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This is an illustration of the shortest vector problem (basis vectors in blue, shortest vector in red). In the SVP, a basis of a vector space V and a norm N (often L 2) are given for a lattice L and one must find the shortest non-zero vector in V, as measured by N, in L.
was thought to have infinitely many solutions that were called geometrically real. These solutions are the unit vectors that form the surface of a unit sphere. A geometrically real quaternion is one that can be written as a linear combination of i, j and k, such that the squares of the coefficients add up to one. Hamilton demonstrated that ...
For example, the collection of all possible linear combinations of the vectors on the left-hand side is called their span, and the equations have a solution just when the right-hand vector is within that span. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is ...
In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set. [1]
A standard basis consists of the vectors e i which contain a 1 in the i-th slot and zeros elsewhere. This vector space is the coproduct (or direct sum) of countably many copies of the vector space F. Note the role of the finiteness condition here.
sum of three equal lengthed vectors. Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of triangle geometry. It is also referred to as Sylvester's (triangle) problem in literature
“In clinical practice, I have never seen fluctuating annual cholesterol levels in patients not on lipid-lowering medications,” he says. ( Lipids are fatty compounds, and cholesterol is a type ...
A generalized eigenvalue problem (second sense) is the problem of finding a (nonzero) vector v that obeys = where A and B are matrices. If v obeys this equation, with some λ , then we call v the generalized eigenvector of A and B (in the second sense), and λ is called the generalized eigenvalue of A and B (in the second sense) which ...
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