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noun
adj
- (linear algebra, of a function in two variables) Linear (preserving linear combinations) in each variable.
- (complex analysis, physics, engineering) Of or pertaining to a Möbius transformation (type of conformal map representable as the ratio of two linear functions).
- linear with respect to each of two variables or positions
adj
- (linear algebra) Being the space of all linear functionals of (some other space).
- (grammar) Pertaining to a grammatical number in certain languages that refers to two of something, such as a pair of shoes.
- Pertaining to two, pertaining to a pair of.
- (category theory) Being the dual of some other category; containing the same objects but with source and target reversed for all morphisms.
- (mathematics, physics) Exhibiting duality.
- Characterized by having two (usually equivalent) components.
- having more than one decidedly dissimilar aspects or qualities
- consisting of or involving two parts or components usually in pairs
- a grammatical number category referring to two items or units as opposed to one item (singular) or more than two items (plural)
noun
- (geometry) Of a regular polyhedron with V vertices and F faces, the regular polyhedron having F vertices and V faces.
- (wrestling) A head-to-head match or meet between two teams, such as two high schools or colleges.
- (grammar) The dual number.
- (mathematics) Of a vector in an inner product space, the linear functional corresponding to taking the inner product with that vector. The set of all duals is a vector space called the dual space.
- Of an item that is one of a pair, the other item in the pair.
verb
adj
- (linear algebra, of a system of linear equations) Having more equations than variables.
- (of a problem or question) Having more constraints or causes than necessary to determine a solution or result.
- (usually psychoanalysis) Determined by multiple causes in such a way that any of the causes on its own would be sufficient to account for the effect.
verb
adj
- (linear algebra, of matrix) Having no inverse.
- Being the only one of the kind; unique.
- Being only one of a larger population; single, individual.
- (set theory, of a cardinal number) Not equal to its own cofinality.
- Distinguished by superiority: peerless, unmatched, eminent, exceptional, extraordinary.
- (linear algebra, of transformation) Having the property that the matrix of coefficients of the new variables has a determinant equal to zero.
- (chiefly law) Each; individual.
- (grammar) Referring to only one thing or person.
- Out of the ordinary; curious.
- unusual or striking
- the single one of its kind
- being a single and separate person or thing
- beyond or deviating from the usual or expected
- grammatical number category referring to a single item or unit
- composed of one member, set, or kind
noun
noun
- (linear algebra) A term at any position in a matrix.
- (Midlands) A passageway between terraced houses that provides a means of entering a back garden or yard.
- (uncountable) Permission to enter.
- A small group formed within a church, especially Episcopal, for simple dinner and fellowship, and to help facilitate new friendships
- A record made in a log, diary or anything similarly organized; (computing) a datum in a database.
- The act of entering.
- A doorway that provides a means of entering a building.
- An item in a list, such as an article in a dictionary or encyclopedia.
- The exhibition or depositing of a ship's papers at the customhouse, to procure licence to land goods; or the giving an account of a ship's cargo to the officer of the customs, and obtaining his permission to land the goods.
- (hunting) The introduction of new hounds into a pack.
- (insurance) The start of an insurance contract.
- (law) The act of taking possession.
- A small room immediately inside the front door of a house or other building, often having an access to a stairway and leading on to other rooms
- (music) The point when a musician starts to play or sing; entrance.
- something that provides access (to get in or get out)
- something (manuscripts or architectural plans and models or estimates or works of art of all genres etc.) submitted for the judgment of others (as in a competition)
- an item inserted in a written record
- the act of beginning something new
- the act of entering
- a written record of a commercial transaction
adj
noun
- (taxonomy) A taxonomic designation (such as of a subspecies) consisting of more than two terms.
- (linguistics, Sinology) A type of term consisting of multiple parts.
- (algebra, strict sense) An expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as a_nxⁿ+a_n-1xⁿ⁻¹+...+a_0x⁰.
- a mathematical function that is the sum of a number of terms
noun
- (mathematics, functional analysis) Of a bounded linear operator A, the set of scalar values λ such that the operator A—λI, where I denotes the identity operator, does not have a bounded inverse; intended as a generalisation of the linear algebra sense.
- (mathematics, linear algebra) The set of eigenvalues of a matrix.
- (psychology, education, usually with the) The autism spectrum.
- Specifically, a range of colours representing light (electromagnetic radiation) of contiguous frequencies; hence electromagnetic spectrum, visible spectrum, ultraviolet spectrum, etc.
- A range; a continuous, infinite, one-dimensional set, possibly bounded by extremes.
- The image of something seen that persists after the eyes are closed.
- (chemistry) The pattern of absorption or emission of radiation produced by a substance when subjected to energy (radiation, heat, electricity, etc.).
- (commutative algebra, algebraic geometry) An abstract object in mathematics created from a commutative ring R and denoted operatorname Spec(R) or operatorname SpecR and said to be the spectrum of R; useful in the study of such rings for providing a geometric object which encodes many of the properties R, and in modern geometry for generalizing the notion of an algebraic variety to that of an affine scheme. Formally, the set of all prime ideals R equipped with the Zariski topology and augmented with a sheaf of rings called the structure sheaf, generated by the B-sheaf on the distinguished open sets D_f which assigns the localization of R at f to each set D_f, regarded as a ring of functions on D_f. See Spectrum of a ring on Wikipedia.Wikipedia
- an ordered array of the components of an emission or wave
- a broad range of related objects or values or qualities or ideas or activities
adj
- (linear algebra, by specialization, of a system of linear equations) Such that all the constant terms are zero.
- (ring theory, of an element of a graded ring) Belonging to one of the summands of the grading (if the ring is graded over the natural numbers and the element is in the kth summand, it is said to be homogeneous of degree k; if the ring is graded over a commutative monoid I, and the element is an element of the ith summand, it is said to be of grade i)
- (of a linear differential equation) Having its degree-zero term equal to zero; admitting the trivial solution.
- (algebra, of a polynomial) Such that all its nonzero terms have the same degree.
- Having the same composition throughout; of uniform make-up.
- (probability theory, Fourier analysis, of a distribution S on Euclidean n-space (or on ℝⁿmathbf 0)) Informally: Determined by its restriction to the unit sphere. Formally: Such that, for all real t>0 and test functions ϕ( mathbf x), the equality S[t⁻ⁿϕ( mathbf x/t)]=t^(mS)[ϕ( mathbf x)] holds for some fixed real or complex m.
- Of the same kind; alike, similar.
- (of a linear map f between vector spaces graded by a commutative monoid I) Which respects the grading of its domain and codomain. Formally: Satisfying f(V_j)⊆W_i+j for fixed i (called the degree or grade of f), V_j the jth component of the grading of f 's domain, W_k the kth component of the grading of f 's codomain, and + representing the monoid operation in I.
- (geometry, of a space equipped with a group action) Informally: Everywhere the same, uniform, in the sense that any point can be moved to any other (via the group action) while respecting the structure of the space. Formally: Such that the group action is transitively and acts by automorphisms on the space (some authors also require that the action be faithful).
- (set theory, order theory, of a relation) Holding between a set and itself; being an endorelation.
- (of a first-order differential equation) Capable of being written in the form f(x,y) mathop dy=g(x,y) mathop dx where f and g are homogeneous functions of the same degree as each other.
- (mathematics) In any of several technical senses uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.).
- (geometry) Of or relating to homogeneous coordinates.
- The function f(x,y)#61;x²#43;x²ʸ#43;y² is not homogeneous on all of #92;mathbb#123;R#125;² because f(2,2)#61;16#92;neq 2ᵏ#42;3#61;2ᵏf(1,1) for any k, but f is homogeneous on the subspace of #92;mathbb#123;R#125;² spanned by (1,0) because f(#92;alphax,#92;alphay)#61;#92;alphax²#61;#92;alpha²f(x,y) for all (x,y)#92;in#92;operatorname#123;Span#125;#92;#123;(1,0)#92;#125;.
- (mathematical analysis, generalizing the case of polynomial functions, of a function f) Such that if each of f 's inputs are multiplied by the same scalar, f 's output is multiplied by the same scalar to some fixed power (called the degree of homogeneity or degree of f). (Formally and more generally, of a partial function f between vector spaces whose domain is a linear cone) Satisfying the equality f(s mathbf x)=sᵏᶠ(
- (of a general differential equation) Homogeneous as a function of the dependent variable and its derivatives.
- (chemistry) In the same state of matter.
- all of the same or similar kind or nature
noun
- (mathematics, linear algebra) A relation between generators of a module.
- (Jungian psychology) An archetypal pairing of contrasexual opposites, symbolizing the communication of the conscious and unconscious minds.
- (astronomy, astrology) An alignment of three celestial bodies (for example, the Sun, Earth, and Moon) such that one body is directly between the other two, such as occurs at an eclipse.
- (zoology) The association of two protozoa end-to-end or laterally for the purpose of asexual exchange of genetic material.
- (Gnosticism) Complementary female–male pairings of the emanations known as Aeons.
- (genetics) The pairing of chromosomes in meiosis.
- (medicine) The fusion of some or all of the organs.
- the straight line configuration of 3 celestial bodies (as the sun and earth and moon) in a gravitational system
noun
- (linear algebra) The number of elements of any basis of a vector space.
- (computing) Any of the independent ranges of indices in a multidimensional array.
- A construct whereby objects or individuals can be distinguished.
- (physics) One of the physical properties that are regarded as fundamental measures of a physical quantity, such as mass, length and time.
- (geometry) The number of independent coordinates needed to specify uniquely the location of a point in a space; also, any of such independent coordinates.
- (science fiction, fantasy) A universe or plane of existence.
- A single aspect of a given thing.
- A measure of spatial extent in a particular direction, such as height, width or breadth, or depth.
- the magnitude of something in a particular direction (especially length or width or height)
- a construct whereby objects or individuals can be distinguished
- one of three Cartesian coordinates that determine a position in space
- magnitude or extent
verb
adj
- (linear algebra, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.
- (module theory, of a module) In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules.
- (representation theory, of a linear representation of a group or algebra) Being a direct sum of simple representations (also known as irreducible representations).
- (ring theory, of an algebra or ring) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.
- (category theory, most generally, of an abelian category) Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects.
- (Lie theory, of a Lie algebra) Being a direct sum of simple Lie algebras.
- (group theory, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial.
- (of a ring, somewhat proscribed) Semiprimitive: having trivial Jacobson radical.
adj
- (linear algebra, of a matrix) Which commutes with its conjugate transpose.
- (topology, of a topology or topological space) In which disjoint closed sets can be separated by disjoint neighborhoods.
- (complex analysis, of a family of continuous functions) Which is pre-compact.
- (commutative algebra, of a domain) Integrally closed: equal its own integral closure in its field of fractions.
- (functional analysis, of a Hilbert space operator) Which commutes with its adjoint.
- (probability theory, statistics, of a distribution, random variable, etc.) Which has a very specific bell curve shape; that is or has the qualities of a normal distribution.
- (physics, of a mode in an oscillating system) In which all parts of an object vibrate at the same frequency (a normal mode).
- (rail transport, of points) In the default position, set for the most frequently used route.
- (chemistry) Of, relating to, or being a solution containing one equivalent weight of solute per litre of solution.
- (category theory, of a category) Which contains only normal morphisms.
- (organic chemistry) Describing a straight chain isomer of an aliphatic hydrocarbon, or an aliphatic compound in which a substituent is in the 1- position of such a hydrocarbon.
- (fandom slang, sarcastic, with “about”) Fervently interested in a subject; obsessed.
- (algebraic geometry, of a variety or scheme) Such that the local ring at every point is an integrally closed domain.
- (category theory, of a morphism) Which is the kernel or cokernel of some morphism, respectively.
- (number theory, of a real number) In whose representation in a given base b ≥ 2, for every positive integer n, the bⁿ possible strings of n digits follow a uniform distribution.
- Usual, healthy; not sick or ill or unlike oneself.
- (set theory, of a function from the ordinals to the ordinals) Which is strictly monotonically increasing and continuous with respect to the order topology.
- (algebra, of a field extension of a field K) Which is the splitting field of a family of polynomials in K.
- (algebra, of a subgroup) With cosets which form a group.
- (commutative algebra, of a ring) Such that all of its localizations at prime ideals are integrally closed domains.
- (education, of a school) Teaching teachers how to teach; teaching teachers the norms of education.
- According to norms or rules or to a regular pattern.
- (geometry) Perpendicular to a tangent of a curve or tangent plane of a surface.
- in accordance with scientific laws
- conforming with or constituting a norm or standard or level or type or social norm; not abnormal
- forming a right angle
- being approximately average or within certain limits in e.g. intelligence and development
noun
- (geometry, countable) A line or vector that is perpendicular to another line, surface, or plane.
- (medicine, countable) A person who is healthy, normal, as opposed to one who is morbid.
- (slang, countable) A person who is normal, who fits into mainstream society, as opposed to those who live alternative lifestyles.
- (countable, uncountable) The usual state.
- something regarded as a normative example
adj
- (algebra, commutative algebra, of a ring element in a ring B relative to a subring A) Being the root of some monic polynomial in A.
- Constituting a whole together with other parts or factors; not omittable or removable.
- (mathematics) Relating to integration (“the process of finding the integral [noun] of a function”).
- (mathematics) Of, pertaining to, or being an integer.
- constituting the undiminished entirety; lacking nothing essential especially not damaged
- of or denoted by an integer
- existing as an essential constituent or characteristic
noun
- (mathematics) One of the two fundamental operations of calculus (the other being differentiation), whereby a function's displacement, area, volume, or other qualities arising from the study of infinitesimal change are quantified, usually defined as a limiting process on a sequence of partial sums. Denoted using a long s: ∫, or a variant thereof.
- (mathematics) A definite integral: the result of the application of such an operation onto a function and a suitable subset of the function's domain: either a number or positive or negative infinity. In the former case, the integral is said to be finite or to converge; in the latter, the integral is said to diverge. In notation, the domain of integration is indicated either below the sign, or, if it is an interval, with its endpoints as sub- and super-scripts, and the function being integrated forming part of the integrand (or, generally, differential form) appearing in front of the integral sign.
- (specifically) Any of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.
- (mathematics) An indefinite integral: the result of the application of such an operation onto a function together with an indefinite domain, yielding a function; a function's antiderivative;
- the result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)
adj
- (algebra, number theory, of a number) Which is a root of some polynomial whose coefficients are rational.
- of or relating to algebra
- Of, or relating to, algebra.
- (algebra, of a field) Whose every element is a root of some polynomial whose coefficients are rational.
- (chess, of notation) Describing squares by file (referred to in intrinsic order rather than by the piece starting on that file) and rank, both with reference to a fixed point rather than a player-dependent perspective.
- (mathematics, of an expression, equation, or function) Containing only numbers, letters, and arithmetic operators.
noun
- (algebra) A polynomial with two terms.
- (taxonomy) A scientific name at the rank of species, with two terms: a generic name and a specific name.
- (algebra) A quantity expressed as the sum or difference of two terms.
- (mathematics) a quantity expressed as a sum or difference of two terms; a polynomial with two terms
adj
adj
- (algebra, of an algebraic structure) Having a commutative operation.
- (mathematics, of a binary operation) Such that the order in which the operands are taken does not affect their image under the operation.
- Relating to exchange; interchangeable.
- (mathematics, of a diagram of morphisms) Such that any two sequences of morphisms with the same initial and final positions compose to the same morphism.
- (of a binary operation) independent of order; as in e.g.: ‘a x b’ = ‘b x a’
noun
- (linear algebra) a vector with the size given by the product of two vectors computed as the product of the magnitudes of the vectors and the sine of the angle between their directions, and directed perpendicular to the given two vectors, with positive orientation.
- a vector that is the product of two other vectors
noun
adj
name
verb
- (algebra, transitive and intransitive, acts on a polynomial) To factor into linear factors.
- (transitive, intransitive, slang) To leave.
- (sports, especially baseball) For both teams involved in a doubleheader to win one game each and lose another.
- (intransitive, of a couple) To separate.
- (of an object which expresses the relationship between algebraic structures, particularly a short exact sequence) To contain an object which may be so expressed.
- (transitive) To share; to divide.
- To be broken; to be dashed to pieces.
- (ambitransitive) To (cause to) break up; to throw into discord.
- (intransitive, of something solid, particularly wood) To break along the grain fully or partly along a more or less straight line.
- (intransitive) To burst out laughing.
- (transitive, ergative, of something solid) To divide fully or partly along a more or less straight line.
- (intransitive, politics) To vote for candidates of opposite parties.
- (generally, of an algebraic structure) To be expressable as a direct sum of sub-modules, -algebras, etc.
- come open suddenly and violently, as if from internal pressure
- discontinue an association or relation; go different ways
- go one's own way; move apart
- separate or cut with a tool, such as a sharp instrument
- separate into parts or portions
adj
- (stock exchange, historical, of quotations) Given in sixteenths rather than eighths.
- (London stock exchange) Designating ordinary stock that has been divided into preferred ordinary and deferred ordinary.
- Divided.
- (algebra, of a short exact sequence) Having the middle object (group, module, etc.) equal to the direct sum of the others.
- (of coffee) Comprising half decaffeinated and half caffeinated espresso.
- (stock exchange, of an order, sale, etc.) Divided so as to be done or executed part at one time or price and part at another time or price.
- (especially of wood) cut or ripped longitudinally with the grain
- having been divided; having the unity destroyed
noun
- A crack or longitudinal fissure.
- A dessert or confection resembling a banana split.
- (bowling) A result of a first throw that leaves two or more pins standing with one or more pins between them knocked down.
- (baseball, slang) A split-finger fastball.
- A bottle of wine containing 37.5 centiliters, half the volume of a standard 75-centiliter bottle; a demi.
- (bodybuilding) A workout routine as seen by its distribution of muscle groups or the extent and manner they are targeted in a microcycle.
- A breach or separation, as in a political party; a division.
- A unit of measure used for champagne or other spirits: 18.75 centiliters or one quarter of a standard 75-centiliter bottle. Commercially comparable to ¹⁄₂₀ (US) gallon, which is ¹⁄₂ of a fifth.
- (gymnastics, cheerleading, dance, usually in the phrase "to do the splits") A maneuver of spreading or sliding the feet apart until the legs are flat on the floor 180 degrees apart, either sideways to the body or with one leg in front and one behind, thus lowering the body completely to the floor in an upright position.
- (systematics) The division of a single taxon into two or more taxa; as opposed to a lump.
- A piece that is split off, or made thin, by splitting; a splinter; a fragment.
- (construction) A tear resulting from tensile stresses.
- (music) A recording containing songs by multiple artists; a split single or split album.
- (gambling) A division of a stake happening when two cards of the kind on which the stake is laid are dealt in the same turn.
- A split shot or split stroke.
- (athletics, speedrunning) The elapsed time at specific intermediate points in a race or speedrun.
- (leather manufacture) One of the sections of a skin made by dividing it into two or more thicknesses.
- a promised or claimed share of loot or money
- a dessert of sliced fruit and ice cream covered with whipped cream and cherries and nuts
- (tenpin bowling) a divided formation of pins left standing after the first bowl
- a lengthwise crack in wood
- a bottle containing half the usual amount
- the act of rending or ripping or splitting something
- extending the legs at right angles to the trunk (one in front and the other in back)
- an increase in the number of outstanding shares of a corporation without changing the shareholders' equity
- an opening made forcibly as by pulling apart
- division of a group into opposing factions
noun
adj
- (linear algebra, of a function in two variables) Linear (preserving linear combinations) in each variable.
- (complex analysis, physics, engineering) Of or pertaining to a Möbius transformation (type of conformal map representable as the ratio of two linear functions).
- linear with respect to each of two variables or positions
noun
- (linear algebra) A term at any position in a matrix.
- (Midlands) A passageway between terraced houses that provides a means of entering a back garden or yard.
- (uncountable) Permission to enter.
- A small group formed within a church, especially Episcopal, for simple dinner and fellowship, and to help facilitate new friendships
- A record made in a log, diary or anything similarly organized; (computing) a datum in a database.
- The act of entering.
- A doorway that provides a means of entering a building.
- An item in a list, such as an article in a dictionary or encyclopedia.
- The exhibition or depositing of a ship's papers at the customhouse, to procure licence to land goods; or the giving an account of a ship's cargo to the officer of the customs, and obtaining his permission to land the goods.
- (hunting) The introduction of new hounds into a pack.
- (insurance) The start of an insurance contract.
- (law) The act of taking possession.
- A small room immediately inside the front door of a house or other building, often having an access to a stairway and leading on to other rooms
- (music) The point when a musician starts to play or sing; entrance.
- something that provides access (to get in or get out)
- something (manuscripts or architectural plans and models or estimates or works of art of all genres etc.) submitted for the judgment of others (as in a competition)
- an item inserted in a written record
- the act of beginning something new
- the act of entering
- a written record of a commercial transaction
noun
- (mathematics, functional analysis) Of a bounded linear operator A, the set of scalar values λ such that the operator A—λI, where I denotes the identity operator, does not have a bounded inverse; intended as a generalisation of the linear algebra sense.
- (mathematics, linear algebra) The set of eigenvalues of a matrix.
- (psychology, education, usually with the) The autism spectrum.
- Specifically, a range of colours representing light (electromagnetic radiation) of contiguous frequencies; hence electromagnetic spectrum, visible spectrum, ultraviolet spectrum, etc.
- A range; a continuous, infinite, one-dimensional set, possibly bounded by extremes.
- The image of something seen that persists after the eyes are closed.
- (chemistry) The pattern of absorption or emission of radiation produced by a substance when subjected to energy (radiation, heat, electricity, etc.).
- (commutative algebra, algebraic geometry) An abstract object in mathematics created from a commutative ring R and denoted operatorname Spec(R) or operatorname SpecR and said to be the spectrum of R; useful in the study of such rings for providing a geometric object which encodes many of the properties R, and in modern geometry for generalizing the notion of an algebraic variety to that of an affine scheme. Formally, the set of all prime ideals R equipped with the Zariski topology and augmented with a sheaf of rings called the structure sheaf, generated by the B-sheaf on the distinguished open sets D_f which assigns the localization of R at f to each set D_f, regarded as a ring of functions on D_f. See Spectrum of a ring on Wikipedia.Wikipedia
- an ordered array of the components of an emission or wave
- a broad range of related objects or values or qualities or ideas or activities
noun
- (mathematics, linear algebra) A relation between generators of a module.
- (Jungian psychology) An archetypal pairing of contrasexual opposites, symbolizing the communication of the conscious and unconscious minds.
- (astronomy, astrology) An alignment of three celestial bodies (for example, the Sun, Earth, and Moon) such that one body is directly between the other two, such as occurs at an eclipse.
- (zoology) The association of two protozoa end-to-end or laterally for the purpose of asexual exchange of genetic material.
- (Gnosticism) Complementary female–male pairings of the emanations known as Aeons.
- (genetics) The pairing of chromosomes in meiosis.
- (medicine) The fusion of some or all of the organs.
- the straight line configuration of 3 celestial bodies (as the sun and earth and moon) in a gravitational system
noun
- (linear algebra) The number of elements of any basis of a vector space.
- (computing) Any of the independent ranges of indices in a multidimensional array.
- A construct whereby objects or individuals can be distinguished.
- (physics) One of the physical properties that are regarded as fundamental measures of a physical quantity, such as mass, length and time.
- (geometry) The number of independent coordinates needed to specify uniquely the location of a point in a space; also, any of such independent coordinates.
- (science fiction, fantasy) A universe or plane of existence.
- A single aspect of a given thing.
- A measure of spatial extent in a particular direction, such as height, width or breadth, or depth.
- the magnitude of something in a particular direction (especially length or width or height)
- a construct whereby objects or individuals can be distinguished
- one of three Cartesian coordinates that determine a position in space
- magnitude or extent
verb
noun
- (algebra) A polynomial with two terms.
- (taxonomy) A scientific name at the rank of species, with two terms: a generic name and a specific name.
- (algebra) A quantity expressed as the sum or difference of two terms.
- (mathematics) a quantity expressed as a sum or difference of two terms; a polynomial with two terms
adj
noun
- (linear algebra) a vector with the size given by the product of two vectors computed as the product of the magnitudes of the vectors and the sine of the angle between their directions, and directed perpendicular to the given two vectors, with positive orientation.
- a vector that is the product of two other vectors
noun
adj
name
verb
- (algebra, transitive and intransitive, acts on a polynomial) To factor into linear factors.
- (transitive, intransitive, slang) To leave.
- (sports, especially baseball) For both teams involved in a doubleheader to win one game each and lose another.
- (intransitive, of a couple) To separate.
- (of an object which expresses the relationship between algebraic structures, particularly a short exact sequence) To contain an object which may be so expressed.
- (transitive) To share; to divide.
- To be broken; to be dashed to pieces.
- (ambitransitive) To (cause to) break up; to throw into discord.
- (intransitive, of something solid, particularly wood) To break along the grain fully or partly along a more or less straight line.
- (intransitive) To burst out laughing.
- (transitive, ergative, of something solid) To divide fully or partly along a more or less straight line.
- (intransitive, politics) To vote for candidates of opposite parties.
- (generally, of an algebraic structure) To be expressable as a direct sum of sub-modules, -algebras, etc.
- come open suddenly and violently, as if from internal pressure
- discontinue an association or relation; go different ways
- go one's own way; move apart
- separate or cut with a tool, such as a sharp instrument
- separate into parts or portions
adj
- (stock exchange, historical, of quotations) Given in sixteenths rather than eighths.
- (London stock exchange) Designating ordinary stock that has been divided into preferred ordinary and deferred ordinary.
- Divided.
- (algebra, of a short exact sequence) Having the middle object (group, module, etc.) equal to the direct sum of the others.
- (of coffee) Comprising half decaffeinated and half caffeinated espresso.
- (stock exchange, of an order, sale, etc.) Divided so as to be done or executed part at one time or price and part at another time or price.
- (especially of wood) cut or ripped longitudinally with the grain
- having been divided; having the unity destroyed
noun
- A crack or longitudinal fissure.
- A dessert or confection resembling a banana split.
- (bowling) A result of a first throw that leaves two or more pins standing with one or more pins between them knocked down.
- (baseball, slang) A split-finger fastball.
- A bottle of wine containing 37.5 centiliters, half the volume of a standard 75-centiliter bottle; a demi.
- (bodybuilding) A workout routine as seen by its distribution of muscle groups or the extent and manner they are targeted in a microcycle.
- A breach or separation, as in a political party; a division.
- A unit of measure used for champagne or other spirits: 18.75 centiliters or one quarter of a standard 75-centiliter bottle. Commercially comparable to ¹⁄₂₀ (US) gallon, which is ¹⁄₂ of a fifth.
- (gymnastics, cheerleading, dance, usually in the phrase "to do the splits") A maneuver of spreading or sliding the feet apart until the legs are flat on the floor 180 degrees apart, either sideways to the body or with one leg in front and one behind, thus lowering the body completely to the floor in an upright position.
- (systematics) The division of a single taxon into two or more taxa; as opposed to a lump.
- A piece that is split off, or made thin, by splitting; a splinter; a fragment.
- (construction) A tear resulting from tensile stresses.
- (music) A recording containing songs by multiple artists; a split single or split album.
- (gambling) A division of a stake happening when two cards of the kind on which the stake is laid are dealt in the same turn.
- A split shot or split stroke.
- (athletics, speedrunning) The elapsed time at specific intermediate points in a race or speedrun.
- (leather manufacture) One of the sections of a skin made by dividing it into two or more thicknesses.
- a promised or claimed share of loot or money
- a dessert of sliced fruit and ice cream covered with whipped cream and cherries and nuts
- (tenpin bowling) a divided formation of pins left standing after the first bowl
- a lengthwise crack in wood
- a bottle containing half the usual amount
- the act of rending or ripping or splitting something
- extending the legs at right angles to the trunk (one in front and the other in back)
- an increase in the number of outstanding shares of a corporation without changing the shareholders' equity
- an opening made forcibly as by pulling apart
- division of a group into opposing factions
adj
- (linear algebra) Being the space of all linear functionals of (some other space).
- (grammar) Pertaining to a grammatical number in certain languages that refers to two of something, such as a pair of shoes.
- Pertaining to two, pertaining to a pair of.
- (category theory) Being the dual of some other category; containing the same objects but with source and target reversed for all morphisms.
- (mathematics, physics) Exhibiting duality.
- Characterized by having two (usually equivalent) components.
- having more than one decidedly dissimilar aspects or qualities
- consisting of or involving two parts or components usually in pairs
- a grammatical number category referring to two items or units as opposed to one item (singular) or more than two items (plural)
noun
- (geometry) Of a regular polyhedron with V vertices and F faces, the regular polyhedron having F vertices and V faces.
- (wrestling) A head-to-head match or meet between two teams, such as two high schools or colleges.
- (grammar) The dual number.
- (mathematics) Of a vector in an inner product space, the linear functional corresponding to taking the inner product with that vector. The set of all duals is a vector space called the dual space.
- Of an item that is one of a pair, the other item in the pair.
verb
adj
- (linear algebra, of a system of linear equations) Having more equations than variables.
- (of a problem or question) Having more constraints or causes than necessary to determine a solution or result.
- (usually psychoanalysis) Determined by multiple causes in such a way that any of the causes on its own would be sufficient to account for the effect.
verb
adj
- (linear algebra, of matrix) Having no inverse.
- Being the only one of the kind; unique.
- Being only one of a larger population; single, individual.
- (set theory, of a cardinal number) Not equal to its own cofinality.
- Distinguished by superiority: peerless, unmatched, eminent, exceptional, extraordinary.
- (linear algebra, of transformation) Having the property that the matrix of coefficients of the new variables has a determinant equal to zero.
- (chiefly law) Each; individual.
- (grammar) Referring to only one thing or person.
- Out of the ordinary; curious.
- unusual or striking
- the single one of its kind
- being a single and separate person or thing
- beyond or deviating from the usual or expected
- grammatical number category referring to a single item or unit
- composed of one member, set, or kind
noun
adj
noun
- (taxonomy) A taxonomic designation (such as of a subspecies) consisting of more than two terms.
- (linguistics, Sinology) A type of term consisting of multiple parts.
- (algebra, strict sense) An expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as a_nxⁿ+a_n-1xⁿ⁻¹+...+a_0x⁰.
- a mathematical function that is the sum of a number of terms
adj
- (linear algebra, by specialization, of a system of linear equations) Such that all the constant terms are zero.
- (ring theory, of an element of a graded ring) Belonging to one of the summands of the grading (if the ring is graded over the natural numbers and the element is in the kth summand, it is said to be homogeneous of degree k; if the ring is graded over a commutative monoid I, and the element is an element of the ith summand, it is said to be of grade i)
- (of a linear differential equation) Having its degree-zero term equal to zero; admitting the trivial solution.
- (algebra, of a polynomial) Such that all its nonzero terms have the same degree.
- Having the same composition throughout; of uniform make-up.
- (probability theory, Fourier analysis, of a distribution S on Euclidean n-space (or on ℝⁿmathbf 0)) Informally: Determined by its restriction to the unit sphere. Formally: Such that, for all real t>0 and test functions ϕ( mathbf x), the equality S[t⁻ⁿϕ( mathbf x/t)]=t^(mS)[ϕ( mathbf x)] holds for some fixed real or complex m.
- Of the same kind; alike, similar.
- (of a linear map f between vector spaces graded by a commutative monoid I) Which respects the grading of its domain and codomain. Formally: Satisfying f(V_j)⊆W_i+j for fixed i (called the degree or grade of f), V_j the jth component of the grading of f 's domain, W_k the kth component of the grading of f 's codomain, and + representing the monoid operation in I.
- (geometry, of a space equipped with a group action) Informally: Everywhere the same, uniform, in the sense that any point can be moved to any other (via the group action) while respecting the structure of the space. Formally: Such that the group action is transitively and acts by automorphisms on the space (some authors also require that the action be faithful).
- (set theory, order theory, of a relation) Holding between a set and itself; being an endorelation.
- (of a first-order differential equation) Capable of being written in the form f(x,y) mathop dy=g(x,y) mathop dx where f and g are homogeneous functions of the same degree as each other.
- (mathematics) In any of several technical senses uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.).
- (geometry) Of or relating to homogeneous coordinates.
- The function f(x,y)#61;x²#43;x²ʸ#43;y² is not homogeneous on all of #92;mathbb#123;R#125;² because f(2,2)#61;16#92;neq 2ᵏ#42;3#61;2ᵏf(1,1) for any k, but f is homogeneous on the subspace of #92;mathbb#123;R#125;² spanned by (1,0) because f(#92;alphax,#92;alphay)#61;#92;alphax²#61;#92;alpha²f(x,y) for all (x,y)#92;in#92;operatorname#123;Span#125;#92;#123;(1,0)#92;#125;.
- (mathematical analysis, generalizing the case of polynomial functions, of a function f) Such that if each of f 's inputs are multiplied by the same scalar, f 's output is multiplied by the same scalar to some fixed power (called the degree of homogeneity or degree of f). (Formally and more generally, of a partial function f between vector spaces whose domain is a linear cone) Satisfying the equality f(s mathbf x)=sᵏᶠ(
- (of a general differential equation) Homogeneous as a function of the dependent variable and its derivatives.
- (chemistry) In the same state of matter.
- all of the same or similar kind or nature
noun
adj
- (linear algebra, of a function in two variables) Linear (preserving linear combinations) in each variable.
- (complex analysis, physics, engineering) Of or pertaining to a Möbius transformation (type of conformal map representable as the ratio of two linear functions).
- linear with respect to each of two variables or positions
adj
- (linear algebra, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.
- (module theory, of a module) In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules.
- (representation theory, of a linear representation of a group or algebra) Being a direct sum of simple representations (also known as irreducible representations).
- (ring theory, of an algebra or ring) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.
- (category theory, most generally, of an abelian category) Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects.
- (Lie theory, of a Lie algebra) Being a direct sum of simple Lie algebras.
- (group theory, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial.
- (of a ring, somewhat proscribed) Semiprimitive: having trivial Jacobson radical.
adj
- (linear algebra, of a matrix) Which commutes with its conjugate transpose.
- (topology, of a topology or topological space) In which disjoint closed sets can be separated by disjoint neighborhoods.
- (complex analysis, of a family of continuous functions) Which is pre-compact.
- (commutative algebra, of a domain) Integrally closed: equal its own integral closure in its field of fractions.
- (functional analysis, of a Hilbert space operator) Which commutes with its adjoint.
- (probability theory, statistics, of a distribution, random variable, etc.) Which has a very specific bell curve shape; that is or has the qualities of a normal distribution.
- (physics, of a mode in an oscillating system) In which all parts of an object vibrate at the same frequency (a normal mode).
- (rail transport, of points) In the default position, set for the most frequently used route.
- (chemistry) Of, relating to, or being a solution containing one equivalent weight of solute per litre of solution.
- (category theory, of a category) Which contains only normal morphisms.
- (organic chemistry) Describing a straight chain isomer of an aliphatic hydrocarbon, or an aliphatic compound in which a substituent is in the 1- position of such a hydrocarbon.
- (fandom slang, sarcastic, with “about”) Fervently interested in a subject; obsessed.
- (algebraic geometry, of a variety or scheme) Such that the local ring at every point is an integrally closed domain.
- (category theory, of a morphism) Which is the kernel or cokernel of some morphism, respectively.
- (number theory, of a real number) In whose representation in a given base b ≥ 2, for every positive integer n, the bⁿ possible strings of n digits follow a uniform distribution.
- Usual, healthy; not sick or ill or unlike oneself.
- (set theory, of a function from the ordinals to the ordinals) Which is strictly monotonically increasing and continuous with respect to the order topology.
- (algebra, of a field extension of a field K) Which is the splitting field of a family of polynomials in K.
- (algebra, of a subgroup) With cosets which form a group.
- (commutative algebra, of a ring) Such that all of its localizations at prime ideals are integrally closed domains.
- (education, of a school) Teaching teachers how to teach; teaching teachers the norms of education.
- According to norms or rules or to a regular pattern.
- (geometry) Perpendicular to a tangent of a curve or tangent plane of a surface.
- in accordance with scientific laws
- conforming with or constituting a norm or standard or level or type or social norm; not abnormal
- forming a right angle
- being approximately average or within certain limits in e.g. intelligence and development
noun
- (geometry, countable) A line or vector that is perpendicular to another line, surface, or plane.
- (medicine, countable) A person who is healthy, normal, as opposed to one who is morbid.
- (slang, countable) A person who is normal, who fits into mainstream society, as opposed to those who live alternative lifestyles.
- (countable, uncountable) The usual state.
- something regarded as a normative example
adj
- (algebra, commutative algebra, of a ring element in a ring B relative to a subring A) Being the root of some monic polynomial in A.
- Constituting a whole together with other parts or factors; not omittable or removable.
- (mathematics) Relating to integration (“the process of finding the integral [noun] of a function”).
- (mathematics) Of, pertaining to, or being an integer.
- constituting the undiminished entirety; lacking nothing essential especially not damaged
- of or denoted by an integer
- existing as an essential constituent or characteristic
noun
- (mathematics) One of the two fundamental operations of calculus (the other being differentiation), whereby a function's displacement, area, volume, or other qualities arising from the study of infinitesimal change are quantified, usually defined as a limiting process on a sequence of partial sums. Denoted using a long s: ∫, or a variant thereof.
- (mathematics) A definite integral: the result of the application of such an operation onto a function and a suitable subset of the function's domain: either a number or positive or negative infinity. In the former case, the integral is said to be finite or to converge; in the latter, the integral is said to diverge. In notation, the domain of integration is indicated either below the sign, or, if it is an interval, with its endpoints as sub- and super-scripts, and the function being integrated forming part of the integrand (or, generally, differential form) appearing in front of the integral sign.
- (specifically) Any of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.
- (mathematics) An indefinite integral: the result of the application of such an operation onto a function together with an indefinite domain, yielding a function; a function's antiderivative;
- the result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)
adj
- (algebra, number theory, of a number) Which is a root of some polynomial whose coefficients are rational.
- of or relating to algebra
- Of, or relating to, algebra.
- (algebra, of a field) Whose every element is a root of some polynomial whose coefficients are rational.
- (chess, of notation) Describing squares by file (referred to in intrinsic order rather than by the piece starting on that file) and rank, both with reference to a fixed point rather than a player-dependent perspective.
- (mathematics, of an expression, equation, or function) Containing only numbers, letters, and arithmetic operators.
adj
- (algebra, of an algebraic structure) Having a commutative operation.
- (mathematics, of a binary operation) Such that the order in which the operands are taken does not affect their image under the operation.
- Relating to exchange; interchangeable.
- (mathematics, of a diagram of morphisms) Such that any two sequences of morphisms with the same initial and final positions compose to the same morphism.
- (of a binary operation) independent of order; as in e.g.: ‘a x b’ = ‘b x a’