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Mathematical ordering of a partial order
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial
Linear_extension
Mathematical function, in linear algebra
In mathematics, and more specifically in linear algebra, a linear map (or linear mapping) is a particular kind of function between vector spaces, which
Linear_map
Mathematical method in functional analysis
below. The resulting extension remains linear and bounded, and is thus continuous, which makes it a continuous linear extension. This procedure is known
Continuous_linear_extension
Theorem on extension of bounded linear functionals
the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space
Hahn–Banach_theorem
Order whose elements are all comparable
a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. For delimitation
Total_order
Topics referred to by the same term
Kolmogorov extension theorem, in probability theory Linear extension, in order theory Sheaf extension, in algebraic geometry Tietze extension theorem, in
Extension
Mathematical set with an ordering
case that x ≤ ∗ y . {\displaystyle x\leq ^{*}y.} A linear extension is an extension that is also a linear (that is, total) order. As a classic example, the
Partially_ordered_set
Method of curve fitting
In mathematics, linear interpolation (sometimes lerp) is a method of curve fitting using linear polynomials to construct new data points within the range
Linear_interpolation
Type of algebraic field extension
a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits into linear factors over
Normal_extension
Linear map from a vector space to its field of scalars
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars
Linear_form
Node ordering for directed acyclic graphs
science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge (u,v) from vertex
Topological_sorting
Unsolved problem on partial orders
relationship, a ≤ b, has three linear extensions, a ≤ b ≤ c, a ≤ c ≤ b, and c ≤ a ≤ b. In all three of these extensions, a is earlier than b. However,
1/3–2/3_conjecture
Class of statistical models
generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model
Generalized_linear_model
Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Calculus of stochastic differential equations
{E} \left[\int _{0}^{t}H_{s}^{2}\,ds\right].} By a continuous linear extension, the integral extends uniquely to all predictable integrands satisfying
Itô_calculus
Statistical modeling method
In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory
Linear_regression
Statistical model
statistics, a generalized linear mixed model (GLMM) is an extension to the generalized linear model (GLM) in which the linear predictor contains random
Generalized linear mixed model
Generalized_linear_mixed_model
≤ y in P if and only if x has an earlier position than y in every linear extension of the realizer. Series-parallel partial orders have order dimension
Series-parallel_partial_order
Method to solve optimization problems
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical
Linear_programming
Construction in order theory
with itself. The lexicographic combination of two total orders is a linear extension of their product order, and thus the product order is a subrelation
Product_order
Topics referred to by the same term
up linear in Wiktionary, the free dictionary. Linearity is a property of various things in mathematics, physics, and electronics. Linear, linearly, or
Linear_(disambiguation)
Type of algebraic field extension
if and only if it has distinct roots in any extension of F. That is, if it is the product of distinct linear factors X - a in some algebraic closure of
Separable_extension
Function between topological vector spaces
related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector
Continuous_linear_operator
measure linear position and velocity using a flexible cable and spring-loaded spool. Other common names include string pot, cable-extension transducer
String_potentiometer
Area of mathematics
contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of
Functional_analysis
Representation of mathematical space
{K}}} onto a simplex in L {\displaystyle {\mathcal {L}}} . By affine-linear extension on the simplices, f {\displaystyle f} induces a map between the geometric
Triangulation_(topology)
Algebra associated to any vector space
Calculus of Extension, Internet Archive Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre – Ein neuer Zweig der Mathematik (in German) (The Linear Extension
Exterior_algebra
Australian artist (1925–1987)
1961–1964 – Graham completed a new series of paintings referred to as Linear Extensions. 1964–1973 – Graham conducted experimental studies based on a new
Peter_Benjamin_Graham
Use of high-frequency, alternating polarity, electrical current in medical operations
process of vaporization can be used to ablate tissue targets, or, by linear extension, used to transect or cut tissue. While the processes of vaporization/
Electrosurgery
Algebraic structure in linear algebra
In mathematics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled")
Vector_space
Mathematical formula for the number of Young tableaux
tableaux are their linear extensions. Robert Proctor has given a generalization of the hook length formula to count linear extensions of a larger class
Hook_length_formula
Construction of a larger algebraic field by "adding elements" to a smaller field
completely factors into linear factors over L. Every algebraic extension F/K admits a normal closure L, which is an extension field of F such that L /
Field_extension
Construction in group theory
the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector
Projective_linear_group
integers satisfying the following Riemann bilinear relations: the real linear extension αR:Cg × Cg→R of α satisfies αR(iv, iw)=αR(v, w) for all (v, w) in Cg
Riemann_form
Mathematical measure for partial orders
in all of the linear extensions, if any such t {\displaystyle t} exists. In other words, that the intersection of those linear extensions equals P {\displaystyle
Order_dimension
System of resource-aware logic
linear logic (that is linear logic with weakening, an extension rather than a fragment) was shown to be decidable, in 1995. Many variations of linear
Linear_logic
Induced map between the dual spaces of the two vector spaces
In linear algebra and functional analysis, the transpose or algebraic adjoint of a linear map between two vector spaces, defined over the same field, is
Transpose_of_a_linear_map
Directed graph with no directed cycles
family of topological orderings of a DAG is the same as the family of linear extensions of the reachability relation for the DAG, so any two graphs representing
Directed_acyclic_graph
Actuator that creates motion in a straight line
A linear actuator is an actuator that creates linear motion (i.e., in a straight line), in contrast to the circular motion of a conventional electric motor
Linear_actuator
Study of differential field extensions induced by linear differential equations
differential field extension generated by the solutions of a linear differential equation, using the differential Galois group of the field extension. A major goal
Picard–Vessiot_theory
Numerical ordering with a margin of error
comparable pairs, the partial orders that have the largest number of linear extensions are semiorders. Semiorders are known to obey the 1/3–2/3 conjecture:
Semiorder
Order-preserving mathematical function
analysis on a topological vector space X {\displaystyle X} , a (possibly non-linear) operator T : X → X ∗ {\displaystyle T:X\rightarrow X^{*}} is said to be
Monotonic_function
Generalised alphabetical order
well. The lexicographical order of two totally ordered sets is thus a linear extension of their product order. One can define similarly the lexicographic
Lexicographic_order
Filename extension for a native executable program
In Windows, OS/2, and DOS, .exe is the filename extension for a file that is runnable as a native executable computer program. Such a file is sometimes
.exe
Function with a smaller domain
}_{\operatorname {domain} f}=f.} A linear extension (respectively, continuous extension, etc.) of a function f {\displaystyle f} is an extension of f {\displaystyle
Restriction_(mathematics)
Boundary condition for generalized functions
{\textstyle C^{1}} -domain, the trace operator can be defined as continuous linear extension of the operator T : C ∞ ( Ω ¯ ) → L p ( ∂ Ω ) {\displaystyle T:C^{\infty
Trace_operator
Group of matrices with determinant 1
In mathematics, the special linear group SL ( n , R ) {\displaystyle \operatorname {SL} (n,R)} of degree n {\displaystyle n} over a commutative ring
Special_linear_group
Archaeological horizon of Neolithic Europe
from the German Linearbandkeramik, it is also known as the Linear Band Ware, Linear Ware, Linear Ceramics or Incised Ware culture, falling within the Danubian I
Linear_Pottery_culture
Mathematical function
finite field extension L/K, which is a K-linear map from L onto K. Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can
Field_trace
Extension of linear logic
Noncommutative logic is an extension of linear logic that combines the commutative connectives of linear logic with the noncommutative multiplicative
Noncommutative_logic
Claim preclusion in law
judicata does not restrict the appeals process, which is considered a linear extension of the same lawsuit as the suit travels up (and back down) the appellate
Res_judicata
Group of 𝑛 × 𝑛 invertible matrices
In mathematics, the general linear group of degree n {\displaystyle n} is the set of n × n {\displaystyle n\times n} invertible matrices, together with
General_linear_group
Visual depiction of a partially ordered set
found in linear time, if such a diagram exists. In particular, if the input poset is a graded poset, it is possible to determine in linear time whether
Hasse_diagram
Glossary of terms used in branch of mathematics
has length 1, etc. Linear. See total order. Linear extension. A linear extension of a partial order is an extension that is a linear order, or total order
Glossary_of_order_theory
Partially ordered set with alternatingly-related elements
fences. A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been
Fence_(mathematics)
Conjugate transpose of an operator in infinite dimensions
In mathematics, specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint)
Hermitian_adjoint
Special subset of a partially ordered set
A linear (ultra)filter is an (ultra)filter on the lattice of vector subspaces of a given vector space, ordered by inclusion. Explicitly, a linear filter
Filter_(mathematics)
Type of ordering of a set
precisely if its covering relation is empty. The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic
Dense_order
Inequality for the number of extensions of partial orders to linear orders
the Fishburn–Shepp inequality, is an inequality for the number of linear extensions of finite partial orders. The inequality was conjectured by Ivan Rival
XYZ_inequality
Set of vectors used to define coordinates
can be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components
Basis_(linear_algebra)
Mathematical proposition equivalent to the axiom of choice
of all linearly independent subsets of V. Since V is not the zero vector space, there exists a nonzero element v of V, so P contains the linearly independent
Zorn's_lemma
Type of function in linear algebra
theorem – Theorem on extension of bounded linear functionalsPages displaying short descriptions of redirect targets Linear functional – Linear map from a vector
Sublinear_function
Logical problem studied in computer science
non-linear arithmetic over the real numbers involving transcendental functions. This fact motivates an extension of the SMT problem to non-linear theories
Satisfiability modulo theories
Satisfiability_modulo_theories
TVS X then Y has the extension property from M to X if every continuous linear map f : M → Y has a continuous linear extension to all of X. If X and
Vector-valued Hahn–Banach theorems
Vector-valued_Hahn–Banach_theorems
Dimension of the extension field viewed as a vector space over the base field
by observing that the elements 1, T, T2, etc., are linearly independent over C. The field extension C(T2) also has infinite degree over C. However, if
Degree_of_a_field_extension
Linear operator defined on a dense linear subspace
an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at Discontinuous linear map § General existence
Unbounded_operator
Least squares approximation of linear functions to data
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems
Linear_least_squares
optimization problem has low extension complexity, this could potentially be used to devise efficient algorithms for the problem, using linear programming on its
Extension_complexity
In mathematics, vector subspace
specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is
Linear_subspace
Force needed to pull a spring grows linearly with distance
as a linear spring. The rod has length L and cross-sectional area A. Its tensile stress σ is linearly proportional to its fractional extension or strain
Hooke's_law
Notation for quantum states
Bra–ket notation or Dirac notation is a mathematical notation for linear algebra and linear operators on complex vector spaces together with their dual spaces
Bra–ket_notation
Video board software standard
(VESA 1.x Non-Linear) VESA BIOS Extension Serial Control Interface Standard at the Wayback Machine (archived 2005-12-21) VESA BIOS Extension/Accelerator
VESA_BIOS_Extensions
{\displaystyle \Omega ^{\circ }(n)={\binom {n}{p}}.} There is only one linear extension (the identity mapping), and both polynomials have leading term 1 p
Order_polynomial
Polymer
Linear low-density polyethylene (LLDPE) is one of several varieties of polyethylene (PE). PE's, the dominant synthetic polymer, are produced in many forms
Linear low-density polyethylene
Linear_low-density_polyethylene
Type of derivative in differential geometry
pullback of tensor fields. For covariant tensors, this is just the multi-linear extension of the pullback map ( Φ X t ) p ∗ : T Φ X t ( p ) ∗ M → T p ∗ M , (
Lie_derivative
Concept of extending human lifespan
lifespan. Moreover, the very notion of a "life-extension factor" that could apply across taxa presumes a linear response rarely seen in biology." There are
Life_extension
Algorithm for generating pseudo-randomized numbers
A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear
Linear_congruential_generator
Circuit simulation software
software, produced by semiconductor manufacturer Analog Devices (originally by Linear Technology). It is the most widely distributed and used SPICE software in
LTspice
Under-construction maglev high-speed train line in Japan
Yamanashi test track (see below). The trainsets are popularly known in Japan as linear motor car (リニアモーターカー, rinia mōtā kā), though there are many technical variations
Chūō_Shinkansen
In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named
Friedrichs_extension
Magnetic tape data storage technology
Linear Tape-Open (LTO), also known as the LTO Ultrium format, is a magnetic tape data storage technology used for backup, data archiving, and data transfer
Linear_Tape-Open
x\in F\cap K.} A linear functional ψ : E → R {\displaystyle \psi :E\to \mathbb {R} } is called a K {\displaystyle K} -positive extension of ϕ {\displaystyle
M._Riesz_extension_theorem
Statistical model for a binary dependent variable
Stationery Office. OCLC 808240121. Theil, Henri (1969). "A Multinomial Extension of the Linear Logit Model". International Economic Review. 10 (3): 251–59. doi:10
Logistic_regression
and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3? The Dittert conjecture concerning the maximum
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
by the number of linear extensions of the corresponding partially ordered set. This connection between the number of linear extensions and volume can be
Order_polytope
American mathematician (1914–2005)
1963, Dantzig's Linear Programming and Extensions was published by Princeton University Press. It quickly became a standard text in linear programming. Dantzig
George_Dantzig
District of Callao, Peru
The area in territorial conflict with the district of Lima has a linear extension of 0.35 kilometers and is mainly located in a section of Moreau street
Callao_District
forking extension of a type is an extension of that type that is not free[clarification needed] whereas a non-forking extension is an extension that is
Forking_extension
All bases of a vector space have equally many elements
bases have the same cardinality. As a basis is a generating set that is linearly independent, the dimension theorem is a consequence of the following theorem
Dimension theorem for vector spaces
Dimension_theorem_for_vector_spaces
Projection of data onto lower-dimensional manifolds
potentially existing across non-linear manifolds (non-affine subspaces) which cannot be adequately captured by linear decomposition methods, onto lower-dimensional
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Measure of non-compactness Banach–Mazur theorem Bounded linear operator Continuous linear extension Compact operator Approximation property Invariant subspace
List of functional analysis topics
List_of_functional_analysis_topics
Partial converse of Taylor's theorem
Theorem on extension of bounded linear functionals Whitney covering lemma, on covering open sets by cubes, used by Whitney to prove the extension theorem
Whitney_extension_theorem
Locally compact topological group with an invariant averaging operation
well-defined norm-1 linear form on the subspace Ru + Y taking tu + y to t. By the Hahn–Banach theorem, this form admits a norm-1 linear extension to all of ℓ∞(Z)
Amenable_group
Subset whose closure is the whole space
range is contained within Y . {\displaystyle Y.} See also Continuous linear extension. A topological space X {\displaystyle X} is hyperconnected if and only
Dense_set
Dynamic data structure
dynamic hashing such as Larson's Linear Hashing with Partial Extensions, Linear Hashing with Priority Splitting, Linear Hashing with Partial Expansions
Linear_hashing
Arithmetic operation
second through fourth conditions trivially imply that f is a linear function on [−1, 0]. The linear approximation to natural tetration function x e {\displaystyle
Tetration
Web browser developed by Google
as the Google Chrome Extensions Gallery. Some extensions focus on providing accessibility features. Google Tone is an extension developed by Google that
Google_Chrome
Complexity class
using k colors are there for a particular graph G? How many different linear extensions are there for a given partially ordered set, or, equivalently, how
♯P-complete
Property of elements related by inequalities
partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order
Comparability
{\displaystyle f} be a linear form on M . {\displaystyle M.} Then f {\displaystyle f} has an extension to a continuous positive linear form on X {\displaystyle
Positive_linear_functional
LINEAR EXTENSION
LINEAR EXTENSION
Surname or Lastname
Swedish
Swedish : ornamental name from lind ‘lime tree’ + either the German suffix -er denoting an inhabitant, or the surname suffix -ér, derived from the Latin adjectival ending -er(i)us.English (mainly southeastern) : variant of Lind 2.German : habitational name from any of numerous places called Linden or Lindern, named with German Linden ‘lime trees’.
Surname or Lastname
English
English : variant of Lanier 1.Dutch : variant of Leonard.Jewish (western Ashkenazic) : name taken by someone who was good at chanting the Pentateuch at public worship in the synagogue or who regularly did so, from West Yiddish layner ‘reader’ (a derivative of West Yiddish laynen ‘to read’, which comes ultimately from Latin legere ‘to read’).Jewish (Ashkenazic) : occupational name for a flax grower or merchant, from German Lein ‘flax’ + agent suffix -er.
Male
Scandinavian
Scandinavian form of Old Norse Einarr, EINAR means "lone warrior."
Male
English
Irish Anglicized form of Gaelic Fionnbarr, FINBAR means "fair-headed."
Boy/Male
Sikh
Love unending
Female
English
English name probably derived from Germanic lindi, LINDA means "serpent."Â In some cases, it may have been derived from the Spanish word for "pretty."
Male
Greek
(ΑἰνÎας) Variant spelling of Greek AineÃas, AINEAS means "praiseworthy."
Surname or Lastname
English
English : habitational name from Lingart, Lancashire, or Lingards Wood in Marsden, West Yorkshire, both named from Old English līn ‘flax’ + garðr ‘enclosure’.
Female
Scottish
Variant spelling of Scottish Lilias, LILEAS means "lily."
Surname or Lastname
English
English : variant of Lingard.French : occupational name for a maker of or dealer in linen goods, from Old French linge ‘linen (goods)’ (see Linge 1).
Female
English
Variant spelling of English Linsey, LINSAY means "Lincoln's wetlands."
Boy/Male
Hindu
The Sun
Boy/Male
Hindu
Lingam
Male
Yiddish
 Variant spelling of Yiddish Lieber, LIBER means "beloved." Compare with another form of Liber.
Surname or Lastname
English
English : metronymic from Line.
Boy/Male
Irish
Meaning “â€fair-haired,â€â€ the name has been popular since the sixth century when St. Finbar came to an area of Cork that was being tormented by a serpent. The people begged him to do something to help them. One night he went to where the serpent was sleeping and sprinkled it with holy water. The angry serpent tore and devoured the land until she slithered into the sea at Cork Harbor. The track she left behind filled with water and became the River Lee and that’s why St. Finbar is the patron saint of Cork. It is said that the sun didn’t set for two weeks after Finbar’s death.
Surname or Lastname
English (Cornish)
English (Cornish) : habitational name from a place named with Cornish lan ‘church’. In England this surname is now found chiefly in the southern counties of Wiltshire and Hampshire, and Berkshire; it has no doubt moved there from Cornwall.
Surname or Lastname
English (Devon; of Cornish origin)
English (Devon; of Cornish origin) : topographic name for someone who lived by a menhir, i.e. a tall standing stone erected in prehistoric times (Cornish men ‘stone’ + hir ‘long’).
Girl/Female
Irish
Eimear possessed the “Six Gifts of Womanhood†– “beauty, a gentle voice, sweet words, wisdom, needlework and chastity!†She was bethrothed to the warrior Cuchulainn (read the legend) when they were children and they loved each other very deeply. But Cuchulainn had “a wandering eye†and Eimear endured this, realizing “everything new is fair,†but when he made love to Fand, wife of the sea god Manannan, Eimear confronted the lovers. After seeing the strength of Fand’s love she offered to withdraw. Touched by this display of unselfishness, Fand left Cuchulainn and returned to the sea. When Cuchulainn died Eimear spoke movingly and lovingly at his graveside.
Surname or Lastname
English
English : occupational name for a whitewasher, Middle English limer, lymer, an agent derivative of Old English līm ‘lime’.
LINEAR EXTENSION
LINEAR EXTENSION
Surname or Lastname
English
English : topographic name from Middle English stille ‘calm’, ‘quiet’, + welle ‘spring’, ‘stream’, or possibly a habitational name from a minor place, now lost, of which the first element may have been Old English stigel, stigol ‘stile’, ‘steep place’.
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Tamil, Telugu
Spectator; Wind Blowing Fast
Girl/Female
Tamil
God is gracious
Boy/Male
Tamil
Lord Krishna
Boy/Male
Hindu, Indian, Malayalam, Marathi
Soft; Delicate
Girl/Female
American, Arabic, Chinese, Christian, Jamaican, Muslim
Heaven; Peace; Angels
Boy/Male
Armenian, Australian
Fighter
Male
English
(עֵדֶן) Hebrew unisex name EDEN means "delight" or "place of pleasure." In the bible, this is the name of the garden in which Adam and Eve lived.
Boy/Male
Tamil
Sarvayagyodhipa | ஸரà¯à®µà®¯à®œà¯à®žà¯‹à®¤à¯€à®ªà®¾
Lord of all sacrificial offerings
Girl/Female
Australian, Danish, Finnish, German, Swedish
Grace; Graceful Meadow; Favor
LINEAR EXTENSION
LINEAR EXTENSION
LINEAR EXTENSION
LINEAR EXTENSION
LINEAR EXTENSION
n.
Made of linen; as, linen cloth; a linen stocking.
a.
Like a line; narrow; of the same breadth throughout, except at the extremities; as, a linear leaf.
prep. & adv.
Near.
a.
In the direction of a line; of or pertaining to a line; measured on, or ascertained by, a line; linear; as, lineal magnitude.
a.
Of or pertaining to a line; consisting of lines; in a straight direction; lineal.
a.
Composed of lines; delineated; as, lineal designs.
a.
Descending in a direct line from an ancestor; hereditary; derived from ancestors; -- opposed to collateral; as, a lineal descent or a lineal descendant.
adv.
In a linear manner; with lines.
n.
Alt. of Lingam
n.
A lunar distance.
a.
Of a linear shape.
n.
One who lines, as, a liner of shoes.
n.
A dealer in linen; a linen draper.
n.
One who adjusts things to a line or lines or brings them into line.
n.
A vessel belonging to a regular line of packets; also, a line-of-battle ship; a ship of the line.
v. t.
To mark with a line or lines; to cover with lines; as, to line a copy book.
a.
Linear.
v. t.
To convert into vinegar; to make like vinegar; to render sour or sharp.
a.
Of, pertaining to, or included by, two lines; as, bilinear coordinates.
a.
Formed by right lines; rectilineal; as, a right-lined angle.