Differential topology

In differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

, a critical value of a differentiable function

Differentiable function

In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...

  between differentiable manifold

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

s is the image (value) ƒ(x) in N of a critical point

Critical point (mathematics)

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...

 x in M.

The basic result on critical values is Sard's lemma

Sard's lemma

Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis which asserts that the image of the set of critical points of a smooth function f from one Euclidean space or manifold to another has Lebesgue measure 0 – they form a null set...

. The set of critical values can be quite irregular; but in Morse theory

Morse theory

In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...

 it becomes important to consider real

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

-valued functions on a manifold M, such that the set of critical values is in fact finite. The theory of Morse functions shows that there are many such functions; and that they are even typical, or generic in the sense of Baire category.

Statistics

In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, a critical value is the value corresponding to a given significance level. This cutoff value determines the boundary between those samples resulting in a test statistic that leads to rejecting the null hypothesis

Null hypothesis

The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...

 and those that lead to a decision not to reject the null hypothesis. If the calculated value from the statistical test is greater than the critical value, then the null hypothesis is rejected in favour of the alternative hypothesis, and vice versa.

Complex dynamics

In complex dynamics

Complex dynamics

Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions.-Techniques:*General** Montel's theorem...

 , a critical value  is the image of a critical point.