Euclidean spaceEuclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes.
Euclidean groupIn mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space ; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n). The Euclidean group E(n) comprises all translations, rotations, and reflections of ; and arbitrary finite combinations of them.
Foreign relations of SwitzerlandThe foreign relations of Switzerland are the primary responsibility of the Federal Department of Foreign Affairs (FDFA). Some international relations of Switzerland are handled by other departments of the federal administration of Switzerland. Article 54 of the Swiss Constitution of 1999 declares the safeguarding of Switzerland's independence and welfare as the principal objective of Swiss foreign policy.
HeliosIn ancient Greek religion and mythology, Helios (ˈhiːliəs,_-ɒs; ; Homeric Greek: Ἠέλιος ) is the god and personification of the Sun. His name is also Latinized as Helius, and he is often given the epithets Hyperion ("the one above") and Phaethon ("the shining"). Helios is often depicted in art with a radiant crown and driving a horse-drawn chariot through the sky. He was a guardian of oaths and also the god of sight.
Kivu Ebola epidemicThe Kivu Ebola epidemic was an outbreak of Ebola virus disease (EVD) mainly in eastern Democratic Republic of the Congo (DRC), and in other parts of Central Africa, from 2018 to 2020. Between 1 August 2018 and 25 June 2020 it resulted in 3,470 reported cases. The Kivu outbreak also affected Ituri Province, whose first case was confirmed on 13 August 2018. In November 2018, the outbreak became the biggest Ebola outbreak in the DRC's history, and had become the second-largest Ebola outbreak in recorded history worldwide, behind only the 2013–2016 Western Africa epidemic.
Islamic terrorismIslamic terrorism (also known as Islamist terrorism or radical Islamic terrorism) refers to terrorist acts with religious motivations carried out by fundamentalist militant Islamists and Islamic extremists. Incidents and fatalities from Islamic terrorism have been concentrated in eight Muslim-majority countries (Afghanistan, Egypt, Iraq, Libya, Nigeria, Pakistan, Somalia, and Syria), while four Islamic extremist groups (Islamic State, Boko Haram, the Taliban, and al-Qaeda) were responsible for 74% of all deaths from terrorism in 2015.
Group (mathematics)In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).
SubgroupIn group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is often denoted H ≤ G, read as "H is a subgroup of G". The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G).
BijectionIn mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set; there are no unpaired elements between the two sets. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.
Rotation (mathematics)Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space.
Complex numberIn mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the , and b is called the . The set of complex numbers is denoted by either of the symbols or C.
Rotation matrixIn linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R: If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the above equations become the trigonometric summation angle formulae.
