Classical Hamiltonian quaternionsWilliam Rowan Hamilton invented quaternions, a mathematical entity in 1843. This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition only by the terminology which is used. Hamilton defined a quaternion as the quotient of two directed lines in tridimensional space; or, more generally, as the quotient of two vectors.
Connected spaceIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space is a if it is a connected space when viewed as a subspace of . Some related but stronger conditions are path connected, simply connected, and -connected.
Quasi-projective varietyIn mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in scheme theory, where a quasi-projective scheme is a locally closed subscheme of some projective space. An affine space is a Zariski-open subset of a projective space, and since any closed affine subset can be expressed as an intersection of the projective completion and the affine space embedded in the projective space, this implies that any affine variety is quasiprojective.
Dandelin spheresIn geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres. The Dandelin spheres were discovered in 1822. They are named in honor of the French mathematician Germinal Pierre Dandelin, though Adolphe Quetelet is sometimes given partial credit as well.
Bounded variationIn mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value.
Laplace's equationIn mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as or where is the Laplace operator, is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
BiquaternionIn abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: Biquaternions when the coefficients are complex numbers. Split-biquaternions when the coefficients are split-complex numbers. Dual quaternions when the coefficients are dual numbers.
SubsetIn mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements. The subset relation defines a partial order on sets.
InfinityInfinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.
Ball (mathematics)In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere.
