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Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of ''n'' sets, also known as an ''n''-fold Cartesian product, which can be represented by an ''n''-dimensional array, where each element is an ''n''- tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Examples A deck of cards A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are ''not'' linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''product (mathematics), product''. The multiplication of Natural number, whole numbers may be thought of as Multiplication and repeated addition, repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''. Both numbers can be referred to as ''factors''. :a\times b = \underbrace_ For example, 4 multiplied by 3, often written as 3 \times 4 and spoken as "3 times 4", can be calculated by adding 3 copies of 4 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dot Operator
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a ''product''. The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''. Both numbers can be referred to as ''factors''. :a\times b = \underbrace_ For example, 4 multiplied by 3, often written as 3 \times 4 and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together: :3 \times 4 = 4 + 4 + 4 = 12 Here, 3 (the ''multiplier'') and 4 (the ''multiplicand'') are the ''factors'', and 12 is the ''product''. One of the main properties of multiplication ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular array variable, array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operation (mathematics), operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents function composition, composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hybrid Name (botany)
In botanical nomenclature, a hybrid may be given a hybrid name, which is a special kind of botanical name, but there is no requirement that a hybrid name should be created for plants that are believed to be of hybrid origin. The ''International Code of Nomenclature for algae, fungi, and plants'' (ICNafp) provides the following options in dealing with a hybrid: * A hybrid may get a name; this will usually be the option of choice for naturally occurring hybrids. * A hybrid may also be indicated by a formula listing the parents. Such a formula uses the multiplication sign "×" to link the parents. ** "It is usually preferable to place the names or epithets in a formula in alphabetical order. The direction of a cross may be indicated by including the sexual symbols (♀: female; ♂: male) in the formula, or by placing the female parent first. If a non-alphabetical sequence is used, its basis should be clearly indicated." (H.2A.1) * Grex names can be given to orchid hybrids. A hybrid n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotational Symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids. Formal treatment Formally the rotational symmetry is symmetry with respect to some or all rotations in ''m''-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of ''E''+(''m'') (see Euclidean group). Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crocosmia × Crocosmiiflora
''Crocosmia'' × ''crocosmiiflora'', montbretia, is a garden hybrid of '' C. aurea'' and '' C. pottsii'', first bred in 1880 in France by Victor Lemoine.Parnell, J. and Curtis, T. 2012. ''Webb's An Irish Flora.'' Cork University Press. The name "montbretia" is often found in horticultural literature.Everett, T.H., ed. ''Montbreitia'', In: ''New Illustrated Encyclopedia of Gardening'', Greystone Press, New York, 1960. pp. 1181-1182. Description ''Crocosmia'' × ''crocosmiiflora'' grows to 90 cm high with long sword-shaped leaves, shorter than the flowering stem and arising from the plant base, ribbed and up to 20mm wide. The base is a corm, a swollen underground stem lasting one year. The flowers are up to 5 cm long and coloured deep orange. Cultivation In the United States, ''Crocosmia'' × ''crocosmiiflora'' is considered suitable for planting in hardiness zones 5–9, but in more northerly locations it can be planted in the spring and the corms dug out in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Screen Resolution
The display resolution or display modes of a digital television, computer monitor or display device is the number of distinct pixels in each dimension that can be displayed. It can be an ambiguous term especially as the displayed resolution is controlled by different factors in cathode ray tube (CRT) displays, flat-panel displays (including liquid-crystal displays) and projection displays using fixed picture-element (pixel) arrays. It is usually quoted as ', with the units in pixels: for example, ' means the width is 1024 pixels and the height is 768 pixels. This example would normally be spoken as "ten twenty-four by seven sixty-eight" or "ten twenty-four by seven six eight". One use of the term ''display resolution'' applies to fixed-pixel-array displays such as plasma display panels (PDP), liquid-crystal displays (LCD), Digital Light Processing (DLP) projectors, OLED displays, and similar technologies, and is simply the physical number of columns and rows of pixels creating ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
APL (programming Language)
APL (named after the book ''A Programming Language'') is a programming language developed in the 1960s by Kenneth E. Iverson. Its central datatype is the multidimensional array. It uses a large range of special graphic symbols to represent most functions and operators, leading to very concise code. It has been an important influence on the development of concept modeling, spreadsheets, functional programming, and computer math packages. It has also inspired several other programming languages. History Mathematical notation A mathematical notation for manipulating arrays was developed by Kenneth E. Iverson, starting in 1957 at Harvard University. In 1960, he began work for IBM where he developed this notation with Adin Falkoff and published it in his book ''A Programming Language'' in 1962. The preface states its premise: This notation was used inside IBM for short research reports on computer systems, such as the Burroughs B5000 and its stack mechanism when stack machi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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