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The determination of the upper bound for the number of limit cycles in two-dimensional polynomial vector fields of degree n and an investigation of their relative positions. The first problem is yet unsolved for n = 8. Therefore, this problem is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry.
The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive x-axis. This coordinate representation of free vectors allows their algebraic features to be expressed in a convenient numerical fashion.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
This is an illustration of the closest vector problem (basis vectors in blue, external vector in green, closest vector in red). In CVP, a basis of a vector space V and a metric M (often L 2) are given for a lattice L, as well as a vector v in V but not necessarily in L. It is desired to find the vector in L closest to v (as measured by M).
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) [1] states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. [2] [3] For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in ℝ 3 to every point p on a sphere such that ...
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
Every operator on a non-trivial complex finite dimensional vector space has an eigenvector, solving the invariant subspace problem for these spaces. In the field of mathematics known as functional analysis , the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends ...