Search results
Results From The WOW.Com Content Network
In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. [1] Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
In this definition, the choice of is independent of . In other words, the choice of N {\displaystyle N} is uniformly applicable to all natural numbers m {\displaystyle m} . Hence, one can easily see that uniform convergence is a stronger property than pointwise convergence: the existence of uniform limit implies the existence and equality of ...
This definition is technically called Q-convergence, short for quotient-convergence, and the rates and orders are called rates and orders of Q-convergence when that technical specificity is needed. § R-convergence, below, is an appropriate alternative when this limit does not exist.
Since the topological vector space definition of Cauchy sequence requires only that there be a continuous "subtraction" operation, it can just as well be stated in the context of a topological group: A sequence () in a topological group is a Cauchy sequence if for every open neighbourhood of the identity in there exists some number such that ...
The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. This is the “weak convergence of laws without laws being ...
This concept is often contrasted with uniform convergence.To say that = means that {| () |:} =, where is the common domain of and , and stands for the supremum.That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent.
In physiology, psychology, or psychophysics, a limen or a liminal point is a sensory threshold of a physiological or psychological response. Such points delineate boundaries of perception; that is, a limen defines a sensory threshold beyond which a particular stimulus becomes perceivable, and below which it remains unperceivable.