Mathematical objectA mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs. Typically, a mathematical object can be a value that can be assigned to a variable, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, sets, functions, expressions, geometric objects, transformations of other mathematical objects, and spaces.
Trans-Neptunian objectA trans-Neptunian object (TNO), also written transneptunian object, is any minor planet in the Solar System that orbits the Sun at a greater average distance than Neptune, which has a semi-major axis of 30.1 astronomical units (au). Typically, TNOs are further divided into the classical and resonant objects of the Kuiper belt, the scattered disc and detached objects with the sednoids being the most distant ones. As of October 2020, the catalog of minor planets contains 678 numbered and more than 2,000 unnumbered TNOs.
Compact closed categoryIn , a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the having finite-dimensional vector spaces as s and linear maps as s, with tensor product as the structure. Another example is , the category having sets as objects and relations as morphisms, with .
Closed monoidal categoryIn mathematics, especially in , a closed monoidal category (or a monoidal closed category) is a that is both a and a in such a way that the structures are compatible. A classic example is the , Set, where the monoidal product of sets and is the usual cartesian product , and the internal Hom is the set of functions from to . A non- example is the , K-Vect, over a field . Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another.
Astronomical objectAn astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists within the observable universe. In astronomy, the terms object and body are often used interchangeably. However, an astronomical body or celestial body is a single, tightly bound, contiguous entity, while an astronomical or celestial object is a complex, less cohesively bound structure, which may consist of multiple bodies or even other objects with substructures.
Monoid (category theory)In , a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a (C, ⊗, I) is an M together with two morphisms μ: M ⊗ M → M called multiplication, η: I → M called unit, such that the pentagon and the unitor diagram commute. In the above notation, 1 is the identity morphism of M, I is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the Cop.
Universal propertyIn mathematics, more specifically in , a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties.
Linear mapIn mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a .
Merge sortIn computer science, merge sort (also commonly spelled as mergesort) is an efficient, general-purpose, and comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the relative order of equal elements is the same in the input and output. Merge sort is a divide-and-conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up merge sort appeared in a report by Goldstine and von Neumann as early as 1948.
Bucket sortBucket sort, or bin sort, is a sorting algorithm that works by distributing the elements of an array into a number of buckets. Each bucket is then sorted individually, either using a different sorting algorithm, or by recursively applying the bucket sorting algorithm. It is a distribution sort, a generalization of pigeonhole sort that allows multiple keys per bucket, and is a cousin of radix sort in the most-to-least significant digit flavor. Bucket sort can be implemented with comparisons and therefore can also be considered a comparison sort algorithm.
Bubble sortBubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm that repeatedly steps through the input list element by element, comparing the current element with the one after it, swapping their values if needed. These passes through the list are repeated until no swaps had to be performed during a pass, meaning that the list has become fully sorted. The algorithm, which is a comparison sort, is named for the way the larger elements "bubble" up to the top of the list.
Radix sortIn computer science, radix sort is a non-comparative sorting algorithm. It avoids comparison by creating and distributing elements into buckets according to their radix. For elements with more than one significant digit, this bucketing process is repeated for each digit, while preserving the ordering of the prior step, until all digits have been considered. For this reason, radix sort has also been called bucket sort and digital sort. Radix sort can be applied to data that can be sorted lexicographically, be they integers, words, punch cards, playing cards, or the mail.
