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Dual Number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one-dimensional vector space; the general case of n dimensions leads to the Grassmann numbers. The algebra of dual numbers is a ring that is a local ring since the principal ideal generated by ε is its only maximal ideal
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Dual (grammatical Number)
Dual (abbreviated DU) is a grammatical number that some languages use in addition to singular and plural. When a noun or pronoun appears in dual form, it is interpreted as referring to precisely two of the entities (objects or persons) identified by the noun or pronoun acting as a single unit or in unison. Verbs can also have dual agreement forms in these languages. The dual number existed in Proto-Indo-European, persisted in many of its descendants, such as Ancient Greek
Ancient Greek
and Sanskrit, which have dual forms across nouns, verbs, and adjectives, and Gothic, which used dual forms in pronouns and verbs. It can still be found in a few modern Indo-European languages
Indo-European languages
such as Scottish Gaelic, Slovenian, and Sorbian
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Abstract Algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory
Category theory
is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects
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Slope
In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.[1] Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but it might be from the "m for multiple" in the equation of a straight line "y = mx + b" or "y = mx + c".[2] Slope
Slope
is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line
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Shear Mapping
In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction.[1] This type of mapping is also called shear transformation, transvection, or just shearing. An example is the mapping that takes any point with coordinates ( x , y ) displaystyle (x,y) to the point ( x + 2 y , y ) displaystyle (x+2y,y) . In this case, the displacement is horizontal, the fixed line is the x displaystyle x -axis, and the signed distance is the y displaystyle y coordinate. Note that points on opposite sides of the reference line are displaced in opposite directions. Shear mappings must not be confused with rotations
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Absolute Space And Time
Absolute space and time
Absolute space and time
is a concept in physics and philosophy about the properties of the universe
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Galilean Transformation
In physics, a Galilean transformation
Galilean transformation
is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group (assumed throughout below). Without the translations in space and time the group is the homogeneous Galilean group. The Galilean group is the group of motions of Galilean relativity
Galilean relativity
acting on the four dimensions of space and time, forming the Galilean geometry. This is the passive transformation point of view
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Velocity
The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of its speed and direction of motion (e.g. 7001600000000000000♠60 km/h to the north). Velocity
Velocity
is an important concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity
Velocity
is a physical vector quantity; both magnitude and direction are needed to define it. The scalar absolute value (magnitude) of velocity is called "speed", being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s) or as the SI base unit of (m⋅s−1). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector
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Event (relativity)
In physics, and in particular relativity, an event is the instantaneous physical situation or occurrence associated with a point in spacetime (that is, a specific place and time). For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time.[1] Strictly speaking, the notion of an event is an idealization, in the sense that it specifies a definite time and place, whereas any actual event is bound to have a finite extent, both in time and in space.[2][3] Upon choosing a frame of reference, one can assign coordinates to the event: three spatial coordinates x → = ( x , y , z ) displaystyle vec x =(x,y,z) to describe the location and one time coordinate t displaystyle t to specify the moment at which the event occurs
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Quadratic Equation
In algebra, a quadratic equation (from the Latin
Latin
quadratus for "square") is any equation having the form a x 2 + b x + c = 0 displaystyle ax^ 2 +bx+c=0 where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to 0. If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.[1] Because the quadratic equation involves only one unknown, it is called "univariate"
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Parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits any of several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane which is parallel to another plane that is tangential to the conical surface.[a] A third description is algebraic
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Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction. In Euclidean geometry
Euclidean geometry
a transformation is a one-to-one correspondence between two sets of points or a mapping from one plane to another.[1] A translation can be described as a rigid motion: the other rigid motions are rotations, reflections and glide reflections. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. A translation operator is an operator T δ displaystyle T_ mathbf delta such that T δ f ( v ) = f ( v + δ ) . displaystyle T_ mathbf delta f(mathbf v )=f(mathbf v +mathbf delta )
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Polynomial Ring
In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Polynomial
Polynomial
rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear operator
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Linear Algebra
Linear
Linear
algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , displaystyle a_ 1 x_ 1 +cdots +a_ n x_ n =b, linear functions such as ( x 1 , … , x n ) ↦ a 1 x 1 + … + a n x n , displaystyle (x_ 1 ,ldots ,x_ n )mapsto a_ 1 x_ 1 +ldots +a_ n x_ n , and their representations through matrices and vector spaces.[1][2][3] Linear
Linear
algebra is central to almost all areas of mathematics
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Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may be distinct from the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1. Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions
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