A pseudoEuclideanspace is a finite dimension al real number real vector space together with a degenerate form non degenerate definite bilinear form indefinite quadratic form . Such a quadratic form can ... properties of Euclideanspace. For example a straight line may be perpendicular to itself. Another pseudoEuclideanspace is the plane z x y j consisting of split ... of a vector x in the space is defined as q x . In a pseudoEuclideanspace, unlike in a Euclideanspace, there exist non zero vectors with zero magnitude, and also vectors with negative magnitude. Associated with the quadratic form q is the pseudoEuclidean inner product math langle x, y ... is symmetric, but not positive definite, so it is not a true inner product . Whereas Euclideanspace has a unit sphere , pseudoEuclideanspace has the hypersurface s x q x ... group . Every pseudoEuclideanspace has a linear cone given by x q x 0 . When the pseudoEuclideanspace provides a model for spacetime , the linear cone is called the light cone of the origin. See also Pseudo Riemannian manifold References Walter Noll 1964 Euclidean ... of the space, and 1 &le k n . For true Euclideanspace s one has k n , so the quadratic form is positive definite, rather than indefinite. A very important pseudoEuclideanspace is Minkowski space , which is the mathematical setting in which Albert Einstein s theory of special relativity is conveniently formulated. For Minkowski space, n 4 and k 3 so that math q x x 1 2 x 2 2 x 3 2 x 4 2, math The geometry associated with this pseudo ... Rosenfeld, A History of Non Euclidean Geometry Springer 1988 English translation p.266. cite book last Szekeres first Peter title A course in modern mathematical physics groups, Hilbert space, and differential ... Lorentzian manifolds fr Espace pseudo euclidien nl Pseudo euclidische ruimte ru ... more details
or more directions, then the result is a pseudoEuclideanspace . Smooth manifolds built from such spaces ...Image Coord system CA 0.svg thumb right 250px Every point in three dimensional Euclideanspace is determined by three coordinates. In mathematics , Euclideanspace is the Euclidean plane and three dimensional space of Euclidean geometry , as well as the generalizations of these notions to higher dimension s. The term Euclidean distinguishes these spaces from the curved space s of non Euclidean geometry ... geometry Greek geometry defined the Euclidean plane and Euclidean three dimensional space using certain ... mathematics, it is more common to define Euclideanspace using Cartesian coordinates and the ideas ... dimension s. From the modern viewpoint, there is essentially only one Euclideanspace of each dimension ... real coordinate space . A Point geometry point in Euclideanspace may be identified by a tuple ... . Mathematicians often denote the n dimensional space n dimensional Euclideanspace by math mathbb ..., is to define the Euclidean plane as a two dimensional real number real vector space equipped ... to the points of the Euclidean plane, the addition operation in the vector space corresponds to translation ... mathematicians. A final wrinkle is that Euclideanspace is not technically a vector space ... choosing a preferred basis . Euclidean structure Euclideanspace is more than just a real coordinate space. In order to apply Euclidean geometry one needs to be able to talk about the distances ... theorem . Real coordinate space together with this Euclidean structure is called Euclideanspace and often denoted E sup n sup . Many authors refer to R sup n sup itself as Euclideanspace, with the Euclidean structure being understood . The Euclidean structure makes E sup n sup an inner ... of Euclideanspace are then defined as orientation mathematics orientation preserving linear transformation ... orthogonal matrix special orthogonal matrices . Topology of Euclideanspace Since Euclidean ... more details
In EuclideanSpace Category Euclidean symmetries Category Group theory ... rotation about the translated axis, etc. Thus the conjugacy class within the Euclidean group E 3 ... more details
in Euclideanspace conjugate in SO 4 . Each pair of completely orthogonal planes through O ... there is no Conjugation of isometries in Euclideanspace conjugation by any element of SO 4 ...In mathematics , the group mathematics group of rotations about a fixed point in four dimensional Euclideanspace is denoted SO 4 . The name comes from the fact that it is isomorphic to the special orthogonal group of order 4. In this article Rotation mathematics rotation means rotational displacement . For the sake of uniqueness rotation angles are assumed to be in the segment math 0, pi math except where mentioned or clearly implied by the context otherwise. Geometry of 4D rotations There are two kinds of 4D rotations simple rotations and double rotations. Simple rotations A simple rotation R about a rotation centre O leaves an entire plane A through O axis plane fixed. Every plane B that is completely orthogonal ref Two flat subspaces S sub 1 sub and S sub 2 sub of dimensions M and N of a Euclideanspace S of at least M N dimensions are called completely orthogonal if every line in S1 is orthogonal to every line in S2. If dim S M N then S1 and S2 intersect in a single point O. If dim S M N then S1 and S2 may or may not intersect ... space compact 6 parameter Lie group . Each plane through the rotation centre O is the axis plane ... space orientation preserving Isometry isometric linear mappings of a 4D vector space with inner product over the reals onto itself. With respect to an orthonormal basis linear algebra basis in such a space ... algebra rank one and is of unit Norm mathematics Euclidean norm as a 16D vector if and only if A is indeed ... A point in 4D space with Cartesian coordinates u , x , y , z may be represented by a quaternion ... real part, i.e. math a p math . The Euler Rodrigues formula for 3D rotations Our ordinary 3D space is conveniently treated as the subspace with coordinate system OXYZ of the 4D space with coordinate ... more details
such an inversion. DEFAULTSORT Fixed Points Of Isometry Groups In EuclideanSpace Category Euclidean .... The point groups in two dimensions with respect to any point leave that point fixed. 3D Space Only the trivial isometry group C sub 1 sub leaves the whole space fixed. Plane C sub s sub with respect ... more details
wiktionary Euclidean Euclidian Euclidean or, less commonly, Euclidian relates to Euclid an ancient Greek mathematician , a town or others. It may refer to Geometry Euclideanspace , the two dimensional plane and three dimensional space of Euclidean geometry as well as their higher dimensional generalizations. Euclidean geometry , the study of the properties of Euclidean spaces Non Euclidean geometry , systems of points, lines, and planes analogous to Euclidean geometry but without uniquely determined parallel lines Euclidean distance , the distance between pairs of points in Euclidean spaces Euclidean ball , the set of points within some fixed distance from a center point Number theory Euclidean algorithm , a method for finding greatest common divisors Extended Euclidean algorithm , a method for solving the Diophantine equation ax by d where d is the greatest common divisor of a and b . Euclidean domain , a system of numbers or values with properties similar enough to those of the integers to allow the extended Euclidean algorithm to work Euclid s lemma if a prime number divides a product of two numbers, then it divides at least one of those two numbers. Other Euclidean relation , a property of binary relations related to transitivity Euclidean distance map , a digital image in which each pixel value represents the Euclidean distance to an obstacle Euclidean zoning , a system of land use management modeled after the zoning code of Euclid, Ohio See also Euclid disambiguation Euclid s Elements Euclid s Elements , a 13 book mathematical treatise written by Euclid, that includes both geometry and number theory The Euclidean division of the Intermediate Math League of Eastern Massachusetts disambig Category Mathematical disambiguation ... more details
wiktionary pseudo For the novel with the original title Pseudo Hocus Bogus The prefix pseudo from Greek lying, false is used to mark something as falsity false , fraud ulent, or pretending to be something it is not. Biology prefix In biology and botany taxonomy the prefix pseudo or pseud can indicate a species with a visual similarity to another genus . An example is the Iris species Iris pseudacorus , by having leaves similar to those of Acorus calamus in the Acorus genus, having pseud acorus false acorus in its botanical name . See also lookfrom pseudo Falsehood Pseudorealism Deception Mimicry Pseudo.com Pseudo Blood of Our Own Category Prefixes Category Greek loanwords Category Biology prefixes and suffixes da Pseudo nl Pseudo sk Pseudo ... more details
. By using this formula as distance, Euclideanspace or even any inner product space becomes a metric space . The associated Norm mathematics norm is called the Norm mathematics Euclidean norm Euclidean norm . Older literature refers to the metric as Pythagorean metric . Definition The Euclidean distance ... in EuclideanspaceEuclidean n space , then the distance from p to q , or from q to p is given by NumBlk ... q n p n 2 sqrt sum i 1 n q i p i 2 . math EquationRef 1 The position of a point in a Euclidean n space is a Euclidean vector . So, p and q are Euclidean vectors, starting from the origin of the space, and their tips indicate two points. The Euclidean norm , or Euclidean length , or magnitude of a vector ... line segment from the Origin mathematics origin of the Euclideanspace vector tail , to a point in that space vector tip . If we consider that its length is actually the distance from its tail to its tip, it becomes clear that the Euclidean norm of a vector is just a special case of Euclidean distance the Euclidean distance between its tail and its tip. The distance between points p and q ... Euclideanspace, the distance is math d sqrt p 1 q 1 2 p 2 q 2 2 p 3 q 3 2 . math N dimensions In general, for an n dimensional space, the distance is math d p, q sqrt p 1 q 1 2 p 2 q 2 2 ... p i q i 2 ... p n q n 2 . math Squared Euclidean Distance The standard Euclidean distance can be squared ...In mathematics , the Euclidean distance or Euclidean metric is the ordinary distance between two points ... q mathbf p q 1 p 1, q 2 p 2, cdots, q n p n math In a three dimensional space n 3 , this is an arrow ... at two successive instants of time. The Euclidean distance between p and q is just the Euclidean length ... factor of length, which is the Euclidean distance. In higher dimensions there are other possible norms. Two dimensions In the Euclidean plane , if p p sub 1 sub , p sub 2 sub and q ... becomes math d p, q p 1 q 1 2 p 2 q 2 2 ... p i q i 2 ... p n q n 2. math Squared Euclidean ... more details
In mathematics and especially algebraic topology and homology theory , a Euclidean simplex is a special type of convex set in Euclideanspace . It generalises the idea of a triangle, and is used for Triangulation topology triangulation s. Definition Image Tetrahedron.svg thumb 175px A Euclidean 3 simplex in E sup 3 sup . Let nowrap 1 y sub 0 sub , y sub 1 sub , &hellip , y sub k sub be linearly independent points in Euclidean n space, denoted E sup n sup . Let S be a subset of E sup n sup given by math S left sum i 0 k lambda i bold y i lambda i ge 0 text and sum j 0 k lambda j 1 right . math The set mathematics set S is called a Euclidean k simplex with vertices nowrap 1 y sub 0 sub , y sub 1 sub , &hellip , y sub k sub , and is often denoted as nowrap 1 nowiki nowiki y sub 0 sub , y sub 1 sub , &hellip , y sub k sub nowiki nowiki . Given a point nowrap 1 y in S , the sub i sub give Barycentric coordinate system mathematics barycentric coordinate s on S . ref name ABC Citation first .... year Jan 2009 ISBN 0486462390 ref Examples A Euclidean 0 simplex is a point mathematics point . A Euclidean 1 simplex is a line segment . A Euclidean 2 simplex is a triangle . A Euclidean 3 simplex is a tetrahedron . Standard Euclidean simplex The standard Euclidean k simplex , denoted by sub ... with vertices 1,0,0,0 , 0,1,0,0 , 0,0,1,0 and 0,0,0,1 in E sup 4 sup . Faces Given a Euclidean k simplex nowrap 1 nowiki nowiki y sub 0 sub , y sub 1 sub , &hellip , y sub k sub nowiki nowiki , the Euclidean ... , y sub 1 sub , &hellip , y sub p sub nowiki nowiki . ref name ABC A Euclidean k simplex has faces ... face is the k simplex itself. Examples Consider the standard Euclidean 3 simplex sub 3 sub . The 0 ... face is a 2 dimensional face namely the non standard Euclidean 2 simplex given by the triangle ... face is a 1 dimensional face namely the non standard Euclidean 1 simplex given by the line ... , 0,1,0,0 and 0,0,1,0 . The opposite face is a 0 dimensional face namely the non standard Euclidean ... more details
In mathematics, and especially general topology , the Euclidean topology is an example of a topology given to the set of real number s, denoted by R . To give the set R a topology means to say which subset s of R are open , and to do so in a way that the following axiom s are met ref name CEIT Citation first L. A. last Steen first2 J. A. last2 Seebach title Counterexamples in Topology publisher Dover year 1995 ISBN 048668735X ref The union mathematics union of open sets is an open set. The finite intersection mathematics intersection of open sets is an open set. The set R and the empty set are open sets. Construction The set R and the empty set are required to be open sets, and so we define R and to be open sets in this topology. Given two real numbers, say x and y , with nowrap 1 x y we define an uncountably infinite family of open sets denoted by S sub x , y sub as follows ref name CEIT math S x,y r in bold R x r y . math Along with the set R and the empty set , the sets S sub x , y sub with nowrap 1 x y are used as a basis topology basis for the Euclidean topology. In other words, the open sets of the Euclidean topology are given by the set R , the empty set and the unions and finite intersections of various sets S sub x , y sub for different pairs of x , y . Properties The real line, with this topology, is a T5 space T sub 5 sub space . Given two subsets, say A and B , of R with nowrap 1 font style text decoration overline A font B A font style text decoration overline B font , where font style text decoration overline A font denotes the closure topology closure of A , etc., there exist open sets S sub A sub and S sub B sub with nowrap 1 A S sub A sub and nowrap 1 B S sub B sub such that nowrap 1 S sub A sub S sub B sub . ref name CEIT References Reflist Category Topology es Topolog a euclideana nl Euclidische topologie ... more details
. See also fixed points of isometry groups in EuclideanspaceEuclidean plane isometry Poincar group ...Unreferenced date December 2009 In mathematics , the Euclidean group E n , sometimes called ISO n or similar, is the symmetry group of n dimensional Euclideanspace . Its elements, the isometry isometries associated with the Euclidean Metric mathematics metric , are called Euclidean moves . These group ... s, which together generate E sup sup n . E sup sup n is also called a special Euclidean ... isometry. The Euclidean group for SE 3 is used for the kinematics of a rigid body , in classical mechanics . A rigid body motion is in effect the same as a curve in the Euclidean group. Starting ... to the starting orientation by a Euclidean motion, say f t . Setting t 0, we have f 0 I , the identity ... cannot jump from 1 to &minus 1. The Euclidean groups are not only topological group s, they are Lie ... group The Euclidean group E n is a subgroup of the affine group for n dimensions, and in such a way .... In the terms of Felix Klein s Erlangen programme , we read off from this that Euclidean geometry , the geometry of the Euclidean group of symmetries, is therefore a specialisation of affine geometry . All affine theorems apply. The extra factor in Euclidean geometry is the notion of distance , from ... The Euclidean group is a subgroup of the group of affine transformation s. It has as subgroups ... of images under the isometries is topologically discrete space discrete . E.g. for 1 m n a group ... lattice group lattice s. Examples more general than those are the discrete space group s. Countably ... isometries E sup sub n the whole Euclidean group E n one of these groups in an m dimensional subspace combined with a discrete group of isometries in the orthogonal n m dimensional space one of these groups in an m dimensional subspace combined with another one in the orthogonal n m dimensional space ... that line glide reflection 3 See also Euclidean plane isometry . E 3 6 E sup sup 3 identity 0 translation ... more details
For algebraic number fields whose ring of integers has a Euclidean algorithm Euclidean domain In mathematics , a Euclidean field is an ordered field K for which every non negative element is a square that is, x 0 in K implies that x y sup 2 sup for some y in K . Properties Every Euclidean field is an ordered Pythagorean field , but the converse is not true. Examples The real number s R with the usual operations and ordering form a Euclidean field. The field of real algebraic numbers math mathbb R cap mathbb overline Q math is a Euclidean field. The field of hyperreal number s is an Euclidean field. Counterexamples The rational number s Q with the usual operations and ordering do not form a Euclidean field. For example, 2 is not a square in Q since the square root of 2 is irrational number irrational . The complex number s C do not form a Euclidean field since they cannot be given the structure of an ordered field. External links PlanetMath urlname EuclideanField title Euclidean Field References Refimprove date August 2007 Category Field theory he ... more details
In mathematics , Euclidean relations are a class of binary relation s that satisfy a weakened form of transitive relation transitivity that formalizes Euclid s Common Notion 1 in Euclid s Elements The Elements things which equal the same thing also equal one another. Definition A binary relation R on a set mathematics set X is Euclidean sometimes called right Euclidean if it satisfies the following for every a , b , c in X , if a is related to b and c , then b is related to c . ref name fagin citation title Reasoning About Knowledge first Ronald last Fagin authorlink Ronald Fagin publisher MIT Press year 2003 isbn 9780262562003 page 60 url http books.google.com books?id xHmlRamoszMC&pg PA60 . ref To write this in predicate logic math forall a, b, c in X , a ,R , b land a ,R , c to b ,R , c . math Dually, a relation R on X is left Euclidean if for every a , b , c in X , if b is related to a and c is related to a , then b is related to c math forall a, b, c in X , b ,R , a land c ,R , a to b ,R , c . math Relation to transitivity The property of being Euclidean is different from transitive relation transitivity both the Euclidean property and transitivity infer a relation between b and c from relations between a and b and between a and c , but with different argument orderings in the relations. However, if a relation is symmetric relation symmetric , then the argument orders do not matter, thus a relationship which is both symmetric and transitive is both a right and left Euclidean relation. ref name fagin If a relation is Euclidean and Reflexive relation reflexive , it must also be symmetric and transitive, and hence it must be an equivalence relation . Consequently, equivalence relations are exactly the reflexive Euclidean relations. ref name fagin References reflist Category Mathematical relations Category Euclid Relation ... more details
In mathematics , more specifically in abstract algebra and ring theory , a Euclidean domain also called a Euclidean ring is a Ring mathematics ring that can be endowed with a certain structure &ndash namely a Euclidean function, to be described in detail below &ndash which allows a suitable generalization of the Euclidean algorithm . This generalized Euclidean algorithm can be put to many of the same uses as Euclid s original algorithm in the ring of integer s in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular ... of them B zout identity . Also every ideal in a Euclidean domain is principal ideal principal , which implies a suitable generalization of the Fundamental Theorem of Arithmetic every Euclidean domain is a unique factorization domain . It is important to compare the class of Euclidean domains with the larger ... of a Euclidean domain or, indeed, even of the ring of integers , but knowing an explicit Euclidean ... that the integers and any polynomial ring in one variable over a field are Euclidean domains with respect to easily computable Euclidean functions is of basic importance in computational algebra. So, given an integral domain R , it is often very useful to know that R has a Euclidean function in particular, this implies that R is a PID. However, if there is no obvious Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain. Euclidean domains appear in the following chain of subclass set theory class inclusions ... ideal domain s Euclidean domains field mathematics field s Motivation Consider the set of integer ... theory ordering of some sort defined on the ring. This leads to the concept of a Euclidean domain ... b , we may lift this to r 0 or d r d b . The essential idea behind a Euclidean domain is a ring, any ... for the purpose that the Euclidean algorithm should hold, the range is defined to be the natural ... more details
Euclideanspace that passes through the origin. Examples of subspaces include the solution set to a homogeneous system of linear equations , the subset of Euclideanspace described by a system ... 2005. ref In abstract linear algebra, Euclidean subspaces are important examples of vector space s. In this context, a Euclidean subspace is simply a linear subspace of a Euclideanspace. Note on vectors ..., we regard vectors with n components as point mathematics points in an n dimensional space. That is, we identify the set R sup n sup with n dimensional Euclideanspace . Any subset of R sup n sup ... through the origin, i.e. a copy of a lower dimensional or equi dimensional Euclideanspace sitting ... coordinate system R sup 2 sup . In linear algebra , a Euclidean subspace or subspace of R sup ... space , column space , and row space of a matrix mathematics matrix . ref Linear algebra, as discussed ... . Using this mode of thought, a line in three dimensional space is the same as the set of points on the line, and is therefore just a subset of R sup 3 sup . Definition A Euclidean subspace is a subset ... 3 sup is a three dimensional subspace of itself. In n dimensional space n dimensional space , there are subspaces ... space of a matrix main Null space In linear algebra, a homogeneous system of linear equations can ... of solutions to this equation is known as the null space of the matrix. For example, the subspace of R sup 3 sup described above is the null space of the matrix math A left begin alignat 3 1 && 3 && 2 ... space of some matrix see Algorithms algorithms , below . Linear parametric equations The subset ... n sup . Geometrically, the span is the flat through the origin in n dimensional space determined by the points ..., 0, 1 . Column space and row space main Column space Row space A system of linear parametric equations ... space or image mathematics image of the matrix A . It is precisely the subspace of R sup n sup spanned by the column vectors of A . The row space of a matrix is the subspace spanned by its row vectors ... more details
and associated laws qualify EuclideanspaceEuclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space . Vectors play an important ... as a directed line segment, or arrow, in a Euclideanspace. When it becomes necessary to distinguish it from vectors Vector space as defined elsewhere , this is sometimes referred to as a geometric , spatial , or Euclidean vector. As an arrow in Euclideanspace, a vector possesses a definite initial ... math in space represent the same free vector if they have the same magnitude and direction that is, they are equivalent if the quadrilateral ABB A is a parallelogram . If the Euclideanspace is equipped ... representation may be too cumbersome. Vectors in an n dimensional Euclideanspace can be represented ... Euclideanspace or math mathbb R 3 math , vectors are identified with triples of scalar components ... for example, from 1 to 3 in 3 dimensional Euclideanspace, from 0 to 3 in 4 dimensional spacetime ... Covariance and contravariance of vectors Four vector , a non Euclidean vector in Minkowski space ... , and engineering , a Euclidean vector sometimes called a geometric ref harvnb Ivanov 2001 ref or spatial ... a Magnitude mathematics magnitude or euclidean norm length and direction and can be added according to the parallelogram law of addition. A Euclidean vector is frequently represented by a line ... to distinguish Euclidean vectors from the more general concept in linear algebra of vectors as elements of a vector space . General vectors in this sense are fixed size, ordered collections of items as in the case of Euclidean vectors, but the individual items may not be real number s, and the normal Euclidean concepts of length, distance and angle may not be applicable. A vector space with a definition of these concepts is called an inner product space . In turn, both of these definitions ... at each point of a physical space that is, a vector field . In Cartesian space In the Cartesian coordinate ... more details
About the greatest common divisor the mathematics of spaceEuclidean geometry FOR REASONS OF ACCESSIBILITY ... from Heath 1908 300 . In mathematics , the Euclidean algorithm Ref label a a none also called Euclid ... description of the Euclidean algorithm is in Euclid s Elements c. 300 BC , making it one of the oldest ... abstract algebra ic notions such as Euclidean domain s. The Euclidean algorithm has been generalized ... the world. ref Godfried Toussaint , The Euclidean algorithm generates traditional musical rhythms, Proceedings ... both of them without leaving a remainder . The Euclidean algorithm is based on the principle that the greatest ... Euclidean algorithm reversing the steps in the Euclidean algorithm , the GCD can be expressed as a linear ... is 51. Background Greatest common divisor main Greatest common divisor The Euclidean algorithm calculates ... of the Euclidean algorithm is that it can find the GCD efficiently without having to compute ... with the Euclidean algorithm are recursive. Finally, in infinite descent, ref Rosen, p. 492 ... of smaller solutions must end. The latter argument is used to show that the Euclidean algorithm ... The Euclidean algorithm is iterative, meaning that the answer is found in a series of steps the output ... of the Euclidean algorithm can be proven by a two step argument. ref name Stark, pp. 16 20 ... File Euclidean algorithm 1071 462.gif upright thumb alt Animation in which progressively smaller ..., the Euclidean algorithm can be used to find the greatest common divisor of a ..., the steps are class wikitable id basic Euclidean algorithm style margin left auto margin right auto ... q sub 2 sub 21 r sub 2 sub q sub 2 sub 7 and r sub 2 sub 0 algorithm ends Visualization The Euclidean .... ref name Kimberling 1983 cite journal author Kimberling C year 1983 title A Visual Euclidean ... step k , the Euclidean algorithm computes a quotient q sub k sub and remainder r sub k sub from two ... numbers also states that q sub k sub and r sub k sub are unique, but that is not needed for the Euclidean ... more details
In general topology , the pseudo arc is the simplest nondegenerate Indecomposable continuum hereditarily indecomposable continuum topology continuum . Pseudo arc is an arc like homogeneous continuum. R.H. Bing proved that, in a certain well defined sense, most continua in R sup n sup , n &ge 2, are homeomorphic to the pseudo arc. History In 1920, Bronis aw Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane R sup 2 sup must be a Jordan curve . In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in R sup 2 sup that is homeomorphic to each of its nondegenerate subcontinua must be an arc. In 1922, Knaster described the first ... subcontinua, Moise called his example M a pseudo arc and showed that it was hereditarily indecomposable ... s K , Moise s M , and Bing s B are all homeomorphic. Bing also proved that the pseudo arc is typical among the continua in a Euclideanspace of dimension at least 2 or an infinite dimensional separable Hilbert space . ref The history of the discovery of the pseudo arc is described in harv Nadler 1992 , pp 228&ndash 229. ref Construction The following construction of the pseudo arc follows harv Wayne Lewis 1999 . Chains At the heart of the definition of the pseudo arc is the concept of a chain ... C C 1,C 2, ldots,C n math in a metric space such that math C i cap C j ne emptyset math if and only ... of spaces listed above, the pseudo arc is actually very complex. The concept of a chain being crooked defined below is what endows the pseudo arc with its complexity. Informally, it requires ... D math is crooked in math mathcal C . math Pseudo arc For any collect C of sets, let math C math denote the union of all of the elements of C . That is, let math C bigcup S in C S. math The pseudo arc ... math P bigcap i in mathbb N left mathcal C i right . math Then P is a pseudo arc . Notes references .... 3 1922 , 247 286 Wayne Lewis, The Pseudo Arc , Bol. Soc. Mat. Mexicana, 5 1999 , 25&ndash 77 Edwin ... more details
Euclidean quantum gravity refers to a Wick rotation Wick rotated version of quantum gravity , formulated as a quantum field theory . The manifold s that are used in this formulation are 4 dimensional Riemannian manifold s instead of pseudo Riemannian manifold s. It is also assumed that the manifolds are compact space compact , connected space connected and Boundary topology boundaryless i.e. no Gravitational singularity singularities . Following the usual quantum field theoretic formulation, the vacuum to vacuum amplitude is written as a functional integral QFT functional integral over the metric tensor , which is now the quantum field under consideration. math int mathcal D bold g , mathcal D phi , exp left int d 4x sqrt bold g R mathcal L mathrm matter right math where denotes all the matter fields. See Einstein Hilbert action . References G. W. Gibbons and S. W. Hawking eds. , Euclidean quantum gravity , World Scientific 1993 Theories of gravitation quantum gravity Category Quantum gravity quantum stub fa ru ... more details
for a family of Riemannian metrics on the unit ball in Euclideanspace . The simplest of these is called ... of a triangle is not equal to 180 . The surface of a sphere is not a Euclideanspace, but locally ... & Gilbert N. Lewis 1912 The Space time Manifold of Relativity. The Non Euclidean Geometry of Mechanics ... utilizes non Euclidean geometry to explain instantaneous transport through space and time ... uses non Euclidean geometry to create a brilliant, minimal, Escher like world, where geometry and space ... Gray 1989 Ideas of SpaceEuclidean, Non Euclidean, and Relativistic , 2nd edition, Clarendon Press ... types of geometry center Non Euclidean geometry is either of two specific geometries that are, loosely speaking, obtained by negating the Euclidean parallel postulate , namely hyperbolic geometry ... many geometries which are not Euclidean geometry , but only these two ref this is well ... Euclidean geometries. The essential difference between Euclidean and non Euclidean geometry is the nature ... In Euclidean geometry the lines remain at a constant distance from each other even if extended to infinity .... History This section is linked from Parallel postulate Early history While Euclidean geometry ... known mathematics, non Euclidean geometries were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of the non Euclidean geometries began almost ... role in the later development of non Euclidean geometry. These early attempts at challenging ... were equivalent to the Euclidean postulate V. It is extremely important that these scholars ... on Ibn al Haytham s demonstration. Above, we have demonstrated that Pseudo Tusi s Exposition of Euclid ... of these early attempts made at trying to formulate non Euclidean geometry however provided flawed ... is al Tusi s son, Sadr al Din sometimes known as Pseudo Tusi , who wrote a book on the subject ... postulate. He essentially revised both the Euclidean system of axioms and postulates and the proofs ... more details
Taxobox image Pseudonitzschia seriata.jpg image caption Pseudo nitzschia seriata regnum Chromalveolata phylum Heterokont ophyta classis Diatom Bacillariophyceae ordo Bacillariales familia Bacillariaceae genus Pseudo nitzschia genus authority H. Perag. in H. Perag. and Perag. The genus Pseudo nitzschia includes several species of diatom s known to produce the neurotoxin known as domoic acid , a toxin which is responsible for the human illness called amnesic shellfish poisoning . This genus of phytoplankton is known to form harmful algal bloom s in coastal waters of Canada, California, Oregon, and Washington state. Species that have been shown to produce domoic acid although not all strains are toxigenic ref Bates, S.S. and V.L. Trainer. 2006. The ecology of harmful diatoms. In E. Gran li and J. Turner eds. Ecology of harmful algae. Ecological Studies, Vol. 189. Springer Verlag, Heidelberg, p. 81 93. ref , ref Trainer, V.L., B.M. Hickey, and S.S. Bates. 2008. Toxic diatoms. In P.J. Walsh, S.L. Smith, L.E. Fleming, H. Solo Gabriele, and W.H. Gerwick eds. , Oceans and human health risks and remedies ... ioc details.asp?Algae ID 3 Pseudo nitzschia australis http www.bi.ku.dk ioc details.asp?Algae ID 7 Pseudo nitzschia calliantha http www.bi.ku.dk ioc details.asp?Algae ID 106 Pseudo nitzschia cuspidata http www.bi.ku.dk ioc details.asp?Algae ID 4 Pseudo nitzschia delicatissima http www.bi.ku.dk ioc details.asp?Algae ID 99 Pseudo nitzschia fraudulenta http www.bi.ku.dk ioc details.asp?Algae ID 101 Pseudo nitzschia galaxiae http www.bi.ku.dk ioc details.asp?Algae ID 5 Pseudo nitzschia multiseries http www.bi.ku.dk ioc details.asp?Algae ID 6 Pseudo nitzschia multistriata http www.bi.ku.dk ioc details.asp?Algae ID 8 Pseudo nitzschia pugens http www.bi.ku.dk ioc details.asp?Algae ID 9 Pseudo nitzschia seriata http www.bi.ku.dk ioc details.asp?Algae ID 10 Pseudo nitzschia turgidula References ... The genus Pseudo nitzschia Category Diatoms Diatom stub fr Pseudo nitzschia ... more details
mergeto Media event date March 2011 Multiple issues notability November 2008 refimprove November 2008 original research November 2008 A pseudo event is an event or activity that exists for the sole purpose of the media publicity and serves little to no other function in Real life reality real life . Without the media, nothing meaningful actually occurs at the event, so pseudo events are considered real only after they are viewed through news , advertisement , television or other types of Mass media media . An extremely simple example is sitting for a family portrait the event serves no purpose other than to be viewed through a photograph. Other examples include press conference s, advertisements, media spectacles, and many types of news. The term was coined by the theorist and historian Daniel J. Boorstin in his 1962 book The Image A guide to Pseudo events in America ref cite book last Boorstin first Daniel Joseph title The image A guide to pseudo events in America publisher Vintage location New York year 1961 isbn 0 679 74180 1 ref The celebration is held, Photo Op photographs are taken , the occasion is widely reported . The term is closely related to idea of hyperreality and thus postmodernism , although Boorstin s coinage predates by decades the latter two ideas and the related work of postmodern thinkers such as Jean Baudrillard . In recognizing the differences between a pseudo event and a spontaneous event, Boorstin states characteristics of a pseudo event in his book titled Hidden History. He says that pseudo events are dramatic, planned, repeatable, costly, intellectually planned, social, cause other pseudo events, and that one must be known about it to be considered informed. ref Boorstin, D. Hidden History. Harper & Row, 1987. 279 280. Print. ref A number of video art ists have explored the concept of a pseudo event in their work. The group Ant Farm group Ant Farm especially plays with pseudo events, though not so identified, in their works Media Burn 1975 ... more details
Unreferenced date October 2008 Pseudo Demosthenes is the supposed author s of a number of speeches handed down to us under the name of Demosthenes . They include speech 46, 49 against Timotheus general Timotheus , 50 against Polycles , 52 against Callippos , 53 against Nicostratus , 59 against Neaira hetaera Neaira and perhaps 47, attributed to Apollodorus of Acharnae , follower of Demosthenes. Category Ancient Greek pseudepigrapha Ancient Greece writer stub de Pseudo Demosthenes ... more details
A pseudo photograph is an image , whether made by computer graphics or otherwise howsoever, which appears to be a photograph . Although the term pseudo photograph can be applied regardless of what it depicts, in law its meaning is especially relevant regarding child pornography . In the UK, the Criminal Justice and Public Order Act 1994 amended the Protection of Children Act 1978 so as to define the concept of an indecent pseudo photograph of a child . References http www.statutelaw.gov.uk content.aspx?parentActiveTextDocId 1502057&ActiveTextDocId 1502059 See also Bitmap graphics editor Special effects art stub law term stub Category Digital art Category English law ms Pseudofotograf ... more details
Refimprove date March 2012 Pseudo Phocylides is an apocrypha l work claiming to have been written by Phocylides , a Greek Philosophy Greek philosopher of the 6th century. The text is noticeably Jewish , and depends on the Septuagint , although it does not make direct references to either the Hebrew Bible or Judaism . Textual and linguistic studies point to the work as having originally been written in Greek language Greek , and having originated somewhere between 100BC and 100AD, although the oldest surviving manuscripts date from the 10th century AD. Pseudo Phocylides consists of a series of aphorism s, and these refer indirectly to each of the Noachide Laws , as well as the so called unwritten law s of the Greeks. There are about 250 in total, and these are written as a series of hexameter verses, in the form of a teaching manual each maxim directly commanding the reader to obey it Remain not unmarried, lest you die nameless ref line 175, p. 99, The sentences of Pseudo Phocylides, translated by Pieter Willem van der Horst ref Cut not a youth s masculine procreative faculty ref line 187, p. 101, The sentences of Pseudo Phocylides, translated by Pieter Willem van der Horst ref And let not women imitate the sexual role of men ref line 192, p. 101, The sentences of Pseudo Phocylides, translated by Pieter Willem van der Horst ref Long hair is not fit for men, but for voluptuous women ref line 212, p. 101, The sentences of Pseudo Phocylides, translated by Pieter Willem van der Horst ref Some of the maxims in Pseudo Phocylides were copied directly into one of the Sibylline Oracles , found in Book 2. The text of Pseudo Phocylides is published in volume 2 of Old Testament Pseudepigrapha ... Willem van der Horst Pieter van der Horst , The Sentences of Pseudo Phocylides SVTP 4 Leiden Brill ... poetes phocylide sentences.htm Further reading K. W. Niebuhr, Life and Death in Pseudo Phocylides ... de Pseudo Phokylides fr Pseudo Phocylide ... more details
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