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For example, one definition of the set N {\displaystyle \mathbb {N} } of natural numbers is: 1 is in N . {\displaystyle \mathbb {N} .} If an element n is in N {\displaystyle \mathbb {N} } then n + 1 is in N .
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https://en.wikipedia.org/wiki/Recursive_definition
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{\displaystyle \mathbb {N} .} N {\displaystyle \mathbb {N} } is the intersection of all sets satisfying (1) and (2).There are many sets that satisfy (1) and (2) – for example, the set {1, 1.649, 2, 2.649, 3, 3.649, …} satisfies the definition. However, condition (3) specifies the set of natural numbers by removing the sets with extraneous members.
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https://en.wikipedia.org/wiki/Recursive_definition
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Note that this definition assumes that N {\displaystyle \mathbb {N} } is contained in a larger set (such as the set of real numbers) — in which the operation + is defined. Properties of recursively defined functions and sets can often be proved by an induction principle that follows the recursive definition. For example, the definition of the natural numbers presented here directly implies the principle of mathematical induction for natural numbers: if a property holds of the natural number 0 (or 1), and the property holds of n + 1 whenever it holds of n, then the property holds of all natural numbers (Aczel 1977:742).
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https://en.wikipedia.org/wiki/Recursive_definition
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In mathematics and computer science, a shortest-path tree rooted at a vertex v of a connected, undirected graph G is a spanning tree T of G, such that the path distance from root v to any other vertex u in T is the shortest path distance from v to u in G. In connected graphs where shortest paths are well-defined (i.e. where there are no negative-length cycles), we may construct a shortest-path tree using the following algorithm: Compute dist(u), the shortest-path distance from root v to vertex u in G using Dijkstra's algorithm or Bellman–Ford algorithm. For all non-root vertices u, we can assign to u a parent vertex pu such that pu is connected to u, and that dist(pu) + edge_dist(pu,u) = dist(u). In case multiple choices for pu exist, choose pu for which there exists a shortest path from v to pu with as few edges as possible; this tie-breaking rule is needed to prevent loops when there exist zero-length cycles.
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https://en.wikipedia.org/wiki/Shortest-path_tree
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Construct the shortest-path tree using the edges between each node and its parent.The above algorithm guarantees the existence of shortest-path trees. Like minimum spanning trees, shortest-path trees in general are not unique. In graphs for which all edge weights are equal, shortest path trees coincide with breadth-first search trees.
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https://en.wikipedia.org/wiki/Shortest-path_tree
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In graphs that have negative cycles, the set of shortest simple paths from v to all other vertices do not necessarily form a tree. For simple connected graphs, shortest-path trees can be used to suggest a non-linear relationship between two network centrality measures, closeness and degree. By assuming that the branches of the shortest-path trees are statistically similar for any root node in one network, one may show that the size of the branches depend only on the number of branches connected to the root vertex, i.e. to the degree of the root node. From this one deduces that the inverse of closeness, a length scale associated with each vertex, varies approximately linearly with the logarithm of degree. The relationship is not exact but it captures a correlation between closeness and degree in large number of networks constructed from real data and this success suggests that shortest-path trees can be a useful approximation in network analysis.
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https://en.wikipedia.org/wiki/Shortest-path_tree
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In mathematics and computer science, a splicing rule is a transformation on formal languages which formalises the action of gene splicing in molecular biology. A splicing language is a language generated by iterated application of a splicing rule: the splicing languages form a proper subset of the regular languages.
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https://en.wikipedia.org/wiki/Splicing_rule
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In mathematics and computer science, a stack-sortable permutation (also called a tree permutation) is a permutation whose elements may be sorted by an algorithm whose internal storage is limited to a single stack data structure. The stack-sortable permutations are exactly the permutations that do not contain the permutation pattern 231; they are counted by the Catalan numbers, and may be placed in bijection with many other combinatorial objects with the same counting function including Dyck paths and binary trees.
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https://en.wikipedia.org/wiki/Stack-sortable_permutation
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In mathematics and computer science, a string metric (also known as a string similarity metric or string distance function) is a metric that measures distance ("inverse similarity") between two text strings for approximate string matching or comparison and in fuzzy string searching. A requirement for a string metric (e.g. in contrast to string matching) is fulfillment of the triangle inequality. For example, the strings "Sam" and "Samuel" can be considered to be close. A string metric provides a number indicating an algorithm-specific indication of distance.
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https://en.wikipedia.org/wiki/String_similarity
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The most widely known string metric is a rudimentary one called the Levenshtein distance (also known as edit distance). It operates between two input strings, returning a number equivalent to the number of substitutions and deletions needed in order to transform one input string into another. Simplistic string metrics such as Levenshtein distance have expanded to include phonetic, token, grammatical and character-based methods of statistical comparisons. String metrics are used heavily in information integration and are currently used in areas including fraud detection, fingerprint analysis, plagiarism detection, ontology merging, DNA analysis, RNA analysis, image analysis, evidence-based machine learning, database data deduplication, data mining, incremental search, data integration, malware detection, and semantic knowledge integration.
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https://en.wikipedia.org/wiki/String_similarity
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In mathematics and computer science, an algorithm ( ) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning), achieving automation eventually.
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https://en.wikipedia.org/wiki/Algorithmic_problem
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Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus".In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result.As an effective method, an algorithm can be expressed within a finite amount of space and time, and in a well-defined formal language for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.
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https://en.wikipedia.org/wiki/Algorithmic_problem
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In mathematics and computer science, an algorithmic technique is a general approach for implementing a process or computation.
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https://en.wikipedia.org/wiki/Algorithmic_technique
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In mathematics and computer science, an event structure represents a set of events, some of which can only be performed after another (there is a dependency between the events) and some of which might not be performed together (there is a conflict between the events).
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https://en.wikipedia.org/wiki/Event_structure
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In mathematics and computer science, an unrooted binary tree is an unrooted tree in which each vertex has either one or three neighbors.
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https://en.wikipedia.org/wiki/Unrooted_binary_tree
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In mathematics and computer science, apply is a function that applies a function to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has a role in the study of the denotational semantics of computer programs, because it is a continuous function on complete partial orders.
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https://en.wikipedia.org/wiki/Apply
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Apply is also a continuous function in homotopy theory, and, indeed underpins the entire theory: it allows a homotopy deformation to be viewed as a continuous path in the space of functions. Likewise, valid mutations (refactorings) of computer programs can be seen as those that are "continuous" in the Scott topology. The most general setting for apply is in category theory, where it is right adjoint to currying in closed monoidal categories. A special case of this are the Cartesian closed categories, whose internal language is simply typed lambda calculus.
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https://en.wikipedia.org/wiki/Apply
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In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a computable manner. The field is closely related to constructive analysis and numerical analysis. A notable result is that integration (in the sense of the Riemann integral) is computable.
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https://en.wikipedia.org/wiki/Computable_analysis
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This might be considered surprising as an integral is (loosely speaking) an infinite sum. While this result could be explained by the fact that every computable function from {\displaystyle \mathbb {} to R {\displaystyle \mathbb {R} } is uniformly continuous, the notable thing is that the modulus of continuity can always be computed without being explicitly given. A similarly surprising fact is that differentiation of complex functions is also computable, while the same result is false for real functions.The above motivating results have no counterpart in Bishop's constructive analysis. Instead, it is the stronger form of constructive analysis developed by Brouwer that provides a counterpart in constructive logic.
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https://en.wikipedia.org/wiki/Computable_analysis
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In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithms for primality testing and integer factorization, finding solutions to diophantine equations, and explicit methods in arithmetic geometry. Computational number theory has applications to cryptography, including RSA, elliptic curve cryptography and post-quantum cryptography, and is used to investigate conjectures and open problems in number theory, including the Riemann hypothesis, the Birch and Swinnerton-Dyer conjecture, the ABC conjecture, the modularity conjecture, the Sato-Tate conjecture, and explicit aspects of the Langlands program.
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https://en.wikipedia.org/wiki/Algorithmic_number_theory
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In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes exact computation with expressions containing variables that have no given value and are manipulated as symbols. Software applications that perform symbolic calculations are called computer algebra systems, with the term system alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the language used for the implementation), a dedicated memory manager, a user interface for the input/output of mathematical expressions, a large set of routines to perform usual operations, like simplification of expressions, differentiation using chain rule, polynomial factorization, indefinite integration, etc. Computer algebra is widely used to experiment in mathematics and to design the formulas that are used in numerical programs. It is also used for complete scientific computations, when purely numerical methods fail, as in public key cryptography, or for some non-linear problems.
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https://en.wikipedia.org/wiki/Symbolic_reasoning
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In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network.
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https://en.wikipedia.org/wiki/4-connected_graph
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In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f {\displaystyle f} that takes three arguments creates a nested unary function g {\displaystyle g} , so that the code let x = f ( a , b , c ) {\displaystyle {\text{let }}x=f(a,b,c)} gives x {\displaystyle x} the same value as the code let h = g ( a ) let i = h ( b ) let x = i ( c ) , {\displaystyle {\begin{aligned}{\text{let }}h=g(a)\\{\text{let }}i=h(b)\\{\text{let }}x=i(c),\end{aligned}}} or called in sequence, let x = g ( a ) ( b ) ( c ) . {\displaystyle {\text{let }}x=g(a)(b)(c).} In a more mathematical language, a function that takes two arguments, one from X {\displaystyle X} and one from Y {\displaystyle Y} , and produces outputs in Z , {\displaystyle Z,} by currying is translated into a function that takes a single argument from X {\displaystyle X} and produces as outputs functions from Y {\displaystyle Y} to Z .
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https://en.wikipedia.org/wiki/Currying
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{\displaystyle Z.} This is a natural one-to-one correspondence between these two types of functions, so that the sets together with functions between them form a Cartesian closed category.
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https://en.wikipedia.org/wiki/Currying
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The currying of a function with more than two arguments can then be defined by induction. Currying is related to, but not the same as, partial application. Currying is useful in both practical and theoretical settings.
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https://en.wikipedia.org/wiki/Currying
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In functional programming languages, and many others, it provides a way of automatically managing how arguments are passed to functions and exceptions. In theoretical computer science, it provides a way to study functions with multiple arguments in simpler theoretical models which provide only one argument. The most general setting for the strict notion of currying and uncurrying is in the closed monoidal categories, which underpins a vast generalization of the Curry–Howard correspondence of proofs and programs to a correspondence with many other structures, including quantum mechanics, cobordisms and string theory.
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https://en.wikipedia.org/wiki/Currying
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It was introduced by Gottlob Frege, developed by Moses Schönfinkel, and further developed by Haskell Curry.Uncurrying is the dual transformation to currying, and can be seen as a form of defunctionalization. It takes a function f {\displaystyle f} whose return value is another function g {\displaystyle g} , and yields a new function f ′ {\displaystyle f'} that takes as parameters the arguments for both f {\displaystyle f} and g {\displaystyle g} , and returns, as a result, the application of f {\displaystyle f} and subsequently, g {\displaystyle g} , to those arguments. The process can be iterated.
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https://en.wikipedia.org/wiki/Currying
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In mathematics and computer science, graph edit distance (GED) is a measure of similarity (or dissimilarity) between two graphs. The concept of graph edit distance was first formalized mathematically by Alberto Sanfeliu and King-Sun Fu in 1983. A major application of graph edit distance is in inexact graph matching, such as error-tolerant pattern recognition in machine learning.The graph edit distance between two graphs is related to the string edit distance between strings. With the interpretation of strings as connected, directed acyclic graphs of maximum degree one, classical definitions of edit distance such as Levenshtein distance,Hamming distance and Jaro–Winkler distance may be interpreted as graph edit distances between suitably constrained graphs. Likewise, graph edit distance is also a generalization of tree edit distance between rooted trees.
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https://en.wikipedia.org/wiki/Graph_edit_distance
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In mathematics and computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of packing balls in the Hamming metric into the space of all possible words. It gives an important limitation on the efficiency with which any error-correcting code can utilize the space in which its code words are embedded. A code that attains the Hamming bound is said to be a perfect code.
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https://en.wikipedia.org/wiki/Perfect_code
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In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational objects, such as functions or datatypes, are defined in terms of each other. Mutual recursion is very common in functional programming and in some problem domains, such as recursive descent parsers, where the datatypes are naturally mutually recursive.
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https://en.wikipedia.org/wiki/Mutually_recursive
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In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number of multiplications. Using the form of the shortest addition chain, with multiplication instead of addition, computes the desired exponent (instead of multiple) of the base. (This corresponds to OEIS sequence A003313 (Length of shortest addition chain for n).) Each exponentiation in the chain can be evaluated by multiplying two of the earlier exponentiation results.
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https://en.wikipedia.org/wiki/Addition-chain_exponentiation
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More generally, addition-chain exponentiation may also refer to exponentiation by non-minimal addition chains constructed by a variety of algorithms (since a shortest addition chain is very difficult to find). The shortest addition-chain algorithm requires no more multiplications than binary exponentiation and usually less. The first example of where it does better is for a15, where the binary method needs six multiplications but the shortest addition chain requires only five: a 15 = a × ( a × 2 ) 2 {\displaystyle a^{15}=a\times (a\times ^{2})^{2}\!}
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https://en.wikipedia.org/wiki/Addition-chain_exponentiation
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(binary, 6 multiplications) a 15 = ( 2 × a ) 3 {\displaystyle a^{15}=(^{2}\times a)^{3}\!} (shortest addition chain, 5 multiplications). a 15 = a 3 × ( 2 ) 2 {\displaystyle a^{15}=a^{3}\times (^{2})^{2}\!}
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https://en.wikipedia.org/wiki/Addition-chain_exponentiation
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(also shortest addition chain, 5 multiplications).On the other hand, the determination of a shortest addition chain is hard: no efficient optimal methods are currently known for arbitrary exponents, and the related problem of finding a shortest addition chain for a given set of exponents has been proven NP-complete. Even given a shortest chain, addition-chain exponentiation requires more memory than the binary method, because it must potentially store many previous exponents from the chain. So in practice, shortest addition-chain exponentiation is primarily used for small fixed exponents for which a shortest chain can be pre-computed and is not too large.
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https://en.wikipedia.org/wiki/Addition-chain_exponentiation
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There are also several methods to approximate a shortest addition chain, and which often require fewer multiplications than binary exponentiation; binary exponentiation itself is a suboptimal addition-chain algorithm. The optimal algorithm choice depends on the context (such as the relative cost of the multiplication and the number of times a given exponent is re-used).The problem of finding the shortest addition chain cannot be solved by dynamic programming, because it does not satisfy the assumption of optimal substructure. That is, it is not sufficient to decompose the power into smaller powers, each of which is computed minimally, since the addition chains for the smaller powers may be related (to share computations). For example, in the shortest addition chain for a15 above, the subproblem for a6 must be computed as (a3)2 since a3 is re-used (as opposed to, say, a6 = a2(a2)2, which also requires three multiplies).
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https://en.wikipedia.org/wiki/Addition-chain_exponentiation
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In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for some values. In other words, evaluating the polynomial P ( x 1 , x 2 ) = 2 x 1 x 2 + x 1 3 + 4 {\displaystyle P(x_{1},x_{2})=2x_{1}x_{2}+x_{1}^{3}+4} at x 1 = 2 , x 2 = 3 {\displaystyle x_{1}=2,x_{2}=3} consists of computing P ( 2 , 3 ) = 2 ⋅ 2 ⋅ 3 + 2 3 + 4 = 24. {\displaystyle P(2,3)=2\cdot 2\cdot 3+2^{3}+4=24.}
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https://en.wikipedia.org/wiki/Polynomial_evaluation
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See also Polynomial ring § Polynomial evaluation For evaluating the univariate polynomial a n x n + a n − 1 x n − 1 + ⋯ + a 0 , {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0},} the most naive method would use n {\displaystyle n} multiplications to compute a n x n {\displaystyle a_{n}x^{n}} , use n − 1 {\displaystyle n-1} multiplications to compute a n − 1 x n − 1 {\displaystyle a_{n-1}x^{n-1}} and so on for a total of n ( n + 1 ) 2 {\displaystyle {\tfrac {n(n+1)}{2}}} multiplications and n {\displaystyle n} additions. Using better methods, such as Horner's rule, this can be reduced to n {\displaystyle n} multiplications and n {\displaystyle n} additions. If some preprocessing is allowed, even more savings are possible.
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https://en.wikipedia.org/wiki/Polynomial_evaluation
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In mathematics and computer science, symbolic-numeric computation is the use of software that combines symbolic and numeric methods to solve problems.
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https://en.wikipedia.org/wiki/Symbolic-numeric_computation
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In mathematics and computer science, the BIT predicate, sometimes written BIT ( i , j ) {\displaystyle {\text{BIT}}(i,j)} , is a predicate that tests whether the j {\displaystyle j} th bit of the number i {\displaystyle i} (starting from the least significant digit) is 1, when i {\displaystyle i} is written as a binary number. Its mathematical applications include modeling the membership relation of hereditarily finite sets, and defining the adjacency relation of the Rado graph. In computer science, it is used for efficient representations of set data structures using bit vectors, in defining the private information retrieval problem from communication complexity, and in descriptive complexity theory to formulate logical descriptions of complexity classes.
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https://en.wikipedia.org/wiki/BIT_predicate
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In mathematics and computer science, the Entscheidungsproblem (German for 'decision problem'; pronounced ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "yes" or "no" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms.
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https://en.wikipedia.org/wiki/Entscheidungsproblem
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In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined in a feedback-free manner (called a "wreath product" or "cascade"). Krohn and Rhodes found a general decomposition for finite automata. The authors discovered and proved an unexpected major result in finite semigroup theory, revealing a deep connection between finite automata and semigroups.
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https://en.wikipedia.org/wiki/Krohn–Rhodes_theory
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In mathematics and computer science, the binary Goppa code is an error-correcting code that belongs to the class of general Goppa codes originally described by Valerii Denisovich Goppa, but the binary structure gives it several mathematical advantages over non-binary variants, also providing a better fit for common usage in computers and telecommunication. Binary Goppa codes have interesting properties suitable for cryptography in McEliece-like cryptosystems and similar setups.
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https://en.wikipedia.org/wiki/Binary_Goppa_code
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In mathematics and computer science, the critical exponent of a finite or infinite sequence of symbols over a finite alphabet describes the largest number of times a contiguous subsequence can be repeated. For example, the critical exponent of "Mississippi" is 7/3, as it contains the string "ississi", which is of length 7 and period 3. If w is an infinite word over the alphabet A and x is a finite word over A, then x is said to occur in w with exponent α, for positive real α, if there is a factor y of w with y = xax0 where x0 is a prefix of x, a is the integer part of α, and the length |y| = α |x|: we say that y is an α-power. The word w is α-power-free if it contains no factors which are β-powers for any β ≥ α.The critical exponent for w is the supremum of the α for which w has α-powers, or equivalently the infimum of the α for which w is α-power-free.
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https://en.wikipedia.org/wiki/Critical_exponent_of_a_word
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In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x).For example (floor), ⌊2.4⌋ = 2, ⌊−2.4⌋ = −3, and for ceiling; ⌈2.4⌉ = 3, and ⌈−2.4⌉ = −2. Historically, the floor of x has been–and still is–called the integral part or integer part of x, often denoted (as well as a variety of other notations). However, the same term, integer part, is also used for truncation towards zero, which differs from the floor function for negative numbers. For n an integer, ⌊n⌋ = ⌈n⌉ = = n.
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https://en.wikipedia.org/wiki/Floor_function
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In mathematics and computer science, the gradations of applicable meaning of a fuzzy concept are described in terms of quantitative relationships defined by logical operators. Such an approach is sometimes called "degree-theoretic semantics" by logicians and philosophers, but the more usual term is fuzzy logic or many-valued logic. The novelty of fuzzy logic is, that it "breaks with the traditional principle that formalisation should correct and avoid, but not compromise with, vagueness". The basic idea of fuzzy logic is that a real number is assigned to each statement written in a language, within a range from 0 to 1, where 1 means that the statement is completely true, and 0 means that the statement is completely false, while values less than 1 but greater than 0 represent that the statements are "partly true", to a given, quantifiable extent.
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https://en.wikipedia.org/wiki/Fuzzy_concept
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Susan Haack comments: "Whereas in classical set theory an object either is or is not a member of a given set, in fuzzy set theory membership is a matter of degree; the degree of membership of an object in a fuzzy set is represented by some real number between 0 and 1, with 0 denoting no membership and 1 full membership." "Truth" in this mathematical context usually means simply that "something is the case", or that "something is applicable". This makes it possible to analyze a distribution of statements for their truth-content, identify data patterns, make inferences and predictions, and model how processes operate. Petr Hájek claimed that "fuzzy logic is not just some "applied logic", but may bring "new light to classical logical problems", and therefore might be well classified as a distinct branch of "philosophical logic" similar to e.g. modal logics.
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https://en.wikipedia.org/wiki/Fuzzy_concept
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In mathematics and computer science, the middle-square method is a method of generating pseudorandom numbers. In practice it is a highly flawed method for many practical purposes, since its period is usually very short and it has some severe weaknesses; repeated enough times, the middle-square method will either begin repeatedly generating the same number or cycle to a previous number in the sequence and loop indefinitely.
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https://en.wikipedia.org/wiki/Middle-square_method
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In mathematics and computer science, the pinwheel scheduling problem is a problem in real-time scheduling with repeating tasks of unit length and hard constraints on the time between repetitions.
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https://en.wikipedia.org/wiki/Pinwheel_scheduling
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In mathematics and computer science, the probabilistic automaton (PA) is a generalization of the nondeterministic finite automaton; it includes the probability of a given transition into the transition function, turning it into a transition matrix. Thus, the probabilistic automaton also generalizes the concepts of a Markov chain and of a subshift of finite type. The languages recognized by probabilistic automata are called stochastic languages; these include the regular languages as a subset.
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https://en.wikipedia.org/wiki/Probabilistic_finite_automata
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The number of stochastic languages is uncountable. The concept was introduced by Michael O. Rabin in 1963; a certain special case is sometimes known as the Rabin automaton (not to be confused with the subclass of ω-automata also referred to as Rabin automata). In recent years, a variant has been formulated in terms of quantum probabilities, the quantum finite automaton.
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https://en.wikipedia.org/wiki/Probabilistic_finite_automata
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In mathematics and computer science, the probabilistic method is used to prove the existence of mathematical objects with desired combinatorial properties. The proofs are probabilistic — they work by showing that a random object, chosen from some probability distribution, has the desired properties with positive probability. Consequently, they are nonconstructive — they don't explicitly describe an efficient method for computing the desired objects. The method of conditional probabilities (Spencer 1987), (Raghavan 1988) converts such a proof, in a "very precise sense", into an efficient deterministic algorithm, one that is guaranteed to compute an object with the desired properties.
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https://en.wikipedia.org/wiki/Method_of_conditional_probabilities
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That is, the method derandomizes the proof. The basic idea is to replace each random choice in a random experiment by a deterministic choice, so as to keep the conditional probability of failure, given the choices so far, below 1. The method is particularly relevant in the context of randomized rounding (which uses the probabilistic method to design approximation algorithms). When applying the method of conditional probabilities, the technical term pessimistic estimator refers to a quantity used in place of the true conditional probability (or conditional expectation) underlying the proof.
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https://en.wikipedia.org/wiki/Method_of_conditional_probabilities
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In mathematics and computer science, the right quotient (or simply quotient) of a language L 1 {\displaystyle L_{1}} with respect to language L 2 {\displaystyle L_{2}} is the language consisting of strings w such that wx is in L 1 {\displaystyle L_{1}} for some string x in L 2 {\displaystyle L_{2}} . Formally: In other words, we take all the strings in L 1 {\displaystyle L_{1}} that have a suffix in L 2 {\displaystyle L_{2}} , and remove this suffix. Similarly, the left quotient of L 1 {\displaystyle L_{1}} with respect to L 2 {\displaystyle L_{2}} is the language consisting of strings w such that xw is in L 1 {\displaystyle L_{1}} for some string x in L 2 {\displaystyle L_{2}} . Formally: In other words, we take all the strings in L 1 {\displaystyle L_{1}} that have a prefix in L 2 {\displaystyle L_{2}} , and remove this prefix. Note that the operands of ∖ {\displaystyle \backslash } are in reverse order: the first operand is L 2 {\displaystyle L_{2}} and L 1 {\displaystyle L_{1}} is second.
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https://en.wikipedia.org/wiki/Quotient_of_a_formal_language
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In mathematics and computer science, the sorting numbers are a sequence of numbers introduced in 1950 by Hugo Steinhaus for the analysis of comparison sort algorithms. These numbers give the worst-case number of comparisons used by both binary insertion sort and merge sort. However, there are other algorithms that use fewer comparisons.
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https://en.wikipedia.org/wiki/Sorting_number
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In mathematics and computer science, the syntactic monoid M ( L ) {\displaystyle M(L)} of a formal language L {\displaystyle L} is the smallest monoid that recognizes the language L {\displaystyle L} .
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https://en.wikipedia.org/wiki/Syntactic_semigroup
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In mathematics and computer science, trace theory aims to provide a concrete mathematical underpinning for the study of concurrent computation and process calculi. The underpinning is provided by an algebraic definition of the free partially commutative monoid or trace monoid, or equivalently, the history monoid, which provides a concrete algebraic foundation, analogous to the way that the free monoid provides the underpinning for formal languages. The power of trace theory stems from the fact that the algebra of dependency graphs (such as Petri nets) is isomorphic to that of trace monoids, and thus, one can apply both algebraic formal language tools, as well as tools from graph theory. While the trace monoid had been studied by Pierre Cartier and Dominique Foata for its combinatorics in the 1960s, trace theory was first formulated by Antoni Mazurkiewicz in the 1970s, in an attempt to evade some of the problems in the theory of concurrent computation, including the problems of interleaving and non-deterministic choice with regards to refinement in process calculi.
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https://en.wikipedia.org/wiki/Trace_theory
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In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
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https://en.wikipedia.org/wiki/Trunc_(command)
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In mathematics and computing, Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains no other instances of "11" before the end. The Fibonacci code is closely related to the Zeckendorf representation, a positional numeral system that uses Zeckendorf's theorem and has the property that no number has a representation with consecutive 1s. The Fibonacci code word for a particular integer is exactly the integer's Zeckendorf representation with the order of its digits reversed and an additional "1" appended to the end.
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https://en.wikipedia.org/wiki/Fibonacci_coding
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In mathematics and computing, a radix point or radix character is a symbol used in the display of numbers to separate the integer part of the value from its fractional part. In English and many other languages (including many that are written right-to-left), the integer part is at the left of the radix point, and the fraction part at the right of it.A radix point is most often used in decimal (base 10) notation, when it is more commonly called the decimal point (the prefix deci- implying base 10). In English-speaking countries, the decimal point is usually a small dot (.) placed either on the baseline, or halfway between the baseline and the top of the digits (·) In many other countries, the radix point is a comma (,) placed on the baseline.These conventions are generally used both in machine displays (printing, computer monitors) and in handwriting.
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https://en.wikipedia.org/wiki/Decimal_comma
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It is important to know which notation is being used when working in different software programs. The respective ISO standard defines both the comma and the small dot as decimal markers, but does not explicitly define universal radix marks for bases other than 10. Fractional numbers are rarely displayed in other number bases, but, when they are, a radix character may be used for the same purpose. When used with the binary (base 2) representation, it may be called "binary point".
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https://en.wikipedia.org/wiki/Decimal_comma
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In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f, from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number x such that f(x) = 0. As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound).Solving an equation f(x) = g(x) is the same as finding the roots of the function h(x) = f(x) – g(x). Thus root-finding algorithms allow solving any equation defined by continuous functions.
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https://en.wikipedia.org/wiki/Root-finding_algorithms
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However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists. Most numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converges towards the root as its limit. They require one or more initial guesses of the root as starting values, then each iteration of the algorithm produces a successively more accurate approximation to the root.
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https://en.wikipedia.org/wiki/Root-finding_algorithms
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Since the iteration must be stopped at some point, these methods produce an approximation to the root, not an exact solution. Many methods compute subsequent values by evaluating an auxiliary function on the preceding values.
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https://en.wikipedia.org/wiki/Root-finding_algorithms
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The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points, and for converging rapidly to these fixed points. The behavior of general root-finding algorithms is studied in numerical analysis. However, for polynomials, root-finding study belongs generally to computer algebra, since algebraic properties of polynomials are fundamental for the most efficient algorithms.
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https://en.wikipedia.org/wiki/Root-finding_algorithms
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The efficiency of an algorithm may depend dramatically on the characteristics of the given functions. For example, many algorithms use the derivative of the input function, while others work on every continuous function. In general, numerical algorithms are not guaranteed to find all the roots of a function, so failing to find a root does not prove that there is no root. However, for polynomials, there are specific algorithms that use algebraic properties for certifying that no root is missed, and locating the roots in separate intervals (or disks for complex roots) that are small enough to ensure the convergence of numerical methods (typically Newton's method) to the unique root so located.
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https://en.wikipedia.org/wiki/Root-finding_algorithms
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In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index. That is, the ith row contains only i elements.
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https://en.wikipedia.org/wiki/Triangular_array
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In mathematics and computing, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. These minimization problems arise especially in least squares curve fitting. The LMA interpolates between the Gauss–Newton algorithm (GNA) and the method of gradient descent.
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https://en.wikipedia.org/wiki/Levenberg–Marquardt_nonlinear_least_squares_fitting_algorithm
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The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. For well-behaved functions and reasonable starting parameters, the LMA tends to be slower than the GNA. LMA can also be viewed as Gauss–Newton using a trust region approach.
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https://en.wikipedia.org/wiki/Levenberg–Marquardt_nonlinear_least_squares_fitting_algorithm
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The algorithm was first published in 1944 by Kenneth Levenberg, while working at the Frankford Army Arsenal. It was rediscovered in 1963 by Donald Marquardt, who worked as a statistician at DuPont, and independently by Girard, Wynne and Morrison.The LMA is used in many software applications for solving generic curve-fitting problems. By using the Gauss–Newton algorithm it often converges faster than first-order methods. However, like other iterative optimization algorithms, the LMA finds only a local minimum, which is not necessarily the global minimum.
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https://en.wikipedia.org/wiki/Levenberg–Marquardt_nonlinear_least_squares_fitting_algorithm
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In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" (or alternatively "a"–"f") to represent values from ten to fifteen. Software developers and system designers widely use hexadecimal numbers because they provide a human-friendly representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble).
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https://en.wikipedia.org/wiki/Hex_format
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For example, an 8-bit byte can have values ranging from 00000000 to 11111111 (0 to 255 decimal) in binary form, which can be conveniently represented as 00 to FF in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example, the decimal value 24,779 would be expressed in hexadecimal as 60CB16.
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https://en.wikipedia.org/wiki/Hex_format
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In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix 0x is used in C, which would denote this value as 0x60CB. Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.
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https://en.wikipedia.org/wiki/Hex_format
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In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" (or alternatively "a"–"f") to represent values from ten to fifteen. Software developers and system designers widely use hexadecimal numbers because they provide a human-friendly representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble).
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https://en.wikipedia.org/wiki/Hex_digit
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For example, an 8-bit byte can have values ranging from 00000000 to 11111111 (0 to 255 decimal) in binary form, which can be conveniently represented as 00 to FF in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example, the decimal value 30,227 would be expressed in hexadecimal as 761316.
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https://en.wikipedia.org/wiki/Hex_digit
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In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix 0x is used in C, which would denote this value as 0x7613. Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.
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https://en.wikipedia.org/wiki/Hex_digit
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In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (or mechanism) for addition throughout the whole range. For a given number of places half of the possible representations of numbers encode the positive numbers, the other half represents their respective additive inverses. The pairs of mutually additive inverse numbers are called complements. Thus subtraction of any number is implemented by adding its complement.
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https://en.wikipedia.org/wiki/Method_of_complements
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Changing the sign of any number is encoded by generating its complement, which can be done by a very simple and efficient algorithm. This method was commonly used in mechanical calculators and is still used in modern computers. The generalized concept of the radix complement (as described below) is also valuable in number theory, such as in Midy's theorem.
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https://en.wikipedia.org/wiki/Method_of_complements
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The nines' complement of a number given in decimal representation is formed by replacing each digit with nine minus that digit. To subtract a decimal number y (the subtrahend) from another number x (the minuend) two methods may be used: In the first method the nines' complement of x is added to y. Then the nines' complement of the result obtained is formed to produce the desired result. In the second method the nines' complement of y is added to x and one is added to the sum.
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https://en.wikipedia.org/wiki/Method_of_complements
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The leftmost digit '1' of the result is then discarded. Discarding the leftmost '1' is especially convenient on calculators or computers that use a fixed number of digits: there is nowhere for it to go so it is simply lost during the calculation. The nines' complement plus one is known as the ten's complement. The method of complements can be extended to other number bases (radices); in particular, it is used on most digital computers to perform subtraction, represent negative numbers in base 2 or binary arithmetic and test underflow and overflow in calculation.
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https://en.wikipedia.org/wiki/Method_of_complements
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In mathematics and computing, universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property (see definition below). This guarantees a low number of collisions in expectation, even if the data is chosen by an adversary. Many universal families are known (for hashing integers, vectors, strings), and their evaluation is often very efficient. Universal hashing has numerous uses in computer science, for example in implementations of hash tables, randomized algorithms, and cryptography.
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https://en.wikipedia.org/wiki/Universal_hashing
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In mathematics and control theory, H2, or H-square is a Hardy space with square norm. It is a subspace of L2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space.
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https://en.wikipedia.org/wiki/H_square
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In mathematics and economics, a corner solution is a special solution to an agent's maximization problem in which the quantity of one of the arguments in the maximized function is zero. In non-technical terms, a corner solution is when the chooser is either unwilling or unable to make a trade-off between goods.
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https://en.wikipedia.org/wiki/Corner_solution
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In mathematics and economics, the arc elasticity is the elasticity of one variable with respect to another between two given points. It is the ratio of the percentage change of one of the variables between the two points to the percentage change of the other variable. It contrasts with the point elasticity, which is the limit of the arc elasticity as the distance between the two points approaches zero and which hence is defined at a single point rather than for a pair of points. Like the point elasticity, the arc elasticity can vary in value depending on the starting point. For example, the arc elasticity of supply of a product with respect to the product's price could be large when the starting and ending prices are both low, but could be small when they are both high.20%/10%=2
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https://en.wikipedia.org/wiki/Arc_elasticity
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In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. The envelope theorem is an important tool for comparative statics of optimization models.The term envelope derives from describing the graph of the value function as the "upper envelope" of the graphs of the parameterized family of functions { f ( x , ⋅ ) } x ∈ X {\displaystyle \left\{f\left(x,\cdot \right)\right\}_{x\in X}} that are optimized.
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https://en.wikipedia.org/wiki/Envelope_Theorem
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In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.In the 1920s A.N. Tolstoi was one of the first to study the transportation problem mathematically. In 1930, in the collection Transportation Planning Volume I for the National Commissariat of Transportation of the Soviet Union, he published a paper "Methods of Finding the Minimal Kilometrage in Cargo-transportation in space".Major advances were made in the field during World War II by the Soviet mathematician and economist Leonid Kantorovich. Consequently, the problem as it is stated is sometimes known as the Monge–Kantorovich transportation problem. The linear programming formulation of the transportation problem is also known as the Hitchcock–Koopmans transportation problem.
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https://en.wikipedia.org/wiki/Optimal_transport_problem
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In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics. These codes are named in honor of Marcel J. E. Golay whose 1949 paper introducing them has been called, by E. R. Berlekamp, the "best single published page" in coding theory.There are two closely related binary Golay codes. The extended binary Golay code, G24 (sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 7-bit errors can be detected. The other, the perfect binary Golay code, G23, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit). In standard coding notation, the codes have parameters and , corresponding to the length of the codewords, the dimension of the code, and the minimum Hamming distance between two codewords, respectively.
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https://en.wikipedia.org/wiki/Extended_binary_Golay_code
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In mathematics and electronics, Cavity perturbation theory describes methods for derivation of perturbation formulae for performance changes of a cavity resonator. These performance changes are assumed to be caused by either introduction of a small foreign object into the cavity, or a small deformation of its boundary. Various mathematical methods can be used to study the characteristics of cavities, which are important in the field of microwave systems, and more generally in the field of electro magnetism. There are many industrial applications for cavity resonators, including microwave ovens, microwave communication systems, and remote imaging systems using electro magnetic waves. How a resonant cavity performs can affect the amount of energy that is required to make it resonate, or the relative stability or instability of the system.
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https://en.wikipedia.org/wiki/Cavity_Perturbation_Theory
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In mathematics and empirical science, quantification (or quantitation) is the act of counting and measuring that maps human sense observations and experiences into quantities. Quantification in this sense is fundamental to the scientific method.
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https://en.wikipedia.org/wiki/Quantification_(science)
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In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined by the property that the Kobayashi pseudometric is a metric. Kobayashi hyperbolicity of a complex manifold X implies that every holomorphic map from the complex line C to X is constant.
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https://en.wikipedia.org/wiki/Kobayashi_metric
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In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.
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https://en.wikipedia.org/wiki/Kähler_manifold
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Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics. Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics.
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https://en.wikipedia.org/wiki/Kähler_manifold
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In mathematics and especially game theory, the airport problem is a type of fair division problem in which it is decided how to distribute the cost of an airport runway among different players who need runways of different lengths. The problem was introduced by S. C. Littlechild and G. Owen in 1973. Their proposed solution is: Divide the cost of providing the minimum level of required facility for the smallest type of aircraft equally among the number of landings of all aircraft Divide the incremental cost of providing the minimum level of required facility for the second smallest type of aircraft (above the cost of the smallest type) equally among the number of landings of all but the smallest type of aircraft. Continue thus until finally the incremental cost of the largest type of aircraft is divided equally among the number of landings made by the largest aircraft type.The authors note that the resulting set of landing charges is the Shapley value for an appropriately defined game.
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https://en.wikipedia.org/wiki/Airport_problem
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In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric functions introduced by Richard Stanley (1984) in his study of the symmetric group of permutations. Formally, the Stanley symmetric function Fw(x1, x2, ...) indexed by a permutation w is defined as a sum of certain fundamental quasisymmetric functions. Each summand corresponds to a reduced decomposition of w, that is, to a way of writing w as a product of a minimal possible number of adjacent transpositions.
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https://en.wikipedia.org/wiki/Stanley_symmetric_function
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They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation w0 = n(n − 1)...21 (written here in one-line notation) has exactly ( n 2 ) ! 1 n − 1 ⋅ 3 n − 2 ⋅ 5 n − 3 ⋯ ( 2 n − 3 ) 1 {\displaystyle {\frac {{\binom {n}{2}}! }{1^{n-1}\cdot 3^{n-2}\cdot 5^{n-3}\cdots (2n-3)^{1}}}} reduced decompositions. (Here ( n 2 ) {\displaystyle {\binom {n}{2}}} denotes the binomial coefficient n(n − 1)/2 and ! denotes the factorial.)
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https://en.wikipedia.org/wiki/Stanley_symmetric_function
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In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers π i , j {\displaystyle \pi _{i,j}} (with positive integer indices i and j) that is nonincreasing in both indices. This means that π i , j ≥ π i , j + 1 {\displaystyle \pi _{i,j}\geq \pi _{i,j+1}} and π i , j ≥ π i + 1 , j {\displaystyle \pi _{i,j}\geq \pi _{i+1,j}} for all i and j.Moreover, only finitely many of the π i , j {\displaystyle \pi _{i,j}} may be nonzero. Plane partitions are a generalization of partitions of an integer. A plane partition may be represented visually by the placement of a stack of π i , j {\displaystyle \pi _{i,j}} unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture.
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https://en.wikipedia.org/wiki/Plane_partition
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The image has matrix form 4 4 3 2 1 4 3 1 1 3 2 1 1 {\displaystyle {\begin{matrix}4&4&3&2&1\\4&3&1&1\\3&2&1\\1\end{matrix}}} Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subset P {\displaystyle {\mathcal {P}}} of positive integer lattice points (i, j, k) in N 3 {\displaystyle \mathbb {N} ^{3}} , such that if (r, s, t) lies in P {\displaystyle {\mathcal {P}}} and if ( i , j , k ) {\displaystyle (i,j,k)} satisfies 1 ≤ i ≤ r {\displaystyle 1\leq i\leq r} , 1 ≤ j ≤ s {\displaystyle 1\leq j\leq s} , and 1 ≤ k ≤ t {\displaystyle 1\leq k\leq t} , then (i, j, k) also lies in P {\displaystyle {\mathcal {P}}} . The sum of a plane partition is n = ∑ i , j π i , j .
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https://en.wikipedia.org/wiki/Plane_partition
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{\displaystyle n=\sum _{i,j}\pi _{i,j}.} The sum describes the number of cubes of which the plane partition consists. Much interest in plane partitions concerns the enumeration of plane partitions in various classes.
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https://en.wikipedia.org/wiki/Plane_partition
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The number of plane partitions with sum n is denoted by PL(n). For example, there are six plane partitions with sum 3 3 2 1 1 1 1 2 1 1 1 1 1 1 1 {\displaystyle {\begin{matrix}3\end{matrix}}\qquad {\begin{matrix}2&1\end{matrix}}\qquad {\begin{matrix}1&1&1\end{matrix}}\qquad {\begin{matrix}2\\1\end{matrix}}\qquad {\begin{matrix}1&1\\1\end{matrix}}\qquad {\begin{matrix}1\\1\\1\end{matrix}}} so PL(3) = 6. Plane partitions may be classified by how symmetric they are. Many symmetric classes of plane partitions are enumerated by simple product formulas.
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https://en.wikipedia.org/wiki/Plane_partition
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In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certain sequence and the first n of them are summed, then one more is included to give the sum of the first n+1 of them, etc. If only finitely many numbers are included, the key issue is usually to find a simple expression for the value of the sum, or to require the sum to be less than a certain value, or to determine whether the sum is ever an integer. For an infinite series of reciprocals, the issues are twofold: First, does the sequence of sums diverge—that is, does it eventually exceed any given number—or does it converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of positive integers is said to be large if the sum of its reciprocals diverges, and small if it converges.) Second, if it converges, what is a simple expression for the value it converges to, is that value rational or irrational, and is that value algebraic or transcendental?
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https://en.wikipedia.org/wiki/Sum_of_reciprocals
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In mathematics and functional analysis a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators. One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors.
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https://en.wikipedia.org/wiki/Decomposable_operator
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