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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 numberin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manual Of Style/Mathematics
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Manual may refer to: Instructions * User guide * Owner's manual * Instruction manual (gaming) * Online help *Procedures manual *Handbook 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 * Instruction (other) * Tutorial In education, a tutorial 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] |
Ordered Pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unordered pair'', denoted , always equals the unordered pair . Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another. In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ... [...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 termin ... [...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 Gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (mathematics)
In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. 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 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 is a point, called the center of projection, then t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interface (computer Science)
A communication protocol is a system of rules that allows two or more entities of a communications system to transmit information via any variation of a physical quantity. The protocol defines the rules, syntax, semantics, and synchronization of communication and possible error recovery methods. Protocols may be implemented by hardware, software, or a combination of both. Communicating systems use well-defined formats for exchanging various messages. Each message has an exact meaning intended to elicit a response from a range of possible responses predetermined for that particular situation. The specified behavior is typically independent of how it is to be implemented. Communication protocols have to be agreed upon by the parties involved. To reach an agreement, a protocol may be developed into a technical standard. A programming language describes the same for computations, so there is a close analogy between protocols and programming languages: ''protocols are to communica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implementation
Implementation is the realization of an application, execution of a plan, idea, scientific modelling, model, design, specification, Standardization, standard, algorithm, policy, or the Management, administration or management of a process or Goal, objective. Industry-specific definitions Information technology In the information technology industry, implementation refers to the post-sales process of guiding a client from purchase to use of the software or hardware that was purchased. This includes requirements analysis, scope analysis, customizations, systems integrations, user policies, user training and delivery. These steps are often overseen by a project manager using project management methodologies. Software Implementations involve several professionals that are relatively new to the knowledge based economy such as Business analysis, business analysts, software implementation specialists, solutions architects, and project managers. To implement a system successfully, many ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstraction
Abstraction is a process where general rules and concepts are derived from the use and classifying of specific examples, literal (reality, real or Abstract and concrete, 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: "Sculpture, Sculptural form is a powerful abstraction from actual objects and the three-dimensional space which we construe ... through sensory system, touch and sight." Conceptual abstractions may be made 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 in ... [...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 * Swansea Metropolitan University 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 ... [...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] |
Chinese Remainder Theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). The theorem is sometimes called Sunzi's theorem. Both names of the theorem refer to its earliest known statement that appeared in '' Sunzi Suanjing'', a Chinese manuscript written during the 3rd to 5th century CE. This first statement was restricted to the following example: If one knows that the remainder of ''n'' divided by 3 is 2, the remainder of ''n'' divided by 5 is 3, and the remainder of ''n'' divided by 7 is 2, then with no other information, one can determine the remainder of ''n'' divided by 105 (the product of 3, 5, and 7) without knowing the value of ''n''. In this example, the remainder is 23. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonconstructive Proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or ''pure existence theorem''), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof. A constructive proof may also refer to the stronger concept of a proof that is valid in constructive mathematics. Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice. Constructivism also induces a different meaning for some terminology (for example, the term "or" has a stronge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
