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Grassmann
Hermann Günther Grassmann (, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguistics, linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. His work preceded and exceeded the concept which is now known as a vector space. He introduced the Grassmannian, the space which parameterizes all Dimension, ''k''-dimensional linear subspaces of an ''n''-dimensional vector space ''V''. In linguistics he helped free language history and structure from each other. Biography Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an Ordination, ordained Minister (Christianity), minister who taught mathematics and physics at the Stettin Gymnasium (Germany), Gymnasium, where Hermann was educated. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a field (mathematics), field K that has a differentiable structure. For example, the Grassmannian \mathbf_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf(V) of one dimension lower than V. When V is a real number, real or complex number, complex vector space, Grassmannians are compact space, compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsingular projective algebraic variety. The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to \mathbf_2(\mathbf^4), parameterizing them by what are now called Plücker coordinates. (See below.) Herma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra of a complex vector space. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a Path integral formulation, path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed. Informal discussion Grassmann numbers are generated by anti-commuting elements or objects. The idea of anti-commuting objects arises in multiple areas of mathematics: they are typically seen in differential geometry, where the differential forms are anti-commuting. Differential forms are normally defined in terms of derivatives on a manifold; however, one can contemplate the situation where one "forgets" or "ignores" the existence of any underlying manifold, and "forgets" or "ignores" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann's Law
Grassmann's law, named after its discoverer Hermann Grassmann, is a dissimilatory phonological process in Ancient Greek and Sanskrit which states that if an Aspiration (phonetics), aspirated consonant is followed by another aspirated consonant in the next syllable, the first one loses the aspiration. The descriptive linguistics, descriptive version was given for Sanskrit by Pāṇini. Here are some examples in Greek of the effects of Grassmann's law: * 'I sacrifice (an animal)'; 'it was sacrificed' * 'hair'; 'hairs' * 'to bury (present)'; 'a grave' In reduplication, which forms the perfect tense in both Greek and Sanskrit, if the initial consonant is aspirated, the prepended consonant is unaspirated by Grassmann's law. For instance 'I grow' : 'I have grown'. The fact that deaspiration in Greek took place after the change of Proto-Indo-European language, Proto-Indo-European to (PIE ''*bʰn̥ǵʰús'' > (''pakhús'') not ''bakhús'' but Sanskrit (''bahú'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade (geometry), blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude (mathematics), magnitude of a bivector, -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the Parallelepiped#Parallelotope, parallelotope defined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Color Space
A color space is a specific organization of colors. In combination with color profiling supported by various physical devices, it supports reproducible representations of colorwhether such representation entails an analog or a digital representation. A color space may be arbitrary, i.e. with physically realized colors assigned to a set of physical color swatches with corresponding assigned color names (including discrete numbers infor examplethe Pantone collection), or structured with mathematical rigor (as with the NCS System, Adobe RGB and sRGB). A "color space" is a useful conceptual tool for understanding the color capabilities of a particular device or digital file. When trying to reproduce color on another device, color spaces can show whether shadow/highlight detail and color saturation can be retained, and by how much either will be compromised. A "color model" is an abstract mathematical model describing the way colors can be represented as tuples of numbers (e. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann's Laws (color Science)
Grassmann's laws describe empirical results about how the perception of mixtures of colored lights (i.e., lights that co-stimulate the same area on the retina) composed of different spectral power distributions can be algebraically related to one another in a color matching context. Discovered by Hermann Grassmann these "laws" are actually principles used to predict color match responses to a good approximation under photopic and mesopic vision. A number of studies have examined how and why they provide poor predictions under specific conditions. Modern interpretation The four laws are described in modern texts with varying degrees of algebraic notation and are summarized as follows (the precise numbering and corollary definitions can vary across sources): ;First law: Two colored lights appear different if they differ in either dominant wavelength, luminance or ''purity''. Corollary: For every colored light there exists a light with a complementary color such that a mixture of b ... [...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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivector
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations. Geometrically, a simple bivector can be interpreted as characterizing a directed plane segment (or oriented plane segment), much as vectors can be thought of as characterizing '' directed line segments''. The bivector has an ''attitude'' (or direction) of the plane spanned by and , has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szczecin
Szczecin ( , , ; ; ; or ) is the capital city, capital and largest city of the West Pomeranian Voivodeship in northwestern Poland. Located near the Baltic Sea and the Poland-Germany border, German border, it is a major port, seaport, the largest city of northwestern Poland, and seventh-largest city of Poland. the population was 391,566. Szczecin is located on the Oder River, south of the Szczecin Lagoon and the Bay of Pomerania. The city is situated along the southwestern shore of Dąbie Lake, on both sides of the Oder and on several large islands between the western and eastern branches of the river. It is also surrounded by dense forests, shrubland and heaths, chiefly the Ueckermünde Heath, Wkrzańska Heath shared with Germany (Ueckermünde) and the Szczecin Landscape Park. Szczecin is adjacent to the Police, West Pomeranian Voivodeship, town of Police and is the urban centre of the Szczecin agglomeration, an extended metropolitan area that includes communities in the St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stettin
Szczecin ( , , ; ; ; or ) is the capital and largest city of the West Pomeranian Voivodeship in northwestern Poland. Located near the Baltic Sea and the German border, it is a major seaport, the largest city of northwestern Poland, and seventh-largest city of Poland. the population was 391,566. Szczecin is located on the Oder River, south of the Szczecin Lagoon and the Bay of Pomerania. The city is situated along the southwestern shore of Dąbie Lake, on both sides of the Oder and on several large islands between the western and eastern branches of the river. It is also surrounded by dense forests, shrubland and heaths, chiefly the Wkrzańska Heath shared with Germany (Ueckermünde) and the Szczecin Landscape Park. Szczecin is adjacent to the town of Police and is the urban centre of the Szczecin agglomeration, an extended metropolitan area that includes communities in the German states of Brandenburg and Mecklenburg-Western Pomerania. The city's recorded histo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classics
Classics, also classical studies or Ancient Greek and Roman studies, is the study of classical antiquity. In the Western world, ''classics'' traditionally refers to the study of Ancient Greek literature, Ancient Greek and Roman literature and their original languages, Ancient Greek and Latin. Classics may also include as secondary subjects Greco-Roman Ancient philosophy, philosophy, Ancient history, history, archaeology, anthropology, classical architecture, architecture, Ancient art, art, Classical mythology, mythology, and society. In Western culture, Western civilization, the study of the Ancient Greek and Roman classics was considered the foundation of the humanities, and they traditionally have been the cornerstone of an elite higher education. Etymology The word ''classics'' is derived from the Latin adjective ''wikt:classicus, classicus'', meaning "belonging to the highest class of Citizenship, citizens." The word was originally used to describe the members of the Patri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate causes of Phenomenon, phenomena, and usually frame their understanding in mathematical terms. They work across a wide range of Physics#Research fields, research fields, spanning all length scales: from atom, sub-atomic and particle physics, through biological physics, to physical cosmology, cosmological length scales encompassing the universe as a whole. The field generally includes two types of physicists: Experimental physics, experimental physicists who specialize in the observation of natural phenomena and the development and analysis of experiments, and Theoretical physics, theoretical physicists who specialize in mathematical modeling of physical systems to rationalize, explain and predict natural phenomena. Physicists can apply their k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
