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Binary relation over a set and itself
In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian
Homogeneous_relation
Relationship between elements of two sets
A binary relation is called a homogeneous relation when X = Y {\displaystyle X=Y} . A binary relation is also called a heterogeneous relation when it is
Binary_relation
Type of binary relation
a < c; and if x = y and y = z then x = z. A homogeneous relation R on the set X is a transitive relation if, for all a, b, c ∈ X, if a R b and b R c,
Transitive_relation
Type of binary relation
A symmetric relation is a type of binary relation. A homogeneous relation R {\displaystyle R} on a set X {\displaystyle X} is symmetric if: for all a
Symmetric_relation
Concept of uniform or non-uniform in an object's composition or attributes
homogeneous polynomials have the same number of factors of a given kind.[citation needed] In the study of binary relations, a homogeneous relation R
Homogeneity_and_heterogeneity
Mathematical concept for comparing objects
mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in
Equivalence_relation
Property of a relation on a set
strongly connected as defined above. Let R {\displaystyle R} be a homogeneous relation. The following are equivalent: R {\displaystyle R} is strongly connected;
Connected_relation
Mathematical set with an ordering
which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially
Partially_ordered_set
Binary relation that relates every element to itself
reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to
Reflexive_relation
Type of logical relation
true if Y ≠ ∅ . {\displaystyle Y\neq \emptyset .} Serial relation — a total homogeneous relation If Y = ∅ ≠ X , {\displaystyle Y=\emptyset \neq X,} then
Total_relation
Relationship between two sets, defined by a set of ordered pairs
relation concept described above is obtained; it is often called homogeneous relation (or endorelation) to distinguish it from its generalization. The
Relation_(mathematics)
Topics referred to by the same term
Binary relation (or diadic relation – a more in-depth treatment of binary relations) Equivalence relation Homogeneous relation Reflexive relation Serial
Relation
Type of binary relation
In mathematics, a binary relation R {\displaystyle R} on a set X {\displaystyle X} is antisymmetric if there is no pair of distinct elements of X {\displaystyle
Antisymmetric_relation
Mathematical ranking of a set
property this "incomparability relation" needs in order to be an equivalence relation. Define also an induced homogeneous relation ≲ {\displaystyle \,\lesssim
Weak_ordering
Pattern defining an infinite sequence of numbers
by the Fibonacci numbers is the canonical example of a homogeneous linear recurrence relation with constant coefficients (see below). The Fibonacci sequence
Recurrence_relation
Relation that relates every element to some element
In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor
Serial_relation
Glossary of terms used in branch of mathematics
relation of an antichain is just the identity relation. Approximates relation. See way-below relation. Antisymmetric relation. A homogeneous relation
Glossary_of_order_theory
Binary relation which never occurs in both directions
In mathematics, an asymmetric relation is a binary relation R {\displaystyle R} on a set X {\displaystyle X} where for all a , b ∈ X , {\displaystyle
Asymmetric_relation
Property that assigns truth values to k-tuples of individuals
refer to R as an n-ary relation over X, called a homogeneous relation. Without this restriction, R is called a heterogeneous relation. When any of Xi is empty
Finitary_relation
Smallest transitive relation containing a given binary relation
mathematics, the transitive closure R+ of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive. For
Transitive_closure
Type of ordinary differential equation
differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written f (
Homogeneous differential equation
Homogeneous_differential_equation
Vertices connected in pairs by edges
simple graph permitting loops G is a homogeneous relation ~ on the vertices of G that is called the adjacency relation of G. Specifically, for each edge
Graph_(discrete_mathematics)
Type of binary relation
In mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty subset
Well-founded_relation
Order whose elements are all comparable
which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies
Total_order
Operation on binary relations
represents a homogeneous relation on A . {\displaystyle A.} Correspondingly, R T ; R {\displaystyle R^{\textsf {T}}\,;R} is the universal relation on B , {\displaystyle
Composition_of_relations
Coordinate system used in projective geometry
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are
Homogeneous_coordinates
Topological space in group theory
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere as one moves through it, with movement given by the action
Homogeneous_space
Differential equation that is linear with respect to the unknown function
differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which
Linear_differential_equation
Mathematical concept for comparing objects
equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) is a homogeneous binary relation that is
Partial_equivalence_relation
Reversal of the order of elements of a binary relation
is both right-invertible and left-invertible. For an invertible homogeneous relation R , {\displaystyle R,} all right and left inverses coincide; this
Converse_relation
Property of segments that have the same length and the same direction
In Euclidean geometry, equipollence is a homogeneous relation between directed line segments. Two segments are said to be equipollent when they have the
Equipollence_(geometry)
Expression in commutative algebra
specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every
Complete homogeneous symmetric polynomial
Complete_homogeneous_symmetric_polynomial
Several equations of degree 1 to be solved simultaneously
to a homogeneous system, then the vector sum u + v is also a solution to the system. If u is a vector representing a solution to a homogeneous system
System_of_linear_equations
Type of mathematical distribution
In mathematics, a homogeneous distribution is a distribution S on Euclidean space Rn or Rn \ {0} that is homogeneous in the sense that, roughly speaking
Homogeneous_distribution
Reflexive and transitive binary relation
mathematics, in particular in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant to suggest
Preorder
Class of mathematical orderings
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset
Well-order
Polynomial whose nonzero terms all have the same degree
above relation is true for infinitely many λ {\displaystyle \lambda } then the polynomial is homogeneous of degree d. In particular, if P is homogeneous then
Homogeneous_polynomial
Dimension of the column space of a matrix
.., Axr are linearly independent. To see why, consider a linear homogeneous relation involving these vectors with scalar coefficients c1, c2, ..., cr:
Rank_(linear_algebra)
Set whose pairs have minima and maxima
b=a\vee b} and dually for the other direction. One can now check that the relation ≤ {\displaystyle \leq } introduced in this way defines a partial ordering
Lattice_(order)
Something that has mass and volume
the mediators of the electric force (photons) possess energy (see Planck relation) and the mediators of the weak force (W and Z bosons) have mass, but neither
Matter
Partial order with joins
corresponding absorption laws. A set S partially ordered by the binary relation ≤ is a meet-semilattice if For all elements x and y of S, the greatest
Semilattice
Topics referred to by the same term
Homogeneous linear transformation Homogeneous model in model theory Homogeneous polynomial Homogeneous relation: binary relation on a set Homogeneous
Homogeneity_(disambiguation)
Type of functional equation (mathematics)
whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other
Differential_equation
Concept in order theory
\wedge )} is then a meet-semilattice. Moreover, we then may define a binary relation ≤ {\displaystyle \,\leq \,} on A, by stating that x ≤ y {\displaystyle
Join_and_meet
Set on which a group acts freely and transitively
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point
Principal_homogeneous_space
Type of random mathematical object
located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second case, the point process
Poisson_point_process
Reasoning about equations with free variables
(Czelakowski 2003). A homogeneous binary relation is found in the power set of X × X for some set X, while a heterogeneous relation is found in the power
Algebraic_logic
Type of differential equation
mechanics. For example, the equilibrium temperature distribution of a homogeneous solid is a harmonic function. It is usually a matter of straightforward
Partial_differential_equation
Mathematical concept for comparing objects
i<j.} Well-founded induction can be used on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded. (Here, by
Well-quasi-ordering
Weak form of the axiom of choice
needed to develop analysis. A homogeneous relation R {\displaystyle R} on X {\displaystyle X} is called a total relation if for every a ∈ X , {\displaystyle
Axiom_of_dependent_choice
Equation for a material's dielectric constant given its atomic polarizability
polarizability α of the material's constituent atoms and/or molecules, or a homogeneous mixture thereof. It is equivalent to the Lorentz–Lorenz equation, which
Clausius–Mossotti_relation
Overview of and topical guide to logic
Dependency relation Directed set Equivalence relation Euclidean relation Homogeneous relation Idempotence Intransitivity Involutive relation Partial equivalence
Outline_of_logic
In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point lies on a line" or
Incidence_(geometry)
Thermodynamic relation about molar heat capacity
a relation between the molar heat capacity at constant pressure and the molar heat capacity at constant volume for an ideal gas. Mayer's relation states
Mayer's_relation
Mathematical relation defining a sequence
recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear
Linear recurrence with constant coefficients
Linear_recurrence_with_constant_coefficients
Identity relating to differential equations
of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation. The relation can be
Abel's_identity
Transformation of a body from a reference configuration to a current configuration
compression) and a rigid body translation. Affine deformations are also called homogeneous deformations. Therefore, an affine deformation has the form x ( X , t
Deformation_(physics)
homogeneous linear three-term recurrence relation (TTRR, the qualifiers "homogeneous linear" are usually taken for granted) is a recurrence relation of
Three-term recurrence relation
Three-term_recurrence_relation
Compact non-orientable two-dimensional manifold
ax + by + cz = 0 in R3 has the homogeneous coordinates (a : b : c). Thus, these coordinates have the equivalence relation (a : b : c) = (da : db : dc) for
Real_projective_plane
Initial estimate or framework to the solution of a mathematical problem
thermodynamics. Another example of an ansatz is to suppose the solution of a homogeneous linear differential equation to take an exponential form, or a power
Ansatz
Embedding of a Grassmannian into projective space
Grassmann generalized Plücker's embedding to arbitrary k and n. The homogeneous coordinates of the image of the Grassmannian G r ( k , V ) {\displaystyle
Plücker_embedding
Random process independent of past history
chain can be proved to be time-homogeneous by Bayes' rule. A necessary and sufficient condition for a time-homogeneous Markov chain to be stationary is
Markov_chain
Existence and uniqueness of solutions to initial value problems
differential equations will possess a single stationary point y = 0. First, the homogeneous linear equation dy/dt = ay ( a < 0 {\displaystyle a<0} ), a stationary
Picard–Lindelöf_theorem
Mathematical concept
topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories. An equivalence relation on a set X {\displaystyle
Equivalence_class
Set theory concept
relation. A prewellordering on a set X {\displaystyle X} is a homogeneous binary relation ≤ {\displaystyle \,\leq \,} on X {\displaystyle X} that satisfies
Prewellordering
Extension of ideas in combinatorics to infinite sets
into m {\displaystyle m} pieces has a homogeneous set of order type λ {\displaystyle \lambda } . A homogeneous set is in this case a subset of κ {\displaystyle
Infinitary_combinatorics
Mathematical relation inside orderings
mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are
Covering_relation
Lowest energy state in quantum chromodynamics
contains some non-zero but homogeneous field which gives rise to these condensates. However, Stanley Mandelstam showed that a homogeneous vacuum field is also
QCD_vacuum
Operation on the subsets of a set
closure, i.e. "the transitive closure R+ of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive." Mathematics
Closure_(mathematics)
Type of geometry
included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers. Several major types of more abstract
Projective_geometry
Relation of degree three
relation between contexts, terms and types. Given homogeneous relations A, B, and C on a set, a ternary relation (A, B, C) can be defined using composition of
Ternary_relation
Type of ordering of a set
comparable. Equivalently, a partial order is dense precisely if its covering relation is empty. The rational numbers as a linearly ordered set are a densely
Dense_order
Branch of mathematics
relations. Suppose that P is a set and that ≤ is a relation on P ('relation on a set' is taken to mean 'relation amongst its inhabitants', i.e. ≤ is a subset
Order_theory
Used to define marginal product and to distinguish allocative efficiency
In economics, a production function gives the technological relation between quantities of physical inputs and quantities of output of goods. The production
Production_function
Direct conversion of temperature differences to electric voltage and vice versa
essentially unobservable for a localized hot or cold spot in a single homogeneous conducting material, since the overall EMFs from the increasing and decreasing
Thermoelectric_effect
Class of numerical techniques
equation. Consider the normalized heat equation in one dimension, with homogeneous Dirichlet boundary conditions { U t = U x x U ( 0 , t ) = U ( 1 , t )
Finite_difference_method
Generalized function whose value is zero everywhere except at zero
δ ( − x ) = δ ( x ) {\displaystyle \delta (-x)=\delta (x)} which is homogeneous of degree −1. The distributional product of δ with x is equal to zero:
Dirac_delta_function
Model of 3D points projected onto planar image via a lens-less aperture
also be represented in homogeneous coordinates. Let x {\displaystyle \mathbf {x} } be a representation of a 3D point in homogeneous coordinates (a 4-dimensional
Pinhole_camera_model
Parameter in differential equations and dynamical systems
an initial value problem. A linear matrix difference equation of the homogeneous (having no constant term) form X t + 1 = A X t {\displaystyle X_{t+1}=AX_{t}}
Initial_condition
Procedure for solving differential equations
space of solutions of the corresponding homogeneous equation Then a particular solution to the non-homogeneous equation is given by where the c i ( x )
Variation_of_parameters
Type of ordinary differential equation
variables Autonomous Coupled / Decoupled Exact Homogeneous / Nonhomogeneous Features Order Operator Notation Relation to processes Difference (discrete analogue)
Bernoulli differential equation
Bernoulli_differential_equation
Alternative mathematical ordering
binary relation, such as "a < b". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation [a,
Cyclic_order
Equation in thermodynamics
first-order homogenous function. Applying Euler's homogeneous function theorem, one finds the following relation: U = T S − p V + ∑ i = 1 I μ i N i {\displaystyle
Gibbs–Duhem_equation
Differential equations involving stochastic processes
variables Autonomous Coupled / Decoupled Exact Homogeneous / Nonhomogeneous Features Order Operator Notation Relation to processes Difference (discrete analogue)
Stochastic differential equation
Stochastic_differential_equation
Visual depiction of a partially ordered set
different meaning: the directed acyclic graph obtained from the covering relation of a partially ordered set, independently of any drawing of that graph
Hasse_diagram
Algebraic variety in a projective space
zero-locus in P n {\displaystyle \mathbb {P} ^{n}} of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety
Projective_variety
Numbers obtained by adding the two previous ones
Fibonacci sequence may also be derived from the recurrence relation, giving a homogeneous linear differential equation: ∑ k = 0 ∞ F k + 2 x k k ! = ∑
Fibonacci_sequence
Mathematical space
vectors ( W 1 , … , W k ) {\displaystyle (W_{1},\dots ,W_{k})} . The homogeneous coordinates of the element w ∈ G r k ( V ) {\displaystyle w\in \mathbf
Grassmannian
Polynomial invariant under variable permutations
of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power
Symmetric_polynomial
Quantum consistency equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics
Yang–Baxter_equation
complex cube root of 1. Euler–Gompertz constant Euler's homogeneous function theorem – A homogeneous function is a linear combination of its partial derivatives
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Generalized alphabetical order
useful property of the degree reverse lexicographical order is that a homogeneous polynomial is a multiple of the least indeterminate if and only if its
Lexicographic_order
Change in a property of a mixture component with respect to amount
i}}.} By Euler's second theorem for homogeneous functions, Z i ¯ {\displaystyle {\bar {Z_{i}}}} is a homogeneous function of degree 0 (i.e., Z i ¯ {\displaystyle
Partial_molar_property
Type of constraint on solutions to differential equations
variables Autonomous Coupled / Decoupled Exact Homogeneous / Nonhomogeneous Features Order Operator Notation Relation to processes Difference (discrete analogue)
Dirichlet_boundary_condition
Mathematical operation in quantum optics, general relativity and other areas of physics
of BCS theory in a homogeneous system. The Bogoliubov transformation is an isomorphism of either the canonical commutation relation algebra or canonical
Bogoliubov_transformation
Heat required to raise the temperature of a given unit of mass of a substance
c_{m}={\frac {C}{m}}={\frac {c_{\text{volumetric}}}{\rho }}.} For pure homogeneous chemical compounds with established molecular or molar mass, or a molar
Specific_heat_capacity
Theory of nucleation
nucleates in contact with a surface. Homogeneous nucleation is much rarer than heterogeneous nucleation. However, homogeneous nucleation is simpler and easier
Classical_nucleation_theory
Theory of relational databases
second relation matching certain conditions, and so forth. A relation of arity n is a set of n‑tuples. Relational algebra operates on homogeneous sets of
Relational_algebra
Manifold with inversion symmetry
the dual space, a homogeneous space for SU(2) and SL(2,C). Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple
Hermitian_symmetric_space
Branch of algebraic geometry
aspects mainly arise in relation to computing intersections of Schubert cycles. Lifted from the Grassmannian, which is a homogeneous space, to the general
Schubert_calculus
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HOMOGENEOUS RELATION
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HOMOGENEOUS RELATION
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