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warehouse, a degenerate dimension is a dimension key (primary key for a dimension table) in the fact table that does not have its own dimension table, because
Degenerate_dimension
Structure that categorizes facts and measures in a data warehouse
transaction item or line item because the degenerate dimension represents the unique identifier of the parent. Degenerate dimensions often play an integral role
Dimension_(data_warehouse)
Topics referred to by the same term
degeneracy, degenerate, degeneration, or degenerative in Wiktionary, the free dictionary. Degeneracy, degenerate, or degeneration may refer to: Degenerate (album)
Degeneracy
Limiting case which is different from the rest of the class
example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar to the case
Degeneracy_(mathematics)
Energy level of a quantum system
{H}}} has a degenerate eigenvalue E n {\displaystyle E_{n}} of degree gn, the eigenstates associated with it form a vector subspace of dimension gn. In such
Degenerate_energy_levels
Geometric space with four dimensions
Four-dimensional (4D) space is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible
Four-dimensional_space
Probability distribution
the n-dimensional space ℝn endowed with the Lebesgue measure, any distribution concentrated on a d-dimensional subspace with d < n is a degenerate distribution
Degenerate_distribution
Concept in linear algebra
finite-dimensional is that the previous map is not an isomorphism. A nondegenerate or nonsingular form is a bilinear form that is not degenerate, meaning
Degenerate_bilinear_form
Geometric model of the physical space
three-dimensional Euclidean space, which is the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces
Three-dimensional_space
Aspect of physical cosmology
during which stars form from collapsing clouds of gas. In the subsequent Degenerate Era, the stars will have burnt out, leaving all stellar-mass objects as
Future of an expanding universe
Future_of_an_expanding_universe
2nd-degree plane curve which is reducible
the conic as the intersection in three-dimensional space of a plane and a double cone, a conic is degenerate if the plane goes through the vertex of
Degenerate_conic
Fundamental space of geometry
the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are
Euclidean_space
Type of group in mathematics
general neither. In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles
Orthogonal_group
Locus of the zeros of a polynomial of degree two
coefficients). For two-dimensional surfaces (dimension D = 2) in three-dimensional space, there are exactly three non-degenerate cases: P ( X ) = { X 0
Quadric
Curve from a cone intersecting a plane
define a conic as a two-dimensional nondegenerate quadric. With this terminology there are no degenerate conics (only degenerate quadrics), but we shall
Conic_section
Infinitely detailed mathematical structure
arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales
Fractal
Square matrix without an inverse
they delineate the boundary between invertible (well-behaved) cases and degenerate (ill-posed) cases. In abstract terms, singular matrices correspond to
Singular_matrix
All numbers between two given numbers
less than the radius. In the 2-dimensional case, a ball is called a disk. If a half-space is taken as a kind of degenerate ball (without a well-defined
Interval_(mathematics)
global dimension 3 in the 1980s. Sklyanin algebras can be grouped into two different types, the non-degenerate Sklyanin algebras and the degenerate Sklyanin
Sklyanin_algebra
Interpretations of extra dimensions
The idea of a fourth dimension has been a factor in the evolution of modern art, but use of concepts relating to higher dimensions has been little discussed
Fourth dimension in literature
Fourth_dimension_in_literature
Isogonal polyhedron with regular faces
in their terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization. Here an abstract polytope is
Uniform_polyhedron
Algebraic structure designed for geometry
majority of engineering applications of geometry). In this model, a degenerate dimension is added to the three Euclidean dimensions to form the algebra
Geometric_algebra
Surface in algebraic geometry
non-degenerate irreducible surface of degree m – 1 in Pm is either a rational normal scroll or the Veronese surface. In projective space of dimension m + n + 1
Rational_normal_scroll
Overview of and topical guide to databases
Early-arriving fact – Measure – Dimension table – one of the set of companion tables to a fact table. Degenerate dimension – dimension key in the fact table that
Outline_of_databases
Mathematical description of spacetime used in relativity
For an overview, Minkowski spacetime is a 4-dimensional real vector space equipped with a non-degenerate, symmetric bilinear form on the tangent space
Minkowski_spacetime
Differentiable manifold with nondegenerate metric tensor
passing through the point p {\displaystyle p} . A metric tensor is a non-degenerate, smooth, symmetric, bilinear map that assigns a real number to pairs of
Pseudo-Riemannian_manifold
Generalization of the one-dimensional normal distribution to higher dimensions
normal distribution is degenerate and does not have a density. More precisely, it does not have a density with respect to k-dimensional Lebesgue measure (which
Multivariate normal distribution
Multivariate_normal_distribution
Forms which matter can take
condensates and Fermionic condensates (both in extreme cold), neutron-degenerate matter (in extreme density), and quark–gluon plasma (at extremely high
State_of_matter
Mathematical object
parameterize a 2-dimensional torus. Rings of constant ξ1 and ξ2 above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when
3-sphere
Attempt to demonstrate the 4th dimension in visual arts
New possibilities opened up by the concept of four-dimensional space (and difficulties involved in trying to visualize it) helped inspire many modern
Fourth_dimension_in_art
Statement about cubic curves in the projective plane
gives us P9. Since degenerate conics are a union of at most two lines, there are always four out of seven points on a degenerate conic that are collinear
Cayley–Bacharach_theorem
Geometric concept of a 2D space with "points at infinity" adjoined
there are no parallel lines. The last condition excludes the so-called degenerate cases (see below). The term "incidence" is used to emphasize the symmetric
Projective_plane
Geometric transformation that preserves lines but not angles nor the origin
points that define a non-degenerate triangle in a plane, or four points that define a non-degenerate tetrahedron in 3-dimensional space, or generally n +
Affine_transformation
Geometric model of the planar projection of the physical universe
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E}
Euclidean_plane
Analyzes the topology of a manifold by studying differentiable functions on that manifold
More precisely, the index of a non-degenerate critical point p {\displaystyle p} of f {\displaystyle f} is the dimension of the largest subspace of the tangent
Morse_theory
{\displaystyle K/k} to the set of isomorphism classes of non-degenerate n {\displaystyle n} -dimensional quadratic forms over K {\displaystyle K} and taking a
Essential_dimension
Polygon with 2 sides and 2 vertices
polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one
Digon
Fundamental object of geometry
As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves
Point_(geometry)
Invariant that plays a role in algebraic geometry and singularity theory
isolated degenerate singularity at 0 will split up into other isolated singularities which are non-degenerate. The number of such isolated non-degenerate singularities
Milnor_number
Pair of diametrically opposite points on a circle, sphere, or hypersphere
choosing non-antipodal points, and degenerate if antipodal points are allowed; for example, a spherical triangle degenerates to an underspecified lune if two
Antipodal_point
Conjecture in linear algebra
Analogously, for points in three-dimensional Euclidean space, the conjecture states that the sixteen vertices of four non-degenerate tetrahedra of four different
Rota's_basis_conjecture
Egyptian artistic movement 1938–1948
L'Art Dégéneré! ("Long Live Degenerate Art!"); it carried thirty-six signatures. The group adopted the term "degenerate" as a badge of honor. It was
Art_et_Liberté
Algebra based on a vector space with a quadratic form
Clifford algebra of real four-dimensional space with a degenerate quadratic form. Let the vector space V be real four-dimensional space R4, and let the quadratic
Clifford_algebra
Number of positive, negative and zero eigenvalues of a metric tensor
quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive,
Metric_signature
Concept in algebraic geometry
two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional
General_position
Collection of models with the same renormalization group flow limit
the lower critical dimension, the universality class becomes degenerate (this dimension is 2 for the Ising model, or for directed percolation, but 1 for
Universality_class
Doughnut-shaped surface of revolution
revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle
Torus
Conformal field theory on a 2D spacetime
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations
Two-dimensional conformal field theory
Two-dimensional_conformal_field_theory
Supergravity in eleven dimensions
In supersymmetry, eleven-dimensional supergravity is the theory of supergravity in the highest number of dimensions allowed for a supersymmetric theory
Eleven-dimensional supergravity
Eleven-dimensional_supergravity
Loss of one degree of freedom in a three-dimensional, three-gimbal mechanism
parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space. The term can be misleading in the sense that none of the
Gimbal_lock
Part of a straight line that is bounded by two distinct end points
statement to make segments congruent. A line segment can be viewed as a degenerate case of an ellipse, in which the semiminor axis goes to zero, the foci
Line_segment
Polygon with an infinite number of sides
either degenerate in which case it has no faithful realizations, or every vertex-faithful realization is faithful. The apeirogon is not degenerate and thus
Apeirogon
Location of a discrete degeneracy between two electronic states
this context. Degenerate points between potential energy surfaces lie in what is called the intersection or seam space with a dimensionality of 3N-8 (where
Conical_intersection
Lowest energy level of a quantum system
more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists
Ground_state
Study of smooth real-valued functions on manifold and their singularities
functions can be approximated by one that is Morse at all but finitely many degenerate times. The degeneracies involve a birth/death transition of critical points
Cerf_theory
Polyhedron with 2 faces
same set of n edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron
Dihedron
Principle in geometry
constraint, so naive dimension counting yields 25 = 32 conics tangent to five given lines, of which 31 must be ascribed to degenerate conics, as described
Five_points_determine_a_conic
Whose values lie in an infinite-dimensional vector space
An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or
Infinite-dimensional vector function
Infinite-dimensional_vector_function
curves, one can also define non-degenerate homotopy. Here, the 1-parameter family of immersions must be non-degenerate (i.e. the curvature may never vanish)
Regular_homotopy
Theories of higher-dimensional general relativity
Higher-dimensional Einstein gravity is any of various physical theories that attempt to generalize to higher dimensions various results of the standard
Higher-dimensional Einstein gravity
Higher-dimensional_Einstein_gravity
Type of algebraic equation
polynomial of two variables over the complex numbers. For suitable non-degenerate choice of F and G, the equation P(X,Y) = 0 will actually define the modular
Modular_equation
Scalar-valued bilinear function
said to be nondegenerate. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with
Bilinear_form
Extension of Laplace's method for approximating integrals
a non-degenerate saddle point: The Morse lemma for real-valued functions generalizes as follows for holomorphic functions: near a non-degenerate saddle
Method_of_steepest_descent
variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety
Chow_variety
Physics term for multiple concepts
may result in dramatic changes in its physical or chemical properties. Degenerate matter Exotic atoms Negative mass would possess some strange properties
Exotic_matter
Type of Borel measure
In mathematics, a Gaussian measure is a Borel measure on finite-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , closely related to
Gaussian_measure
Matrix of second derivatives
called a degenerate critical point of f , {\displaystyle f,} or a non-Morse critical point of f . {\displaystyle f.} Otherwise it is non-degenerate, and called
Hessian_matrix
Mathematical concept
be an alternating bilinear form on an n-dimensional real vector space V, ω ∈ Λ2(V). Then ω is non-degenerate if and only if n is even and ωn/2 = ω ∧
Symplectic_vector_space
regular digon {2} can be considered to be a degenerate regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the
List_of_regular_polytopes
Broad concept generalizing scalars in mathematics and physics
vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Matrix with a multiplicative inverse
In linear algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is
Invertible_matrix
Integer invariant of certain classes of topological manifolds
is an integer invariant which is defined for an oriented manifold M of dimension divisible by four. This invariant of a manifold has been studied in detail
Signature_(topology)
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
"exotic"). The first exotic spheres were constructed by John Milnor (1956) in dimension n = 7 {\displaystyle n=7} as S 3 {\displaystyle S^{3}} -bundles over S
Exotic_sphere
element Face, a 2-dimensional element Cell, a 3-dimensional element Hypercell or Teron, a 4-dimensional element Facet, an (n-1)-dimensional element Ridge
List_of_mathematical_shapes
Mathematical operation on vectors in 3D space
geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E {\displaystyle E} ), and
Cross_product
Brief thermal runaway nuclear fusion in the core of low-mass stars
is exhausted, some of the helium left behind is instead compacted into degenerate matter, supported against gravitational collapse by quantum mechanical
Helium_flash
Shape with three sides
also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has
Triangle
Painting by George Grosz
Grosz was one of the artists outlawed as degenerate. The current painting was shown at the exhibition of Degenerate Art in 1937. Shortly thereafter, it was
Metropolis_(Grosz)
2007 American film
a new edict that bans discussion of heretical topics such as a third dimension or the ruins at Area 33H. Arthur recognizes an image of the ruins from
Flatland (2007 Johnson and Travis film)
Flatland_(2007_Johnson_and_Travis_film)
Point where the derivative of a function is zero or undefined (in certain cases)
which is nonsingular if and only if it is not zero. In this case, a non-degenerate critical point is a local maximum or a local minimum, depending on the
Critical_point_(mathematics)
Group of 𝑛 × 𝑛 invertible matrices
( V ) {\displaystyle \operatorname {O} (V)} , which preserves a non-degenerate quadratic form on V {\displaystyle V} , symplectic group, Sp ( V ) {\displaystyle
General_linear_group
Quantum mechanics principle
in the alkali metal spectra in an external magnetic field, where all degenerate energy levels are separated, is equal to the number of electrons in the
Pauli_exclusion_principle
Removal of alternate vertices
even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge. More generally any vertex-uniform
Alternation_(geometry)
Associative algebra generalizing the Virasoro algebra
\Omega _{-}}} the corresponding fully degenerate representation of the W(N) algebra. The irreducible finite-dimensional representation R Ω {\displaystyle
W-algebra
Kepler–Poinsot polyhedron
same edge arrangement with the great icosahedron, with which it forms a degenerate uniform compound figure. It is the second of four stellations of the dodecahedron
Small_stellated_dodecahedron
Subdivision of a planar object into triangles
extension the subdivision of a higher-dimension geometric object into simplices. Triangulations of a three-dimensional volume would involve subdividing it
Triangulation_(geometry)
Mathematical functions related to Weierstrass's elliptic function
squared cosecant. The Weierstrass sigma function associated to a two-dimensional lattice Λ ⊂ C {\displaystyle \Lambda \subset \mathbb {C} } is defined
Weierstrass sigma, zeta, and eta functions
Weierstrass_sigma,_zeta,_and_eta_functions
Triangle area in terms of side lengths
triangles), or as a special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral). A modern proof, which uses algebra and is quite
Heron's_formula
Unified field theory
electromagnetism based on the idea of a fifth dimension of space beyond the conventional four-dimensional spacetime of general relativity. According to
Kaluza–Klein_theory
Three-dimensional geometric shape
direction and an even number in the other direction. The kaleidocycle has degenerate pairs of coinciding edges in transition, which function as hinges. The
Kaleidocycle
Quantum physics and chemistry phenomenon
parameters cannot become equal in value ("cross") except on a manifold of dimension k − 2 {\displaystyle k-2} . The phenomenon is also known as the von Neumann–Wigner
Avoided_crossing
Property of geometry, also used to generalize the notion of "distance" in metric spaces
of the other two, c = a + b {\displaystyle c=a+b} , the triangle is degenerate, with zero area. The triangle inequality implies a related statement for
Triangle_inequality
Relationship between certain vector spaces
diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because the group has an outer automorphism
Triality
Partition of space by hyperplanes
convenient to allow the degenerate hyperplane, which is the whole space S, to belong to an arrangement. If A contains the degenerate hyperplane, then it has
Arrangement_of_hyperplanes
Kepler orbit with an eccentricity of less than one
and Eris. A radial trajectory can be a double line segment, which is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity
Elliptic_orbit
Theory proposed by Roger Penrose
projective 3-space C P 3 {\displaystyle \mathbb {CP} ^{3}} , the simplest 3-dimensional compact algebraic variety. It has a physical interpretation as the space
Twistor_theory
Compact astronomical body
the new state of matter that results from this balance, called electron-degenerate matter, discovering that it is stable below a certain limiting mass. By
Black_hole
atoms break into their constituents and matter exists as some form of degenerate matter or quark matter. Such states of matter are studied in high-energy
List_of_states_of_matter
Pentagon with all sides equal but the angles may not be equal
and 8. Each family has one degree of freedom, and they intersect at the degenerate pentagon with one angle of 180°. Some of those pentagons can tile in more
Equilateral_pentagon
K-theory of quadratic forms
abelian group of equivalence classes [ ψ ] {\displaystyle [\psi ]} of non-degenerate ε-quadratic forms ψ ∈ Q ϵ ( F ) {\displaystyle \psi \in Q_{\epsilon }(F)}
L-theory
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DEGENERATE DIMENSION
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DEGENERATE DIMENSION
DEGENERATE DIMENSION
DEGENERATE DIMENSION
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