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A vector addition system (VAS) is one of several mathematical modeling languages for the description of distributed systems.Vector addition systems were introduced by Richard M. Karp and Raymond E. Miller in 1969, [1] and generalized to vector addition systems with states (VASS) by John E. Hopcroft and Jean-Jacques Pansiot in 1979. [2]
An alternative definition of the Minkowski difference is sometimes used for computing intersection of convex shapes. [3] This is not equivalent to the previous definition, and is not an inverse of the sum operation. Instead it replaces the vector addition of the Minkowski sum with a vector subtraction. If the two convex shapes intersect, the ...
Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w . In mathematics and physics , a vector space (also called a linear space) is a set whose elements, often called vectors , can be added together and multiplied ...
By definition, a convex combination of an indexed subset {v 1, v 2, …, v D} of a vector space is any weighted average λ 1 v 1 + λ 2 v 2 + … + λ D v D for some indexed set of non-negative real numbers {λ d} satisfying the equation λ 0 + λ 1 + … + λ D = 1. [10] The definition of a convex set implies that the intersection of two ...
Operant conditioning, also called instrumental conditioning, is a learning process where voluntary behaviors are modified by association with the addition (or removal) of reward or aversive stimuli. The frequency or duration of the behavior may increase through reinforcement or decrease through punishment or extinction .
An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a ring homomorphism from the ...
The fundamental example of a linear complex structure is the structure on R 2n coming from the complex structure on C n.That is, the complex n-dimensional space C n is also a real 2n-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number i is not only a complex linear transform of the space, thought of as a complex ...
In mathematics, vector algebra may mean: The operations of vector addition and scalar multiplication of a vector space; The algebraic operations in vector calculus (vector analysis) – including the dot and cross products of 3-dimensional Euclidean space; Algebra over a field – a vector space equipped with a bilinear product