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Gödel Numbering For Sequences
In mathematics, a Gödel numbering for sequences provides an effective way to represent each finite sequence of natural numbers as a single natural number. While a set theoretical embedding is surely possible, the emphasis is on the effectiveness of the functions manipulating such representations of sequences: the operations on sequences (accessing individual members, concatenation) can be "implemented" using total recursive functions, and in fact by primitive recursive functions. It is usually used to build sequential “ data types” in arithmetic-based formalizations of some fundamental notions of mathematics. It is a specific case of the more general idea of Gödel numbering. For example, recursive function theory can be regarded as a formalization of the notion of an algorithm, and can be regarded as a programming language to mimic lists by encoding a sequence of natural numbers in a single natural number. Monk 1976: 56–58 Csirmaz 1994: 99–100 (seonline Gödel numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manual Of Style/Mathematics
{{disambiguation ...
Manual may refer to: Instructions * User guide * Owner's manual * Instruction manual (gaming) * Online help Other uses * Manual (music), a keyboard, as for an organ * Manual (band) * Manual transmission * Manual, a bicycle technique similar to a wheelie, but without the use of pedal torque * Manual, balancing on two wheels in freestyle skateboarding tricks * '' The Manual (How to Have a Number One the Easy Way)'' is a 1988 book by Bill Drummond and Jimmy Cauty See also * Instructions (other) * Tutorial A tutorial, in education, is a method of transferring knowledge and may be used as a part of a learning process. More interactive and specific than a book or a lecture, a tutorial seeks to teach by example and supply the information to complete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairing Function
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. Definition A pairing function is a bijection :\pi:\mathbb \times \mathbb \to \mathbb. More generally, a pairing function on a set ''A'' is a function that maps each pair of elements from ''A'' into an element of ''A'', such that any two pairs of elements of ''A'' are associated with different elements of ''A,'' or a bijection from A^2 to ''A''. Hopcroft and Ullman pairing function Hopcroft and Ullman (1979) define the following pairing function: \langle i, j\rangle := \frac(i+j-2)(i+j-1) + i, where i, j\in\. This is the same as the Cantor pairing function below, shifted to exclude 0 (i.e., i=k_2+1, j=k_1+1, and \langle i, j\rangle - 1 = \pi(k_2,k_1)). Cantor pairing function The Cantor pairing function is a primitive recu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modularity (programming)
Modular programming is a software design technique that emphasizes separating the functionality of a program into independent, interchangeable modules, such that each contains everything necessary to execute only one aspect of the desired functionality. A module interface expresses the elements that are provided and required by the module. The elements defined in the interface are detectable by other modules. The implementation contains the working code that corresponds to the elements declared in the interface. Modular programming is closely related to structured programming and object-oriented programming, all having the same goal of facilitating construction of large software programs and systems by decomposition into smaller pieces, and all originating around the 1960s. While the historical usage of these terms has been inconsistent, "modular programming" now refers to the high-level decomposition of the code of an entire program into pieces: structured programming to the low-l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pattern Matching
In computer science, pattern matching is the act of checking a given sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually has to be exact: "either it will or will not be a match." The patterns generally have the form of either sequences or tree structures. Uses of pattern matching include outputting the locations (if any) of a pattern within a token sequence, to output some component of the matched pattern, and to substitute the matching pattern with some other token sequence (i.e., search and replace). Sequence patterns (e.g., a text string) are often described using regular expressions and matched using techniques such as backtracking. Tree patterns are used in some programming languages as a general tool to process data based on its structure, e.g. C#, F#, Haskell, ML, Python, Ruby, Rust, Scala, Swift and the symbolic mathematics language Mathematica have special syntax for expressing tree p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abuse Of Notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors and confusion at the same time). However, since the concept of formal/syntactical correctness depends on both time and context, certain notations in mathematics that are flagged as abuse in one context could be formally correct in one or more other contexts. Time-dependent abuses of notation may occur when novel notations are introduced to a theory some time before the theory is first formalized; these may be formally corrected by solidifying and/or otherwise improving the theory. ''Abuse of notation'' should be contrasted with ''misuse'' of notation, which does not have the presentational benefits of the former and should be avoided (such as the misuse of constants of integration). A related concept is abuse of language or abuse of termi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Remainder
In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another. The ''modulo operation'' is the operation that produces such a remainder when given a dividend and divisor. Alternatively, a remainder is also what is left after subtracting one number from another, although this is more precisely called the ''difference''. This usage can be found in some elementary textbooks; colloquially it is replaced by the expression "the rest" as in "Give me two dollars back and keep the rest." However, the term "remainder" is still used in this sense when a function is approximated by a series expansion, where the error expression ("the rest") is referred to as the remainder term. Integer division Given an i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (mathematics)
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a projection is also called a ''projection'', even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: * The projection from a point onto a plane or central projection: If ''C'' is a point, called the center of projection, then the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interface (computer Science)
In computing, an interface is a shared boundary across which two or more separate components of a computer system exchange information. The exchange can be between software, computer hardware, peripheral devices, humans, and combinations of these. Some computer hardware devices, such as a touchscreen, can both send and receive data through the interface, while others such as a mouse or microphone may only provide an interface to send data to a given system. Hardware interfaces Hardware interfaces exist in many components, such as the various buses, storage devices, other I/O devices, etc. A hardware interface is described by the mechanical, electrical, and logical signals at the interface and the protocol for sequencing them (sometimes called signaling). See also: A standard interface, such as SCSI, decouples the design and introduction of computing hardware, such as I/O devices, from the design and introduction of other components of a computing system, thereby allowin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implementation
Implementation is the realization of an application, or execution of a plan, idea, model, design, specification, standard, algorithm, or policy. Industry-specific definitions Computer science In computer science, an implementation is a realization of a technical specification or algorithm as a program, software component, or other computer system through computer programming and deployment. Many implementations may exist for a given specification or standard. For example, web browsers contain implementations of World Wide Web Consortium-recommended specifications, and software development tools contain implementations of programming languages. A special case occurs in object-oriented programming, when a concrete class implements an interface; in this case the concrete class is an ''implementation'' of the interface and it includes methods which are ''implementations'' of those methods specified by the interface. Information technology In the information technology during ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstraction
Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or " concrete") signifiers, first principles, or other methods. "An abstraction" is the outcome of this process—a concept that acts as a common noun for all subordinate concepts and connects any related concepts as a ''group'', ''field'', or ''category''.Suzanne K. Langer (1953), ''Feeling and Form: a theory of art developed from Philosophy in a New Key'' p. 90: " Sculptural form is a powerful abstraction from actual objects and the three-dimensional space which we construe ... through touch and sight." Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only those aspects which are relevant for a particular purpose. For example, abstracting a leather soccer ball to the more general idea of a ball selects only the information on ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smu03
Educational institutions * St. Martin's University, Lacey, Washington, United States * St. Matthew's University, the Cayman Islands * St. Michaels University School, Victoria, British Columbia, Canada * Saint Monica University, Cameroon * Samuel Merritt University, Oakland, California, US * Sangmyung University, Seoul and Cheonan, South Korea * Sefako Makgatho Health Sciences University, Limpopo, South Africa * Shanghai Maritime University, China * Sikkim Manipal University, Gangtok, India * Singapore Management University, Singapore * Southern Medical University, Tonghe, Guangzhou, China * Southern Methodist University, Dallas, Texas, US ** SMU Mustangs, athletic teams * Southeastern Massachusetts University, now University of Massachusetts Dartmouth, US Other uses * Scandinavian Monetary Union, defunct * Somray language * Source measure unit, a type of test equipment * Special Mission Unit, a type of military unit * Suburban Multiple Unit (Queensland Rail) * Suriname Men Unit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Specification
A specification often refers to a set of documented requirements to be satisfied by a material, design, product, or service. A specification is often a type of technical standard. There are different types of technical or engineering specifications (specs), and the term is used differently in different technical contexts. They often refer to particular documents, and/or particular information within them. The word ''specification'' is broadly defined as "to state explicitly or in detail" or "to be specific". A requirement specification is a documented requirement, or set of documented requirements, to be satisfied by a given material, design, product, service, etc. It is a common early part of engineering design and product development processes in many fields. A functional specification is a kind of requirement specification, and may show functional block diagrams. A design or product specification describes the features of the ''solutions'' for the Requirement Specification ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
