Tucana is a constellation of stars in the southern sky, named after the toucan, a South American bird. It is one of twelve constellations conceived in the late sixteenth century by Petrus Plancius from the observations of Pieter Dirkszoon Keyser and Frederick de Houtman. Tucana first appeared on a 35-centimetre-diameter (14 in) celestial globe published in 1598 in Amsterdam by Plancius and Jodocus Hondius and was depicted in Johann Bayer's star atlas Uranometria of 1603. French explorer and astronomer Nicolas Louis de Lacaille gave its stars Bayer designations in 1756. The constellations Tucana, Grus, Phoenix and Pavo are collectively known as the "Southern Birds".
In probability theory and statistics, the exponential distribution is the probability distribution that describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. The great advantage of this method is that it allows the optimization to be solved without explicit parameterization in terms of the constraints. As a result, the method of Lagrange multipliers is widely used to solve challenging constrained optimization problems.
Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation, such as an alpha particle, beta particle with neutrino or only a neutrino in the case of electron capture, or a gamma ray or electron in the case of internal conversion. A material containing such unstable nuclei is considered radioactive. Certain highly excited short-lived nuclear states can decay through neutron emission, or more rarely, proton emission.
The Erlang distribution is a two-parameter family of continuous probability distributions with support
. The two parameters are:
In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution.
A blue straggler is a main-sequence star in an open or globular cluster that is more luminous and bluer than stars at the main sequence turnoff point for the cluster. Blue stragglers were first discovered by Allan Sandage in 1953 while performing photometry of the stars in the globular cluster M3. Standard theories of stellar evolution hold that the position of a star on the Hertzsprung–Russell diagram should be determined almost entirely by the initial mass of the star and its age. In a cluster, stars all formed at approximately the same time, and thus in an H–R diagram for a cluster, all stars should lie along a clearly defined curve set by the age of the cluster, with the positions of individual stars on that curve determined solely by their initial mass. With masses two to three times that of the rest of the main-sequence cluster stars, blue stragglers seem to be exceptions to this rule. The resolution of this problem is likely related to interactions between two or more stars in the dense confines of the clusters in which blue stragglers are found.
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v, but, unlike the cross product, the exterior product is associative. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation—a choice of clockwise or counterclockwise.
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the exponential decay constant:
47 Tucanae, 47 Tuc is a globular cluster located in the constellation Tucana. It is about 4.0 ± 0.35 kpc (13,000 ± 1,100 ly) away from Earth, and 120 light years across. 47 Tuc can be seen with the naked eye, with an apparent magnitude of 4.1. It appears about 50 arcminutes across. Due to its far southern location, 18° from the south celestial pole, it was not catalogued by European astronomers until the 1750s, when the cluster was first identified by Nicolas-Louis de Lacaille from South Africa.
The equirectangular projection is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v. This condition can be written as the equation
In probability theory and statistics, the noncentral chi-squared or noncentral
distribution is a generalization of the chi-squared distribution. This distribution often arises in the power analysis of statistical tests in which the null distribution is a chi-squared distribution; important examples of such tests are the likelihood-ratio tests.
Alpha Tucanae is a binary star system in the southern circumpolar constellation of Tucana. With an apparent visual magnitude of 2.86, it can be seen with the naked eye from the southern hemisphere. Using parallax measurements, the distance to this system can be estimated as 200 light-years. A cool star with a surface temperature of 4300 K, it is 424 times as luminous as the sun and 37 times its diameter. It is 2.5 to 3 times as massive. It is unclear what stage of evolution the star is in.
Kappa Tucanae is a quadruple star system in the constellation Tucana. It is approximately 66.6 light years from Earth and the combined apparent magnitude of the system is +4.25.
Beta Tucanae is a group of six stars which appear to be at least loosely bound into a system in the constellation Tucana. Three of the stars are luminous and distinct enough to have been given their own Bayer designations, β¹ Tucanae through β³ Tucanae. The system is approximately 140 light years from Earth.
Eta Tucanae is a star in the constellation Tucana.
Theta Tucanae is a star in the constellation Tucana.
Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution and not used in statistical models directly.
In probability theory and statistics, the Poisson distribution, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
