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Topic: Outer product

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	Outer product - Wikipedia, the free encyclopedia
		Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product.
		Outer product is also a higher-order function in computer programming languages such as APL.
		The outer product is useful in performing transform operations in digital signal processing and digital image processing.
	en.wikipedia.org /wiki/Outer_product (433 words)

	Cross product - Wikipedia, the free encyclopedia
		In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space.
		There is however the wedge product, which has similar properties, except that the wedge product of two vectors is now a 2-vector instead of an ordinary vector.
		In the context of multilinear algebra, it is possible to define a generalized cross product in terms of parity such that the generalized cross product between two vectors of dimension n is a tensor of rank n−2.
	en.wikipedia.org /wiki/Cross_product (1611 words)

	Semidirect product - Wikipedia, the free encyclopedia
		If G is the semidirect product of the normal subgroup N and the subgroup H, and both N and H are finite, then the order of G equals the product of the orders of N and H.
		H, the semidirect product of N and H with respect to φ, as follows: the underlying set is the cartesian product N × H, and the group operation * is given by
		Suppose G is a semidirect product of the normal subgroup N and the subgroup H.
	en.wikipedia.org /wiki/Semidirect_product (1265 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The configuration is such that the three outer container contacting walls are adapted to contact the three walls of an outer packaging container, where two of the three walls of the outer packaging container are substantially parallel to one another and the third wall is disposed between the first two walls and is perpendicular thereto.
		In that case, the outer packaging container contacting surface is provided by at least a portion of the outer surface of the flexible spring transition section 30, in the manner as illustrated otherwise, for example in Figures 2 and 3.
		The unitary product cushioning structure of claim 1, wherein the length of said inner product contacting wall is in the range of 10% to 80% of the length of said outer product supporting region defining wall.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=01/53166.010726&ELEMENT_SET=DECL (7077 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		In one embodiment the hardened outer portions are biscuits and the softer inner portion includes a meat flavor such as beef, pork, bacon, turkey, ham, fish, chicken, lamb, poultry and/or cheese, peanut butter, garlic flavor or a combination thereof.
		The texture of outer portions 102 and 104 is sufficient to provide a cleansing action to the teeth of the pet chewing the treat and inner filling 106 has a flavor and smell sufficient to entice the pet to chew the treat.
		A method in accordance with Claim 17 wherein a texture of the outer portions is sufficiently hard so as to remove at least one of tarter, plaque, and calculus buildup by contact of at least one of the first outer portion and the second outer portion with the teeth of the pet.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=03/92404.031113&ELEMENT_SET=DECL (3785 words)

	Encyclopedia :: Vector (spatial) (Site not responding. Last check: 2007-10-08)
		The dot product of two vectors a and b (sometimes called inner product, or, since its result is a scalar, the scalar product) is denoted by a·b and is defined as:
		The cross product (also vector product or outer product) differs from the dot product primarily in that the result of a cross product of two vectors is a vector.
		Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane.
	www.hallencyclopedia.com /Vector_(spatial) (3054 words)

	Inner_product_space
		In mathematics, an inner product space is a vector space with additional structure, an inner product (also called scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors.
		Inner product spaces generalize Euclidean spaces (with the dot product as the inner product) and are studied in functional analysis.
		For the real case this corresponds to the dot product of the results of directionally differential scaling of the two vectors, with positive scale factors and orthogonal directions of scaling.
	www.brainyencyclopedia.com /encyclopedia/i/in/inner_product_space.html (1814 words)

	PlanetMath: tensor product (vector spaces)
		Historically, the tensor product was called the outer product, and has its origins in the absolute differential calculus (the theory of manifolds).
		This is version 7 of tensor product (vector spaces), born on 2002-02-18, modified 2005-08-02.
		In the book Calculus on Manifolds of Spivak, tensor product is defined similarly, but its stressed that the result of the tensor product of two multilinear functions is again a multilinear function, which i cannot see in the definition here.
	planetmath.org /encyclopedia/TensorProductClassical.html (597 words)

	Outer product - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-08)
		The outer product or wedge product is a non-closed vector product defined in a vector space V over a scalar field F.
		The properties of the outer product "∧" are, for all vectors x, y, z in V
		Similarly, the product of a bivector with a third LI vector gives rise to an oriented volume, generated by sliding the bivector "area" along of the third vector.
	encyclopedia.learnthis.info /o/ou/outer_product.html (432 words)

	Clifford-Grassmann Geometric Algebra (Site not responding. Last check: 2007-10-08)
		The outer product is indicated by the circumflex accent
		The outer product is always anticommutative: any permutation of two vectors changes the sign of the outer product (you can read that the permutation of any two rows or columns changes the sign of the determinant).
		The outer product increases the dimension as soon as the union increases the cardinal of the random event.
	www.terra.es /personal/rgonzal1/whatis.htm (506 words)

	PlanetMath: cross product (Site not responding. Last check: 2007-10-08)
		The cross product produces the vector that would be in a right-handed coordinate system with the plane.
		, then the cross product of two vectors can be obtained by zeroing the real part of the product of the two quaternions.
		This is version 41 of cross product, born on 2001-11-15, modified 2006-03-11.
	planetmath.org /encyclopedia/CrossProduct.html (331 words)

	Bra-ket notation - Wikipedia, the free encyclopedia
		The bra is simply the conjugate transpose (also called the Hermitian conjugate) of the ket and vice versa.
		, from the properties of the inner product (with c* denoting the complex conjugate of c),
		One of the uses of the outer product is to construct projection operators.
	www.sciencedaily.com /encyclopedia/bra_ket_notation (919 words)

	Multiplying Vectors and Matrices
		The outer product of two vectors is a matrix.
		The outer product of a matrix and a vector is a rank three tensor.
		Outer products will be discussed in more detail in Section 3.7.11.
	documents.wolfram.com /v4/MainBook/3.7.5.html (220 words)

	PlanetMath: outer multiplication
		By way of illustration we mention that the outer product of a column vector, i.e.
		We will not pursue this line of thought here, because the topic of algebra structure is best dealt with in the a more abstract context.
		This is version 1 of outer multiplication, born on 2002-05-27.
	www.planetmath.org /encyclopedia/OuterMultiplication.html (212 words)

	The outer product of two arrays (Site not responding. Last check: 2007-10-08)
		are two numeric arrays, their outer product is an array whose dimension vector is got by concatenating their two dimension vectors, (order is important), and whose data vector is got by forming all possible products of elements of the data vector of
		In particular the outer product of two ordinary vectors is a doubly subscripted array (i.e.
		Notice that the outer product operator is of course non-commutative.
	www.math.unm.edu /splus/node37.html (132 words)

	PEP 211 -- Adding A New Outer Product Operator
		This PEP describes a proposal to define "@" (pronounced "across") as a new outer product operator in Python 2.2.
		When applied to sequences (or other iterable objects), this operator will combine their iterators, so that: for (i, j) in S @ T: pass will be equivalent to: for i in S: for j in T: pass Classes will be able to overload this operator using the special methods "__across__", "__racross__", and "__iacross__".
		This could be treated as an error, or by having the outer iterator detect non-restartable inner iterators and cache their values.
	www.python.org /dev/peps/pep-0211 (875 words)

	Products of Quaternions
		The vector terms are a frame-dependent, symmetric product of space with time and does not appear on the stage of physics, but is still a valid measurement.
		The Grassman outer product is antisymmetric and is formed with a commutator.
		The vector terms are an antisymmetric product of space with time and the negative of the cross product.
	www.theworld.com /~sweetser/quaternions/intro/products/products.html (477 words)

	Long Term Memory and the Matrix Model (Site not responding. Last check: 2007-10-08)
		A outer product between two vectors is calculated by multiplying each element in one vector by each element in the other vector (see Figure 8).
		The outer product operation is expressed algebraically by placing the vectors to be multiplied next to each other.
		So the outer product of v and w is written as v w.
	lsa.colorado.edu /~simon/cmc/chapters/Memory/slide12.html (79 words)

	OUTERJOIN (Site not responding. Last check: 2007-10-08)
		The result of a partitioned outer join is a UNION of the outer joins of each of the groups in the logically partitioned table with the table on the other side of the join.
		The NULL values that the partitioned outer join inserts in making the time series dense are handled in the usual way: the SUM function treats them as 0s.
		The outer WHERE clause specifies time at three levels: the days of August 2001, the entire month, and the entire third quarter of 2001.
	www.oracle.com /technology/obe/obe10gdb/bidw/outerjoin/outerjoin.htm (2874 words)

	Bra-ket notation (Site not responding. Last check: 2007-10-08)
		It is so called because the inner product of two states is denoted by a bracket, ‹φψ›, consisting of a left part, ‹φ, called the bra, and a right part, ψ›, called the ket.
		The notation is justified by the Riesz representation theorem, which states that a Hilbert space and its dual space are isometrically isomorphic.
		is a bra and ψ› is a ket, the outer product φ›‹ψ
	www.serebella.com /encyclopedia/article-Bra-ket_notation.html (1637 words)

	Outer product (Site not responding. Last check: 2007-10-08)
		The outer product or wedge product is a non- closed vector product defined in a vector space V over a scalar field F.
		It can be seen as a to n dimensions of the Gibbs vectorial product cross product which can only be defined in spaces of 3 or 7 dimensions.
		Similarly the product of a bivector with third LI vector gives rise to an volume generated by sliding the bivector "area" of the third vector.
	www.freeglossary.com /Outer_product (612 words)

	Vector outer products (Site not responding. Last check: 2007-10-08)
		The outer product is expressed through chaining of nodes.
		Figure 13 shows the principle of node-chaining for the outer product between two vectors, e.g.
		The use of outer products is of importance for e.g.
	pcf1.chembio.ntnu.no /~bka/research/diagram/node12.html (172 words)

	Outer Product Factorization in Clifford (Site not responding. Last check: 2007-10-08)
		When applying Clifford algebra to geometric problems, an important technique is parametric representation of geometric entities, such as planes and spheres in Euclidean and spherical spaces, which occur in the form of homogeneous multivectors.
		Computing a parametric representation of a geometric entity is equivalent to factoring a homogeneous multivector into an outer product of vectors.
		When no signature constraint is imposed on the vectors whose outer product equals the homogeneous multivector, a classical result can be found in the book of Hodge and Pedoe (1953), where a sufficient and necessary condition for the factorability is given and called quadratic Plücker relations (p-relations).
	www.atcminc.com /mPublications/EP/EPATCM99/ATCMP049/abstract.html (189 words)

	The Outer Product
		The second type of vector product, the outer product, is less common.
		Finally, as with the dot product, if the vectors are complex then the outer product is defined as
		There are two pictures that are helpful to keep in mind when forming dot and outer products.
	www.cps.brockport.edu /~little/matlin/node10.html (79 words)

	(WO 01/53166) UNITARY PRODUCT CUSHIONING STRUCTURE (Site not responding. Last check: 2007-10-08)
		It has an outer container contacting wall (20) at each side which is intended to contact an outer packaging container, and a flexible shock absorbing spring section (30) at each such side.
		A product supporting region (16) is bounded by outer product supporting region defining walls (22), inner product contacting walls (24), an upper ridge (26) between them, and a product supporting platform (28).
		Shock absorption support for a shock sensitive product is provided in at least two of three mutually perpendicular directions (32, 36, 36); most embodiments, including end pieces, end caps, trays and covers, and corner pieces, provide shock absorption protection in three mutually perpendicular directions.
	wipo.int /cgi-pct/guest/getbykey5?KEY=01/53166.010726&ELEMENT_SET=DECL (217 words)

	Matrix Multiplication
		The matrix product can be computed with two matrices of different sizes so long as they are compatible.
		The outer product is a way to build a higher rank tensor from those of lower rank.
		One use of this is to combine two vectors to form a matrix as an outer product.
	documents.wolfram.com /v5/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/2.7.html (283 words)

	APPLICATIONS OF THE CLIFFORD-GRASSMANN ALGEBRA TO THE PLANE GEOMETRY (Site not responding. Last check: 2007-10-08)
		The usual inner and outer product of two vectors (their regressive and progressive Grassmann’s products) may be written by means of their Clifford product as the symmetric and antisymmetric part respectively:
		The geometric (Clifford) product is exactly equivalent to the matrix product of the matrix representations of vectors and complex numbers (real 2×2 matrices).
		P, and letters separated by a space indicate Clifford product (sometimes it is reduced to a product of vector by a scalar).
	www.terra.es /personal/rgonzal1/poster/poster.htm (1013 words)

	Section
		For any affected product or container that has a display panel that is normally viewed by the purchaser at the time of the purchase, the warning statement described in Sec.
		If the product or container is normally packaged, wrapped, or otherwise covered when viewed by the purchaser at the time of the purchase the warning statement described in Sec.
		If the outer packaging has a display panel that is normally viewed by the purchaser at the time of the purchase, the warning statement shall appear on such display panel.
	a257.g.akamaitech.net /7/257/2422/14mar20010800/edocket.access.gpo.gov/cfr_2002/julqtr/40cfr82.108.htm (270 words)

	Q calc (Site not responding. Last check: 2007-10-08)
		The Euclidean product "*" and Euclidean division "\" are formed from taking the complement of the first quaternion before carrying out the operation.
		The Grassman outer product ">x<" is better known as the cross product.
		The outer Euclidean product ">*<" is still a mystery, but my money says it is related to that mystery known as spin (since ">x<" deals with angular momentum).
	theworld.com /~sweetser/java/qcalc/qcalc.html (407 words)

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