence of outcomes y t {\displaystyle y_{t}} over the sequence x t {\displaystyle x_{t}} , if ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits , it may be possible to assign a value to a series, called the sum of the series .
A sequence could be a finite sequence from a data source or an infinite sequence from a discrete dynamical system. Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by a recurrence relation or difference equation.
A renewal system is defined to be the set of all infinite concatenations of some fixed finite collection of finite words. Subshifts of finite type are identical to free (non-interacting) one-dimensional Potts models ( n -letter generalizations of Ising models ), with certain nearest-neighbor configurations excluded.
The basic open sets of the product topology are cylinder sets.These can be characterized as: If any finite set of natural number coordinates I={i} is selected, and for each i a particular natural number value v i is selected, then the set of all infinite sequences of natural numbers that have value v i at position i is a basic open set.