For the uninitiated, functional analysis represents a difficult concept to grasp. A form of mathematical analysis at its heart, it relates to the study of vector spaces: a highly complex and constantly evolving method for graphing and understanding everything from physical forces to virtual environments. It has profound applications and a wide range of uses within particle physics, as supercolliders continue to teach us more about the way reality functions at some of its smallest levels. Vectors are also used in computer animation, as an advanced, increasingly popular method for simulating virtual environments.
What are Vectors?
A vector is a one-dimensional geometrical “array,” or a range of values along a line. It has length, but no width or breadth. Conceptually, a wide range of data can be plotted along a vector. Some readily identifiable examples of vectors include the distance between any two points on a plane, or a timeline of human achievements. Essentially, the connection between any two points, in any sense (real or conceptual) can be considered a vector. The only other qualifying factor is that of scale: a vector is essentially infinite along its single dimension, as it is capable of being scaled up or down in value.
What is a Vector Space?
A vector space is a multi-dimensional map resulting from the combination of multiple vectors. For example, a map of the Milky Way galaxy plotted out as X, Y and Z values might be considered a vector space, although the term is typically used to describe something more conceptual — usually, something scalable, representing a virtual topography that grows and changes at different scales. A four-dimensional map of the known universe, including a vector for time, would be more likely considered a true vector space. Vector spaces allow for the visualization of extremely complicated concepts, which are otherwise difficult for most people to comprehend.
What does Functional Analysis Entail?
Vectors, vector fields and vector spaces are functionally scalable. Functional analysis involves the practical application of these concepts, through the use of a mathematical structure that limits what might otherwise be an infinite function. The concept can be difficult to imagine in simple terms, but consider the minute vibrations on the skin of a drum as the drummer strikes it. These can be plotted with a vector space. By limiting the range of the otherwise scalable space to that which the drum is capable of providing, its movements can be studied in detail, and the way in which it produces sound can be better understood. This is a very limited example of how functional analysis may be used; in reality, it is used in fields as diverse as particle physics, genetics, and even behavioral analysis. Recent, high-profile applications involve the study of infectious disease and epidemiology.
Functional Analysis in Psychology
As it applies to human psychology, and in particular the treatment of autism, functional analysis is a slightly different animal. In a way, however, it operates on similar principles as the more familiar definition of its mathematical counterpart. In the use of functional analysis to treat autism and similar disorders, complicated equivalents of vectors are taken into consideration. These factors include things like behavioral patterns, outside stimuli, and destructive thought processes. Functional analysis in psychology looks at how these factors interact with each other on a conceptual and a practical level, in order to help individual patients develop healthy coping strategies, which are tailored to their own personal needs.
The field of functional analysis involves various mathematical methods of limiting vector spaces, and then studying the resulting topography. It is used to help further the understanding of virtually every aspect of the physical universe, as well as mathematical theories, by producing comprehensible models of things which otherwise defy most human efforts at understanding them. As such, this kind of mathematical analysis is an invaluable tool in the advancement of science and technology.
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