Reality Roots

Conjugate Pair

Existence & Form
Time & Space
Frequency & Wavelength

Reality Roots
Shape of Nature
Reality
Nature's Symphony from the Vibrating Waves of TimeSpace
Dancing Entities to the Rhythm of Emergence
"Classical" Systems Waltzing on Space Waves Turbulent Jitterbug Particles in "Quantum" Time

NOTE: Work in process web site -- Please "Reload" site each time you view to receive updates of the latest additions


Click
links below to access pages

Simplified
Discussion
Pages

Celestial Bodies

Life

Detailed
Discussion
Pages

Roots

Emergence

Substance

Entity

Causality


String Theory
Theory Page Difference Page
Theory of EveryThing Difference Theory

What Science understands as Theory of Everything
Theory based in sequence, Strings of Mathematical Quantities of
Functional Transformations as in Set Theory.

(In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example,
the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively
they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts
in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics,
and can be used as a foundation from which nearly all of mathematics can be derived.)

Courtesy Wikipedia

Science enumerates States as Functional Transformations (Matrices)
in Three (3) basic processes...

Injective -- Bijective -- Surjective

Injective Function
Injective

Bijctive Function

Bijective

Surjective Function

surjective

String Theory

 

 

 

 

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence (aka bijective function), which uniquely maps all elements in both domain and codomain to each other, (see figures).

Occasionally, an injective function from X to Y is denoted f: XY, using an arrow with a barbed tail (U+21A3 ↣ rightwards arrow with tail).[1] The set of injective functions from X to Y may be denoted YX using a notation derived from that used for falling factorial powers, since if X and Y are finite sets with respectively m and n elements, the number of injections from X to Y is nm (see the twelvefold way).

A function f that is not injective is sometimes called many-to-one. However, this terminology is also sometimes used to mean "single-valued", i.e., each argument is mapped to at most one value.

A monomorphism is a generalization of an injective function in category theory

Species or classes[edit]

[I]t is plain that our distinct species are nothing but distinct complex ideas, with distinct names annexed to them. It is true every substance that exists has its peculiar constitution, whereon depend those sensible qualities and powers we observe in it; but the ranking of things into species (which is nothing but sorting them under several titles) is done by us according to the ideas that we have of them: which, though sufficient to distinguish them by names, so that we may be able to discourse of them when we have them not present before us; yet if we suppose it to be done by their real internal constitutions, and that things existing are distinguished by nature into species, by real essences, according as we distinguish them into species by names, we shall be liable to great mistakes.

—John Locke, An Essay Concerning Human Understanding
Transformations: In mathematics, particularly in semigroup theory, a transformation is any function f mapping a set X to itself, i.e. f:XX. In other areas of mathematics, a transformation may simply be any function, regardless of domain and codomain. This wider sense shall not be considered in this article; refer instead to the article on function for that sense.

Examples include linear transformations and affine transformations, rotations, reflections and translations. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra, and described explicitly using matrices.

Translation: A translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In other words, if v is a fixed vector, then the translation Tv will work as Tv(p) = p + v.

For the purpose of visualization, consider a browser window. This window, if maximized to full dimensions of the screen, is the reference plane. Imagine one of the corners as the reference point or origin (0, 0).

Consider a point P(xy) in the corresponding plane. Now the axes are shifted from the original axes to a distance (hk) and this is the corresponding reference axes. Now the origin (previous axes) is (xy) and the point P is (XY) and therefore the equations are:

X = xh or x = X + h or h = xX and Y = yk or y = Y + k or k = yY.

Replacing these values or using these equations in the respective equation we obtain the transformed equation or new reference axes, old reference axes, point lying on the plane.

Reflection: A reflection is a map that transforms an object into its mirror image with respect to a "mirror", which is a hyperplane of fixed points in the geometry. For example, a reflection of the small Latin letter p with respect to a vertical line would look like a "q". In order to reflect a planar figure one needs the "mirror" to be a line (axis of reflection or axis of symmetry), while for reflections in the three-dimensional space one would use a plane (the plane of reflection or symmetry) for a mirror. Reflection may be considered as the limiting case of inversion as the radius of the reference circle increases without bound.
Reflection is considered to be an opposite motion since it changes the orientation of the figures it reflects.

 

I

 

 

Glide reflection[edit]

A glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line. Reversing the order of combining gives the same result. Depending on context, we may consider a simple reflection (without translation) as a special case where the translation vector is the zero vector.

Rotation[edit]

Main article: Rotation (geometry)

A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation. You can rotate the object at any degree measure, but 90° and 180° are two of the most common. Rotation by a positive angle rotates the object counterclockwise, whereas rotation by a negative angle rotates the object clockwise.

Scaling[edit]

Main article: Scaling (geometry)

Uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety or dilation. The result of uniform scaling is similar (in the geometric sense) to the original.

More general is scaling with a separate scale factor for each axis direction; a special case is directional scaling (in one direction). Shapes not aligned with the axes may be subject to shear (see below) as a side effect: although the angles between lines parallel to the axes are preserved, other angles are not.

Shear[edit]

Main article: Shear mapping

Shear is a transform that effectively rotates one axis so that the axes are no longer perpendicular. Under shear, a rectangle becomes a parallelogram, and a circle becomes an ellipse. Even if lines parallel to the axes stay the same length, others do not. As a mapping of the plane, it lies in the class of equi-areal mappings.

More generally[edit]

More generally, a transformation in mathematics means a mathematical function (synonyms: map and mapping). A transformation can be an invertible function from a set X to itself, or from X to another set Y. The choice of the term transformation may simply flag that a function's more geometric aspects are being considered (for example, with attention paid to invariants).

A strong nonlinear transformation applied to a plane through the origin
Transformation (before).png Transformation (after).png
Before After

Partial transformations[edit]

The notion of transformation generalized to partial functions. A partial transformation is a function f: AB, where both A and B are subsets of some set X.[5]

COPYWRITE--- 2011
All Rights Reserved