Special functionsSpecial functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special. Many special functions appear as solutions of differential equations or integrals of elementary functions.
Boolean data typeIn computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted true and false) which is intended to represent the two truth values of logic and Boolean algebra. It is named after George Boole, who first defined an algebraic system of logic in the mid 19th century. The Boolean data type is primarily associated with conditional statements, which allow different actions by changing control flow depending on whether a programmer-specified Boolean condition evaluates to true or false.
Non-local variableIn programming language theory, a non-local variable is a variable that is not defined in the local scope. While the term can refer to global variables, it is primarily used in the context of nested and anonymous functions where some variables can be in neither the local nor the global scope. In Lua they are called the upvalues of the function. In the Python 3 example that follows there is a nested function inner defined in the scope of another function outer.
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.
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).
Scala (programming language)Scala (ˈskɑːlə ) is a strong statically typed high-level general-purpose programming language that supports both object-oriented programming and functional programming. Designed to be concise, many of Scala's design decisions are aimed to address criticisms of Java. Scala source code can be compiled to Java bytecode and run on a Java virtual machine (JVM). Scala can also be compiled to JavaScript to run in a browser, or directly to a native executable.
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.
MemoizationIn computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls to pure functions and returning the cached result when the same inputs occur again. Memoization has also been used in other contexts (and for purposes other than speed gains), such as in simple mutually recursive descent parsing. It is a type of caching, distinct from other forms of caching such as buffering and page replacement.
Transformation matrixIn linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation.
Boolean algebraIn mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬.
Partial applicationIn computer science, partial application (or partial function application) refers to the process of fixing a number of arguments to a function, producing another function of smaller arity. Given a function , we might fix (or 'bind') the first argument, producing a function of type . Evaluation of this function might be represented as . Note that the result of partial function application in this case is a function that takes two arguments. Partial application is sometimes incorrectly called currying, which is a related, but distinct concept.
Geometric transformationIn mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both or both — such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations. Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations).
