3D modelingIn 3D computer graphics, 3D modeling is the process of developing a mathematical coordinate-based representation of any surface of an object (inanimate or living) in three dimensions via specialized software by manipulating edges, vertices, and polygons in a simulated 3D space. Three-dimensional (3D) models represent a physical body using a collection of points in 3D space, connected by various geometric entities such as triangles, lines, curved surfaces, etc.
Transformation opticsTransformation optics is a branch of optics which applies metamaterials to produce spatial variations, derived from coordinate transformations, which can direct chosen bandwidths of electromagnetic radiation. This can allow for the construction of new composite artificial devices, which probably could not exist without metamaterials and coordinate transformation. Computing power that became available in the late 1990s enables prescribed quantitative values for the permittivity and permeability, the constitutive parameters, which produce localized spatial variations.
GrammarIn linguistics, the grammar of a natural language is its set of structural rules on speakers' or writers' usage and creation of clauses, phrases, and words. The term can also refer to the study of such rules, a subject that includes phonology, morphology, and syntax, together with phonetics, semantics, and pragmatics. There are two different ways to study grammar right now: traditional grammar and theoretical grammar. Fluent speakers of a language variety or lect have internalised these rules.
Procedural modelingProcedural modeling is an umbrella term for a number of techniques in computer graphics to create 3D models and textures from sets of rules. L-Systems, fractals, and generative modeling are procedural modeling techniques since they apply algorithms for producing scenes. The set of rules may either be embedded into the algorithm, configurable by parameters, or the set of rules is separate from the evaluation engine. The output is called procedural content, which can be used in computer games, films, be uploaded to the internet, or the user may edit the content manually.
Abelian varietyIn mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field.
Algebraic varietyAlgebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly.
Universal grammarUniversal grammar (UG), in modern linguistics, is the theory of the innate biological component of the language faculty, usually credited to Noam Chomsky. The basic postulate of UG is that there are innate constraints on what the grammar of a possible human language could be. When linguistic stimuli are received in the course of language acquisition, children then adopt specific syntactic rules that conform to UG. The advocates of this theory emphasize and partially rely on the poverty of the stimulus (POS) argument and the existence of some universal properties of natural human languages.
Projective varietyIn algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .
Chow varietyIn mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety is the fine moduli variety parametrizing all effective algebraic cycles of dimension and degree in . The Chow variety may be constructed via a Chow embedding into a sufficiently large projective space.
Generalized flag varietyIn mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F.
Construction grammarConstruction grammar (often abbreviated CxG) is a family of theories within the field of cognitive linguistics which posit that constructions, or learned pairings of linguistic patterns with meanings, are the fundamental building blocks of human language. Constructions include words (aardvark, avocado), morphemes (anti-, -ing), fixed expressions and idioms (by and large, jog X's memory), and abstract grammatical rules such as the passive voice (The cat was hit by a car) or the ditransitive (Mary gave Alex the ball).
Constructive solid geometryConstructive solid geometry (CSG; formerly called computational binary solid geometry) is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine simpler objects, potentially generating visually complex objects by combining a few primitive ones. In 3D computer graphics and CAD, CSG is often used in procedural modeling. CSG can also be performed on polygonal meshes, and may or may not be procedural and/or parametric.
Transformational grammarIn linguistics, transformational grammar (TG) or transformational-generative grammar (TGG) is part of the theory of generative grammar, especially of natural languages. It considers grammar to be a system of rules that generate exactly those combinations of words that form grammatical sentences in a given language and involves the use of defined operations (called transformations) to produce new sentences from existing ones. The method is commonly associated with American linguist Noam Chomsky.
Formal grammarIn formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form. A formal grammar is defined as a set of production rules for such strings in a formal language. Formal language theory, the discipline that studies formal grammars and languages, is a branch of applied mathematics.
Code-switchingIn linguistics, code-switching or language alternation occurs when a speaker alternates between two or more languages, or language varieties, in the context of a single conversation or situation. Code-switching is different from plurilingualism in that plurilingualism refers to the ability of an individual to use multiple languages, while code-switching is the act of using multiple languages together. Multilinguals (speakers of more than one language) sometimes use elements of multiple languages when conversing with each other.
Phrase structure rulesPhrase structure rules are a type of rewrite rule used to describe a given language's syntax and are closely associated with the early stages of transformational grammar, proposed by Noam Chomsky in 1957. They are used to break down a natural language sentence into its constituent parts, also known as , including both lexical categories (parts of speech) and phrasal categories. A grammar that uses phrase structure rules is a type of phrase structure grammar.
Merge (linguistics)Merge (usually capitalized) is one of the basic operations in the Minimalist Program, a leading approach to generative syntax, when two syntactic objects are combined to form a new syntactic unit (a set). Merge also has the property of recursion in that it may apply to its own output: the objects combined by Merge are either lexical items or sets that were themselves formed by Merge. This recursive property of Merge has been claimed to be a fundamental characteristic that distinguishes language from other cognitive faculties.
Voice (grammar)In grammar, the voice (aka diathesis) of a verb describes the relationship between the action (or state) that the verb expresses and the participants identified by its arguments (subject, object, etc.). When the subject is the agent or doer of the action, the verb is in the active voice. When the subject is the patient, target or undergoer of the action, the verb is said to be in the passive voice. When the subject both performs and receives the action expressed by the verb, the verb is in the middle voice.
Ambiguous grammarIn computer science, an ambiguous grammar is a context-free grammar for which there exists a string that can have more than one leftmost derivation or parse tree. Every non-empty context-free language admits an ambiguous grammar by introducing e.g. a duplicate rule. A language that only admits ambiguous grammars is called an inherently ambiguous language. Deterministic context-free grammars are always unambiguous, and are an important subclass of unambiguous grammars; there are non-deterministic unambiguous grammars, however.
Affine varietyIn algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime. Some texts call variety any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense).
