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Motor Variable
In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his split-biquaternions. ''Motor variable'' is used here in place of ''split-complex variable'' for euphony and tradition. For example, :f(z) = u(z) + j \ v(z) ,\ z = x + j y ,\ x,y \in R ,\quad j^2 = +1,\quad u(z),v(z) \in R. Functions of a motor variable provide a context to extend real analysis and provide compact representation of mappings of the plane. However, the theory falls well short of function theory on the ordinary complex plane. Nevertheless, some of the aspects of conventional complex analysis have an interpretation given with motor variables, and more generally in hypercomplex analysis. Elementary function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Projective Line Over A Ring
In mathematics, the projective line over a ring is an extension of the concept of projective line over a field (mathematics), field. Given a ring (mathematics), ring ''A'' (with 1), the projective line P1(''A'') over ''A'' consists of points identified by projective coordinates. Let ''A''× be the group of units of ''A''; pairs and from are related when there is a ''u'' in ''A''× such that and . This relation is an equivalence relation. A typical equivalence class is written . , that is, is in the projective line if the right ideal, one-sided ideal generated by ''a'' and ''b'' is all of ''A''. The projective line P1(''A'') is equipped with a homography#Homography groups, group of homographies. The homographies are expressed through use of the matrix ring over ''A'' and its group of units ''V'' as follows: If ''c'' is in Z(''A''×), the center (group theory), center of ''A''×, then the Group action (mathematics), group action of matrix \left(\beginc & 0 \\ 0 & c \end\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Range Of A Function
In mathematics, the range of a function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called ''surjective'' or ''onto''. For any non-surjective function f: X \to Y, the codomain Y and the image \tilde Y are different; however, a new function can be defined with the original function's image as its codomain, \tilde: X \to \tilde where \tilde(x) = f(x). This new function is surjective. Definitions Given two sets and , a binary relation between and is a function (from to ) if for every element in there is exactly one in such that relates to . The sets and are called the '' domain'' and ''codomain'' of , respectively. The ''image'' of the function is the subset of consisting of only those elements of such that there is at least one in with . Usage As the term "range" can have different meanings, it is considered a good practice to define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Connected Component (topology)
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topological space X the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Compactification
{{disambiguation ...
Compactification may refer to: * Compactification (mathematics), making a topological space compact * Compactification (physics), the "curling up" of extra dimensions in string theory See also * Compaction (other) Compaction may refer to: * Soil compaction, for mechanically induced compaction near the ground surface * Compaction of ceramic powders * Compaction (geology), part of the process of lithification involving mechanical dewatering of a sediment by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a ''square''. The term "wikt:oblong, oblong" is used to refer to a non-square rectangle. A rectangle with Vertex (geometry), vertices ''ABCD'' would be denoted as . The word rectangle comes from the Latin ''rectangulus'', which is a combination of ''rectus'' (as an adjective, right, proper) and ''angulus'' (angle). A #Crossed rectangles, crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Unit Disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose distance from ''P'' is less than or equal to one: :\bar D_1(P)=\.\, Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term ''unit disk'' is used for the open unit disk about the origin, D_1(0), with respect to the standard Euclidean metric. It is the interior of a circle of radius 1, centered at the origin. This set can be identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often denoted \mathbb. The open unit disk, the plane, and the upper half-plane The function :f(z)=\frac is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cayley Transform
In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators . Real homography A simple example of a Cayley transform can be done on the real projective line. The Cayley transform here will permute the elements of in sequence. For example, it maps the positive real numbers to the interval ��1, 1 Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions. As a real homography, points are described with projective coordinates, and the mapping is : ,\ 1= \left frac ,\ 1\right\thicksim - 1, \ x + 1= ,\ 1begin1 & 1 \\ -1 & 1 \end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Conformal Map
In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian matrix and determinant, Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (Orthogonal matrix, orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hyperbolic Angle
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrizes the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation. The hyperbola ''xy'' = 1 is rectangular with semi-major axis \sqrt 2, analogous to the circular angle equaling the area of a circular sector in a circle with radius \sqrt 2. Hyperbolic angle is used as the independent variable for the hyperbolic functions sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular (trigonometric) functions by regarding a hyperbolic angle as defining a hyperbolic triangle. The parameter thus becomes one of the most useful in the calculus of real variables. Definition Consider the rectangular hyperbola \textstyle\, and (by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |