be more than one vector of norm equal to 0. These vectors are often called null vectors. Examples The Minkowski space Causal structure light like vectors of Minkowski space are null vectors. In general .... In the Verma module of a Lie algebra there are null vectors. Sources Linear Algebra 4th Edition , S. Lipcshutz ... 521 86153 3 Category Linear algebra Category Zero Category Vectors be x old ca Vector ... more details
wiktionary frak Frak may refer to Frak expletive , a pseudo profanity popularized through the reimagined Battlestar Galactica series Frak , a 1980s computer game FRAK Mutual Fund Ticker of Van Eck Market Vectors Unconventional Oil & Gas ETF See also Frac disambiguation disambig ... more details
Multiple issues unreferenced December 2009 orphan February 2009 context October 2009 Hybrid vectors are Vector molecular biology vector virus es that are genetically engineered to have qualities of more than one vector. DEFAULTSORT Hybrid Vector Category Viruses Category Molecular biology Biotech stub Virus stub ... more details
Silpelit might be A figure from Eduard M rike Eduard Friedrich M rike s poem Elfenlied . A specific type of the fictional mutant, human evolved species of the Japanese anime and manga Elfen Lied . The Silpelit is the offspring of Homo sapiens who had their genes mutated by interaction with Diclonius vectors or from exposure to the vector virus . disambig es Silpelit ... more details
The term range space has multiple meanings in mathematics In linear algebra , it refers to the column space of a matrix, the set of all possible linear combinations of its column vectors. In computational geometry , it refers to a hypergraph , a pair X, R where each r in R is a subset of X. disambig ... more details
In mathematics , a sparse vector is a Euclidean vector vector whose elements are mostly zeros. Mathematically, a sparse vector has a low L0 norm Zero norm . See also Sparse matrix Norm mathematics Compressed sensing Category Vectors Linear algebra stub cs dk vektor pl Wektor rzadki ... more details
by all vectors a sub 1 sub ,..., a sub 26 sub in R sup 25,1 sup such that either all the a sub i ... by 2 norm 0 vectors z and w with inner product 1. So we can write vectors of II sub 25,1 sub ... sup 2 mn . Conway showed that the roots norm 2 vectors having inner product 1 with w 0,0,1 are the simple roots of the reflection group. There are the vectors ,1, sup 2 sup 2 1 for in the Leech lattice ... simple roots correspond to the Leech lattice vectors. The group Aut sup sup II sub 25,1 sub Ref ... simple group. Vectors Every non zero vector of II sub 25,1 sub can be written uniquely as a positive integer multiple of a primitive vector, so to classify all vectors it is sufficient to classify the primitive vectors. Positive norm vectors Any two positive norm vectors with the same norm are conjugate under the automorphism group. Norm zero vectors There are 24 orbits of norm 0 vectors ... group of II sub 25,1 sub is the Leech lattice. Norm 2 vectors There are 121 orbits of vectors ... v . Norm 4 vectors There are 665 orbits of vectors v of norm 4, corresponding to the 665 isomorphism ... of the even vectors of the lattice L is isomorphic to the orthogonal complement of the vector v . Other vectors There are similar but increasingly complicated descriptions of the vectors of norm 2 n for n 3, 4, 5, ..., and the number of orbits of such vectors increases quite rapidly. References ... more details
Unreferenced date December 2009 In the mathematical field of projective geometry , a projective frame is an ordered collection of points in projective space which can be used as reference points to describe any other point in that space. For example Given three distinct points on a projective line , any other point can be described by its cross ratio with these three points. In a projective plane , a projective frame consists of four points, no three of which lie on a projective line. In general, let K P sup n sup denote n dimensional projective space over an arbitrary field K . This is the projectivization of the vector space K sup n 1 sup . Then a projective frame is an n 2 tuple of points in general position in K P sup n sup . Here general position means that no subset of n 1 of these points lies in a hyperplane a projective subspace of dimension n &minus 1 . Sometimes it is convenient to describe a projective frame by n 2 representative vectors v sub 0 sub , v sub 1 sub , ..., v sub n 1 sub in K sup n 1 sup . Such a tuple of vectors defines a projective frame if any subset of n 1 of these vectors is a basis for K sup n 1 sup . The full set of n 2 vectors must satisfy linear dependence relation math lambda 0 v 0 lambda 1 v 1 cdots lambda n v n lambda n 1 v n 1 0. math However, because the subsets of n 1 vectors are linearly independent, the scalars sub j sub must all be nonzero. It follows that the representative vectors can be rescaled so that sub j sub 1 for all j 0,1,..., n 1. This fixes the representative vectors up to an overall scalar multiple. Hence a projective frame is sometimes defined to be a n 2 tuple of vectors which span K sup n 1 sup and sum to zero. Using such a frame, any point p in K P sup n sup may be described by a projective version of barycentric coordinates mathematics barycentric coordinates a collection of n 2 scalars sub j sub which sum to zero, such that p is represented by the vector math mu 0 v 0 mu 1 v 1 cdots mu n v n mu n 1 v ... more details
In linear algebra , orthogonalization is the process of finding a set of orthogonal vector s that span linear algebra span a particular linear subspace subspace . Formally, starting with a linearly independent set of vectors v sub 1 sub ,..., v sub k sub in an inner product space most commonly the Euclidean space R sup n sup , orthogonalization results in a set of Orthogonality orthogonal vectors u sub 1 sub ,..., u sub k sub that generate the same subspace as the vectors v sub 1 sub ,..., v sub k sub . Every vector in the new set is orthogonal to every other vector in the new set and the new set and the old set have the same linear span . In addition, if we want the resulting vectors to all be unit vector s, then the procedure is called orthonormalization . Colloquially, orthogonalization is the process of splitting a problem or system into its distinct components. Orthogonalization algorithms Methods for performing orthogonalization include Gram Schmidt process , which uses Projection linear algebra projection Householder transformation , which uses Reflection mathematics reflection Givens rotation When performing orthogonalization on a computer, the Householder transformation is usually preferred over the Gram Schmidt process since it is more Numerical stability numerically stable , i.e. rounding errors tend to have less serious effects. On the other hand, the Gram Schmidt process produces the jth orthogonalized vector after the jth iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram Schmidt process applicable for iterative method s like the Arnoldi iteration . The Givens rotation is more easily parallelized than Householder transformations. See also Orthogonality Biorthogonal system Category Linear algebra de Orthogonalisierungsverfahren ja ro Ortogonalizare ru zh ... more details
of arthropod disease Vector epidemiology vectors , and involves a tremendous outreach ... Stable flies . Pathogen infection transmitted by insect or other arthropod vectors. Diseases carried by insects and other arthropod vectors affect more than 700 million people every year, and are considered ... fever Vectors Aedes aegypti main vector Aedes albopictus minor vector threatens 50 million people .... Malaria Vectors Anopheles mosquitoes 500 million become severely ill with malaria every year and more than 1 million die. Leishmaniasis Vectors species in the genus Lutzomyia in the New World and Phlebotomus ... Saharan Africa World Health Organization WHO Typhus Vectors mites, fleas and body louse body lice ... vectors the mosquito species Culex , Anopheles , Mansonia , and Aedes affects over 120 million people. Yellow Fever Principal vectors Aedes simpsoni , Aedes africanus A. africanus , and Aedes aegypti ... Vector Mosquitoes, main vectors A. vigilax , Aedes camptorhynchus , and Culex annulirostris Barmah Forest Virus Vector Known vectors Culex annulirostris , Ocleratus vigilax and O. camptorhynchus and Culicoides ... vectors, the most important being Culex tritaeniorhynchus . West Nile virus Vectors vary according ... quinquefasciatus Southeast are the main vectors. Lyme disease Vectors several species of the genus ... Ixodes ticks. Carrion s disease Vectors sandflies of the genus Lutzomyia . Chagas disease Vector assassin bugs of the subfamily Triatominae . The major vectors are species in the genera Triatoma , Rhodnius , and Panstrongylus . Chikungunya Vectors Aedes mosquitoes Human ewingii ehrlichiosis Vector ... Valley Fever RVF Vectors fleas in the genera Aedes and Culex Scrub typhus Vector Harvest mite Chigger ... borne diseases and their principal vectors . Unpublished document WHO VBC 89.967. Geneva World ... more details
of mass vectors, there are still some debates. ref http en.wikipedia.org wiki Two New Sciences ref Many critics believe that mass vectors are in direct contrast to Galileo s classic idea of Falling Bodies. They believe that mass vectors travel through a special media, similar to that of light traveling ... of mass vectors as quantities which are normally affected by the physical world. Reflist Category ... more details
File Linearly independent vectors in R3.svg thumb right Linearly independent vectors in R sup 3 sup . File Linearly dependent vectors in R3.svg thumb right Linearly dependent vectors in a plane in R sup ... if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent . For instance ... dependent end matrix math weights 9, 5, 4 Here the first three vectors are linearly independent but the fourth ... vectors together are linearly dependent. Linear dependence is a property of the family, not of any ... of linear dependence between random variable s. Definition A finite subset of n vectors, v sub 1 sub ..., then the vectors are said to be linearly independent . Alternatively, linear independence can be directly defined as follows a set of vectors is linearly independent if and only if the only representations ... 1, 2, ..., n . A set of vectors is then said to be linearly dependent if it is not linearly independent ... is a linear combination of the rest of the family. A set of vectors which is linearly independent ... two vectors, and it makes the set of vectors linearly dependent , that is, one of the three vectors ... to the linearly independent set. In general, n linearly independent vectors are required to describe any location in n dimensional space. Example I The vectors 1, 1 and &minus 3, 2 in math mathbb ... determinants An alternative method uses the fact that n vectors in math mathbb R n math are linearly dependent if and only if the determinant of the matrix mathematics matrix formed by taking the vectors as its columns is zero. In this case, the matrix formed by the vectors is math A begin bmatrix ... det A 1 cdot2 1 cdot 3 5 ne 0 . , math Since the determinant is non zero, the vectors 1, 1 and &minus 3, 2 are linearly independent. Otherwise, suppose we have m vectors of n coordinates, with m ..., the reverse is true. That is, we can test whether the m vectors are linearly dependent by testing ... more details
Image Lattice reduction.svg thumb right 300px Lattice reduction in two dimensions the black vectors are the given basis for the lattice represented by blue dots , the red vectors are the reduced basis In mathematics, the goal of lattice basis reduction is given an integer lattice group lattice basis as input, to find a basis linear algebra basis with short, nearly orthogonal vectors. This is realized using different algorithms, whose running time is usually at least exponential in the dimension of the lattice. Nearly Orthogonal One measure of nearly orthogonal is the orthogonality defect . This compares the product of the lengths of the basis vectors with the volume of the parallelepiped they define. For perfectly orthogonal basis vectors, these quantities would be the same. Any particular basis of math n math vectors may be represented by a Matrix mathematics matrix math B math , whose columns are the basis vectors math b i, i 1, ldots, n math . In the fully dimensional case where the number of basis vectors is equal to the dimension of the space they occupy, this matrix is square, and the volume of the fundamental parallelepiped is simply the absolute value of the determinant of this matrix math det B math . If the number of vectors is less than the dimension of the underlying space, then volume is math sqrt det B T B math . For a given lattice math Lambda math , this volume is the same up to sign for any basis, and hence is referred to as the determinant of the lattice math det Lambda math or lattice constant math d Lambda math . The orthogonality defect is the product of the basis ... c math where c is some constant depending only on the number of basis vectors and the dimension of the underlying ... dimensions For a basis consisting of just two vectors, there is a simple and efficient method ... step the larger of the two vectors is reduced by adding or subtracting an integer multiple of the smaller ... vectors that do not sum to zero between which the relation is sought. The LLL algorithm for computing ... more details
Cosine similarity is a measure of similarity between two vectors by measuring the cosine of the angle between them. The cosine of 0 is 1, and less than 1 for any other angle the lowest value of the cosine is 1. The cosine of the angle between two vectors thus determines whether two vectors are pointing in roughly the same direction. This is often used to compare documents in text mining . In addition, it is used to measure cohesion within clusters in the field of data mining . ref P. N. Tan, M. Steinbach & V. Kumar, Introduction to Data Mining , , Addison Wesley 2005 , ISBN 0 321 32136 7, chapter 8 page 500. ref Definition The cosine of two vectors can be easily derived by using the Euclidean vector Dot product Euclidean dot product formula math mathbf a cdot mathbf b left mathbf a right left mathbf b right cos theta math Given two Vector geometric vectors of attributes, A and B , the cosine similarity, , is represented using a dot product and Magnitude mathematics Euclidean vectors magnitude as math text similarity cos theta A cdot B over A B frac sum limits i 1 n A i times B i sqrt sum limits i 1 n A i 2 times sqrt sum limits i 1 n B i 2 math The resulting similarity ranges from &minus 1 meaning exactly opposite, to 1 meaning exactly the same, with 0 usually indicating independence, and in between values indicating intermediate similarity or dissimilarity. For text matching, the attribute vectors A and B are usually the tf idf term frequency vectors of the documents. The cosine similarity can be seen as a method of normalizing document length during comparison. In the case ... frequencies tf idf weights cannot be negative. The angle between two term frequency vectors ... the same calculation of similarity, the normalised angle between the vectors can be used as a bounded ... or distance within a set of vectors is important, then which function is used is immaterial as the a resulting ... vector bit vectors . While the formula extends to vectors in general, it has quite different properties ... more details
Unreferenced date December 2009 This page is an overview of the common notations used when working with vectors ... named v . In handwriting, where boldface type is either unavailable or unwieldy, vectors are often ... vectors A rectangular vector is a coordinate vector specified by components that define a rectangle ... vector have different meaning see covariance and contravariance of vectors . Engineering notation ... z sub are the magnitudes of the components of v . Polar vectors A polar vector is a vector in two ... notations Polar vectors can be specified using either ordered pair notation a subset of ordered set notation using only two components or matrix notation, as with rectangular vectors. In these forms, the first ... of v sub 2 sub . To differentiate polar vectors from rectangular vectors, the angle may be prefixed ..., is the angle, and the angle symbol math angle math is optional. Direct notation Polar vectors ... be unwieldy, but is useful for avoiding the confusion with two dimensional rectangular vectors ... math r 5, theta pi over 9 math math r 5, theta 20 circ math Cylindrical vectors A cylindrical vector is an extension of the concept of polar vectors into three dimensions. It is akin to an arrow in the cylindrical ... from the xy plane to the endpoint of the vector. Ordered set and matrix notations Cylindrical vectors are specified like polar vectors, where the second distance component is Concatenation concatenated ... distinguishes cylindrical vectors in this notation from spherical vectors in similar notation ... 9 , z 3 math math rho 5, phi 20 circ , z 3 math Spherical vectors A spherical vector is another method for extending the concept of polar vectors into three dimensions. It is akin to an arrow in the spherical ... exclusive . Ordered set and matrix notations Spherical vectors are specified like polar vectors, where ... vectors from cylindrical ones. A three dimensional spherical vector v can be represented as any of the following ... angle. Direct notation Like polar and cylindrical vectors, spherical vectors can be specified ... more details
. If V is an empty set, then f and g are just constant vectors, and there are no variables over which to integrate. Thus, the equation reduces to a simple inner product of the two vectors. See ... more details
vectors will be 1 or 1. More specifically, the angle between the normal vectors can be computed ... space. If math n m 1 math , then there are an infinite number of normal vectors for each hyperplane ... normal vectors will be minimal, and therefore will be the dihedral angle between the hyperplanes. External ... more details
2 b 3 end bmatrix . math See also Covariance and contravariance of vectors References see also Linear ... Matrices Category Vectors es Vector columna eo Kolumna vektoro fr Vecteur colonne he nl ... more details
two other uses the vectors mainly used in physics and engineering to represent directed quantities mathematical vectors in general Vector mathematics and physics vector In elementary mathematics , physics ... , subtraction , multiplication , and negation have close analogues for vectors, operations ... and associated laws qualify Euclidean space Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space . Vectors play an important ... by vectors. Many other physical quantities can be usefully thought of as vectors. Although ... 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 ... using the same sort of mathematical vector and matrix operations that apply to the other types of vectors ..., distance and angle do not normally apply to these vectors, either rather, what links the values together ... it from vectors Vector space as defined elsewhere , this is sometimes referred to as a geometric ... and engineering Vectors are fundamental in the physical sciences. They can be used to represent any ... it has a magnitude and direction. Vectors also describe many other physical quantities, such as displacement vector displacement , acceleration , momentum , and angular momentum . Other physical vectors, such as the electric field electric and magnetic field , are represented as a system of vectors ... in Cartesian coordinates, one considers primarily bound vectors. A bound vector is determined by the coordinates ... up the positive x axis. The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion. For example, the sum of the vectors ... and affine vectors In the geometrical and physical settings, sometimes it is possible to associate ... more details
Unreferenced stub auto yes date December 2009 A suicide gene , in genetics , will cause a cell biology cell to kill itself through apoptosis . Activation of these genes can be due to many processes, but the main cellular switch to induce apoptosis is the p53 protein. Stimulation or introduction through gene therapy of suicide genes is a potential way of treating cancer or other proliferative diseases.. By introducing the gene into a malignant tumor, the tumor would reduce in size and possibly disappear completely, provided all the individual cells have received a copy of the gene. This depends on the effectiveness of the Vector molecular biology vector usually a virus in reaching all the cells, and distinguishing adequately between malignant cells and those of normal tissue. When the DNA sample in the virus is taken from the patient s own healthy cells, the virus does not need to be able to differentiate between cancer cells and healthy ones. In addition, the advantage is that it is also able to prevent metastasis upon the death of a tumor. Suicide genes are often utilized in biotechnology to assist in molecular cloning. Vectors incorporate suicide genes for an organism such as E. coli . The cloning project focuses on replacing the suicide gene by the desired fragment. Selection of vectors carrying the desired fragment is improved since vectors retaining the suicide gene result in cell death. See also Terminator Technology DEFAULTSORT Suicide Gene Category Genetics Category Programmed cell death Cell biology stub ca Gen su cida es Gen suicida pt Gene suicida ... more details
In linear algebra , the Gramian matrix or Gram matrix or Gramian of a set of vectors math v 1, dots, v n math in an inner product space is the Hermitian matrix of inner product s, whose entries are given by math G ij langle v j, v i rangle math . An important application is to compute linear independence a set of vectors is linearly independent if and only if the Gram determinant Gram determinant the determinant of the Gram matrix is non zero. It is named after J rgen Pedersen Gram . Examples Most commonly, the vectors are elements of an Euclidean space , or are functions in an L2 space L sup 2 sup space , such as continuous functions on a compact interval mathematics interval a , b which are a subspace of L sup 2 sup a , b . Given Real number real valued functions math ell i cdot , ,i 1, dots,n math on the interval math t 0,t f math , the Gram matrix math G G ij math , is given by the Inner product Examples standard inner product on functions math G ij int t 0 t f ell i tau bar ell j tau , d tau. math Given a real matrix A , the matrix A sup T sup A is a Gram matrix of the columns of A , while the matrix AA sup T sup is the Gram matrix of the rows of A . For a general bilinear form B on a finite dimensional vector space over any Field mathematics field we can define a Gram matrix G attached to a set of vectors math v 1, dots, v n math by math G ij B v i,v j , math . The matrix will be symmetric if the bilinear form B is symmetric. Applications If the vectors are centered ... of basis vectors is the overlap matrix . In control theory or more generally systems theory , the controllability ... matrix is the Gramian matrix for some set of vectors. Further, in finite dimensions it determines the vectors up to isomorphism, i.e. any two sets of vectors with the same Gramian matrix must be related ... formed by the vectors. In particular, the vectors are linear independence linearly independent ... nonsingular . The Gram determinant can also be expressed in terms of the exterior product of vectors ... more details
The Linde Buzo Gray algorithm introduced by Yoseph Linde, Andr s Buzo and Robert M. Gray in 1980 is a vector quantization algorithm to derive a good codebook . It is similar to the k means method in data clustering . The algorithm At each iteration, each vector is split into two new vectors. A initial state centroid of the training sequence B initial estimation 1 code book of size 2 C final estimation after Lloyd s algorithm LGA Optimal code book with 2 vectors D initial estimation 2 code book of size 4 E final estimation after Lloyd s algorithm LGA Optimal code book with 4 vectors References The original paper describing the algorithm, as an extension to Lloyd s algorithm Linde, Y., Buzo, A., Robert M. Gray Gray, R.M. , An Algorithm for Vector Quantizer Design , IEEE Transactions on Communications , vol. 28, pp. 84 94, 1980. http ieeexplore.ieee.org xpls abs all.jsp?&arnumber 1094577 Link External links http www.data compression.com vq.html lbg Category Data clustering algorithms Category Machine learning algorithms Category Neural networks algorithm stub ... more details
A trapped null surface is a Set mathematics set of points defined in the context of general relativity as a closed surface on which outward pointing light ray s are actually converging moving inwards . Trapped null surfaces are used in the definition of the apparent horizon which typically surrounds a black hole . Definition We take a Compact space compact , Orientability orientable , Spacetime Spacetime intervals spacelike surface, and find its outward pointing Surface normal normal vectors. The basic picture to think of here is a ball with pins sticking out of it the pins are the normal vectors. Now we look at light rays that are directed outward, along these normal vectors. The rays will either be diverging the usual case one would expect or converging. Intuitively, if the light rays are converging, this means that the light is moving backwards inside of the ball. If all the rays around the entire surface are converging, we say that there is a trapped null surface . See also Apparent horizon Null surface References cite book title The large scale structure of space time author S. W. Hawking and G. F. R. Ellis publisher Cambridge University Press year 1975 This is the gold standard in black holes because of its place in history. It is also quite thorough. cite book title General Relativity author Robert M. Wald publisher University of Chicago Press year 1984 This book is somewhat more up to date. Category Mathematical methods in general relativity Category Black holes ... more details
About genetics a similar sounding computer term Bit array A T DNA binary system is a pair of plasmids consisting of a binary plasmid and a helper plasmid . The two plasmids are used together to produce Genetically modified plant genetically modified plants . They are artificial Vector molecular biology vectors that have both been created from the naturally occurring Ti plasmid found in Agrobacterium tumefaciens . The binary vector is able to DNA replication replicate in both E. coli and Agrobacterium tumefaciens . Systems in which Transfer DNA T DNA and vir genes are located on separate replicons are called T DNA binary systems. T DNA is located on the binary vector the non T DNA region of this vector containing origin s of replication that could function both in E. coli and in Agrobacterium tumefaciens , and antibiotic resistance genes used to select for the presence of the binary vector in bacteria, became known as vector backbone sequences . The replicon containing the vir genes became known as the vir helper. Strains harboring this replicon and a Transfer DNA T DNA are considered disarmed if they do not contain oncogenes that could be transferred to a plant. There are several binary vector systems that differ mainly in the plasmid region that facilitates replication in Agrobacterium . Commonly used binary vectors include pBIN19 pPVP pGreen References T DNA Binary Vectors and Systems Lan Ying Lee and Stanton B. Gelvin Department of Biological Sciences, Purdue University, West Lafayette, Indiana 47907 1392 Category Genetics genetics stub ... more details
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