Representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
MinifloatIn computing, minifloats are floating-point values represented with very few bits. Predictably, they are not well suited for general-purpose numerical calculations. They are used for special purposes, most often in computer graphics, where iterations are small and precision has aesthetic effects. Machine learning also uses similar formats like bfloat16. Additionally, they are frequently encountered as a pedagogical tool in computer-science courses to demonstrate the properties and structures of floating-point arithmetic and IEEE 754 numbers.
Arbitrary-precision arithmeticIn computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision.
Group representationIn the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.
DecimalThe decimal numeral system (also called the base-ten positional numeral system and denary ˈdiːnəri or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (decimal fractions) of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation. A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system.
Fixed point (mathematics)hatnote|1=Fixed points in mathematics are not to be confused with other uses of "fixed point", or stationary points where math|1=f(x) = 0. In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically for functions, a fixed point is an element that is mapped to itself by the function. Formally, c is a fixed point of a function f if c belongs to both the domain and the codomain of f, and f(c) = c.
Floating-point unitA floating-point unit (FPU, colloquially a math coprocessor) is a part of a computer system specially designed to carry out operations on floating-point numbers. Typical operations are addition, subtraction, multiplication, division, and square root. Some FPUs can also perform various transcendental functions such as exponential or trigonometric calculations, but the accuracy can be very low, so that some systems prefer to compute these functions in software.
Irreducible representationIn mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of . Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.
Dynamic rangeDynamic range (abbreviated DR, DNR, or DYR) is the ratio between the largest and smallest values that a certain quantity can assume. It is often used in the context of signals, like sound and light. It is measured either as a ratio or as a base-10 (decibel) or base-2 (doublings, bits or stops) logarithmic value of the difference between the smallest and largest signal values. Electronically reproduced audio and video is often processed to fit the original material with a wide dynamic range into a narrower recorded dynamic range that can more easily be stored and reproduced; this processing is called dynamic range compression.
False precisionFalse precision (also called overprecision, fake precision, misplaced precision and spurious precision) occurs when numerical data are presented in a manner that implies better precision than is justified; since precision is a limit to accuracy (in the ISO definition of accuracy), this often leads to overconfidence in the accuracy, named precision bias. Madsen Pirie defines the term "false precision" in a more general way: when exact numbers are used for notions that cannot be expressed in exact terms.
Fixed-point theorems in infinite-dimensional spacesIn mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations. The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder (a previous result in a different vein, the Banach fixed-point theorem for contraction mappings in complete metric spaces was proved in 1922). Quite a number of further results followed.
NumberA number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number.
