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entire manifold M is hyperbolic, the map f is called an Anosov diffeomorphism. The dynamics of f on a hyperbolic set, or hyperbolic dynamics, exhibits features
Hyperbolic_set
Formalization of the idea of an attractor or repellor in dynamical systems
attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set. The gravitational tidal forces acting
Stable_manifold
Type of non-Euclidean geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate
Hyperbolic_geometry
A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described
Normally hyperbolic invariant manifold
Normally_hyperbolic_invariant_manifold
Mathematical concept
precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group
Hyperbolic_group
Fixed point that does not have any center manifolds
systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the
Hyperbolic_equilibrium_point
Fractal named after mathematician Benoit Mandelbrot
density of hyperbolicity, is one of the most important open problems in complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often
Mandelbrot_set
Topics referred to by the same term
Hyperbolic structure may refer to: Hyperboloid structure Hyperbolic set This disambiguation page lists mathematics articles associated with the same title
Hyperbolic_structure
Concept in mathematics
In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number
Hyperbolic_metric_space
Isometric automorphisms of a hyperbolic space
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous
Hyperbolic_motion
Tiling of hyperbolic 3-space by uniform polyhedra
complete set of hyperbolic uniform honeycombs. More unsolved problems in mathematics In hyperbolic geometry, a uniform honeycomb in hyperbolic space is
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p
List of uniform tilings on the sphere, plane, and hyperbolic plane
List_of_uniform_tilings_on_the_sphere,_plane,_and_hyperbolic_plane
Limiting set in dynamical systems
then this attractor will be of finite dimensions. Cycle detection Hyperbolic set Stable manifold Steady state Wada basin Hidden oscillation Rössler attractor
Attractor
Normalized hyperbolic volume of the complement of a hyperbolic knot
knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume
Hyperbolic_volume
Spacetime manifold
global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It is called hyperbolic in analogy
Globally_hyperbolic_spacetime
boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space. Conceptually
Gromov_boundary
Argument of the hyperbolic functions
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane
Hyperbolic_angle
Infinitely detailed mathematical structure
various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called
Fractal
24 mathematical problems stated in 1982
influential 1982 paper Three-dimensional manifolds, Kleinian groups and hyperbolic geometry published in the Bulletin of the American Mathematical Society
Thurston's_24_questions
Non-Euclidean geometry
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant negative sectional curvature
Hyperbolic_space
Symmetric subdivision in hyperbolic geometry
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
Geometric mean and hyperbolic angle as coordinates in quadrant I
In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane { ( x , y ) : x > 0 , y > 0 } = Q {\displaystyle
Hyperbolic_coordinates
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank.
List_of_regular_polytopes
hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn
Hyperbolic_Dehn_surgery
Lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set
lemma is a lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set. Informally, the theory states that every pseudo-orbit (which one
Shadowing_lemma
Set of points on a line segment with certain topological properties
automorphisms of the Cantor set are hyperbolic motions, particular isometries of the hyperbolic plane. Thus, the Cantor set is a homogeneous space in the
Cantor_set
Definition of a class of dynamical systems
nonwandering set of f, Ω(f), is a hyperbolic set and compact. The set of periodic points of f is dense in Ω(f). For surfaces, hyperbolicity of the nonwandering
Axiom_A
conjectured that the complement of the special set is Mordellic. A variety is algebraically hyperbolic if the special set is empty. Lang conjectured that a variety
Mordellic_variety
Mathematical tree in the hyperbolic plane
A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry. Displaying hierarchical
Hyperbolic_tree
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations. The Euler equations can be formulated
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
Mutation of quaternions where unit vectors square to +1
lectures at Lehigh University in 1900. Like the quaternions, the set of hyperbolic quaternions form a vector space over the real numbers of dimension
Hyperbolic_quaternion
Model of hyperbolic geometry
model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines
Poincaré_disk_model
Covering by shapes without overlaps or gaps
made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for
Tessellation
number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group on those
Cannon–Thurston_map
Two geometries based on axioms closely related to those specifying Euclidean geometry
modified for elliptic geometry to work) and set to work proving a great number of results in hyperbolic geometry. He finally reached a point where he
Non-Euclidean_geometry
Discrete group of Möbius transformations
discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group
Kleinian_group
Linear map that preserves areas
mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with circular rotations, which preserve circles. The squeeze mapping sets the stage
Squeeze_mapping
Behavior in a nonlinear system
hematopoiesis, as appearing in the Mackey-Glass equations. Attractor Hyperbolic set Periodic point Self-oscillation Stable manifold Phase reduction Thomas
Limit_cycle
Type of unbounded quadratic surface-shaped building or work
amount of material. His design, as well as the full set of supporting calculations analyzing the hyperbolic geometry and sizing the network of members, was
Hyperboloid_structure
In mathematics, the complex hyperbolic space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds
Complex_hyperbolic_space
Straight line segment that passes through the centre of a circle
{\displaystyle n} -dimensional object, or a set of scattered points. The diameter of a set is the least upper bound of the set of all distances between pairs of
Diameter
of hyperbolic geometry, Hilbert's arithmetic of ends is a method for endowing a geometric set, the set of ideal points or "ends" of a hyperbolic plane
Hilbert's_arithmetic_of_ends
Upper-half plane model of hyperbolic non-Euclidean geometry
way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using
Poincaré_half-plane_model
Fundamental result in geometry
}\times {r^{2}}} . Lexell's theorem also has a hyperbolic counterpart: instead of circles, the level sets become pairs of curves called hypercycles, and
Sum_of_angles_of_a_triangle
Tiling of the hyperbolic plane
Böröczky tiling) is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. The tiles are congruent
Binary_tiling
Mathematical notation based on the Arabic script
the second word of دالة زائدية "hyperbolic function") is added to the end of trigonometric functions to express hyperbolic functions. This is similar to
Modern Arabic mathematical notation
Modern_Arabic_mathematical_notation
Number in {..., –2, –1, 0, 1, 2, ...}
negative integers. The set of all integers is often denoted by the boldface Z or blackboard bold Z {\displaystyle \mathbb {Z} } . The set of natural numbers
Integer
Topological manifold that is invariant under the action of dynamical system
non-autonomous dynamical systems are known as Lagrangian Coherent Structures. Hyperbolic set Lagrangian coherent structure Spectral submanifold Hirsh M.W., Pugh
Invariant_manifold
A hyperbolic geometric graph (HGG) or hyperbolic geometric network (HGN) is a special type of spatial network where (1) latent coordinates of nodes are
Hyperbolic_geometric_graph
Compact astronomical body
equations describing general relativity, astrophysicist Karl Schwarzschild set out to apply the idea to stars. He assumed spherical symmetry with no spin
Black_hole
Relation of space and time in relativity theory
given a pair of conjugate hyperbolas, two conjugate diameters are hyperbolically orthogonal. This relationship of diameters was described by Apollonius
Hyperbolic_orthogonality
an acylindrically hyperbolic group is a group admitting a non-elementary 'acylindrical' isometric action on some geodesic hyperbolic metric space. This
Acylindrically hyperbolic group
Acylindrically_hyperbolic_group
Shape in hyperbolic geometry
rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has
Ideal_polyhedron
Critical point on a surface graph which is not a local extremum
approximating integrals Maximum and minimum Derivative test Hyperbolic equilibrium point Hyperbolic geometry Minimax theorem Max–min inequality Mountain pass
Saddle_point
In mathematics, relatively hyperbolic groups form an important class of groups of interest for geometric group theory. The main purpose in their study
Relatively_hyperbolic_group
Class of radio navigation systems
Hyperbolic navigation is a class of radio navigation systems in which a navigation receiver instrument is used to determine location based on the difference
Hyperbolic_navigation
Concept in geometry
mapped to each point). In the case of a hyperbolic space, each line has two distinct ideal points. Here, the set of ideal points takes the form of a quadric
Point_at_infinity
All points in the topological closure not belonging to the interior
mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element
Boundary_(topology)
– inverse hyperbolic cosecant function. (Also written as arcsch.) arcosh – inverse hyperbolic cosine function. arcoth – inverse hyperbolic cotangent function
List of mathematical abbreviations
List_of_mathematical_abbreviations
Manifold of dimension 3 equipped with a hyperbolic metric
topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric
Hyperbolic_3-manifold
Parametrizes complex structures on a surface
{\displaystyle S} to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for
Teichmüller_space
Mathematical software
help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first
SnapPea
One-dimensional complex manifold
{\displaystyle \tau } and hence a torus. The set of all Riemann surfaces can be divided into three subsets: hyperbolic, parabolic and elliptic Riemann surfaces
Riemann_surface
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, exponentials, and
CORDIC
Group of real 2×2 matrices with unit determinant
form an open set, as do the hyperbolic elements (excluding ±1). By contrast, the parabolic elements, together with ±1, form a closed set that is not open
SL2(R)
Shape with three sides
discovered in several spaces, as in hyperbolic space and spherical geometry. A triangle in hyperbolic space is called a hyperbolic triangle, and it can be obtained
Triangle
Unique knot with a crossing number of four
Thurston showed that the figure-eight was hyperbolic, by decomposing its complement into two ideal hyperbolic tetrahedra. (Robert Riley and Troels Jørgensen
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Class of compact connected topological spaces
tube twice inside T with twisting, but without self-intersections. The hyperbolic set Λ of the discrete dynamical system (T, f) is the intersection of the
Solenoid_(mathematics)
1952 novel by John Steinbeck
According to critics, Steinbeck's portrayal of good and evil was both hyperbolic and oversimplified, especially in the character of Cathy. Besides criticizing
East_of_Eden_(novel)
Algorithms and methods of plotting the Mandelbrot set on a computing device
and Julia sets It is also possible to estimate the distance of a limitly periodic (i.e., hyperbolic) point to the boundary of the Mandelbrot set. The upper
Plotting algorithms for the Mandelbrot set
Plotting_algorithms_for_the_Mandelbrot_set
quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters. Świątek, Grzegorz; Graczyk, Jacek (1998), The
Fatou_conjecture
Area in mathematics devoted to the study of finitely generated groups
1987 monograph "Hyperbolic groups" that introduced the notion of a hyperbolic group (also known as word-hyperbolic or Gromov-hyperbolic or negatively curved
Geometric_group_theory
Motion of an object with constant proper acceleration in special relativity
Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation
Hyperbolic motion (relativity)
Hyperbolic_motion_(relativity)
dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic periodic orbits and satisfying a transversality
Morse–Smale_system
Planar surface that forms part of the boundary of a solid object
all authors allow the polytope itself and the empty set as faces of a polytope, where the empty set is for consistency given a "dimension" of −1. For any
Face_(geometry)
Plane curve: conic section
cone Hyperbolic cylinder Hyperbolic paraboloid Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Hyperbolic cylinder Hyperbolic paraboloid
Hyperbola
Smallest closed orientable hyperbolic 3-manifold
manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead
Weeks_manifold
Orbital data format
A two-line element set (TLE, or more rarely 2LE) or three-line element set (3LE) is a data format encoding a list of orbital elements of an Earth-orbiting
Two-line_element_set
Type of topological group
finite index), or the hyperbolic plane. Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. A Fuchsian group
Discrete_group
Mandelbar Set
bifurcation from a hyperbolic component of odd period k to a hyperbolic component of period 2k occurs. Much like the Mandelbrot set, the tricorn has many
Tricorn_(mathematics)
Relation between sides of a right triangle
where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles: cosh
Pythagorean_theorem
Mathematical formula involving a given set of operations
trigonometric functions, inverse trigonometric functions, hyperbolic functions, and inverse hyperbolic functions. The fundamental problem of symbolic integration
Closed-form_expression
Three-holed sphere
compact surfaces in various theories. Two important applications are to hyperbolic geometry, where decompositions of closed surfaces into pairs of pants
Pair_of_pants_(mathematics)
Reals with an extra square root of +1 adjoined
algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle
Split-complex_number
Pseudometric of complex manifolds
manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined by the
Kobayashi_metric
Set of spacetime events, light-connected to a given event
To uphold causality, Minkowski restricted spacetime to non-Euclidean hyperbolic geometry. Because signals and other causal influences cannot travel faster
Light_cone
{\displaystyle dimW^{s}+dimW^{u}=n.} Then, M {\displaystyle M} contains a hyperbolic set Λ {\displaystyle \Lambda } , invariant under P {\displaystyle P} , on
Melnikov_distance
Mathematical space
hyperbolic Dehn surgery theorem states: M ( u 1 , u 2 , … , u n ) {\displaystyle M(u_{1},u_{2},\dots ,u_{n})} is hyperbolic as long as a finite set of
3-manifold
Geometric figure which has infinite surface area but finite volume
paper De solido hyperbolico acuto, written in 1643, a truncated acute hyperbolic solid, cut by a plane. Volume 1, part 1 of his Opera geometrica published
Gabriel's_horn
Danish mathematician
being one of the co-discoverers of the ordered structure of the set of volumes of hyperbolic 3-manifolds. Troels Jørgensen at the Mathematics Genealogy Project
Troels_Jørgensen
American columnist, author and lecturer (born 1946)
criticized for rejecting hyperbolic geometry as a satisfactory basis for Wiles' proof, with critics pointing out that axiomatic set theory (rather than Euclidean
Marilyn_vos_Savant
Pictorial representation of symmetry
alternations and some half symmetry version. In the hyperbolic plane [7,3], family produces a parallel set of uniform tilings, and their dual tilings. There
Coxeter–Dynkin_diagram
Discrete subgroup of the real projective special linear group of dimension 2
regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations
Fuchsian_group
Type of mathematical group
{PSL} _{2}(\mathbb {Z} )} . They, and the hyperbolic surface associated to their action on the hyperbolic plane, often exhibit particularly regular behaviour
Arithmetic_Fuchsian_group
Index of articles associated with the same name
regular tiling of the hyperbolic plane" (PDF), Journal de Physique Lettres, 43 (8): 249–252, doi:10.1051/jphyslet:01982004308024900 This set index article includes
Apeirogonal_tiling
Mathematical concept
the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin ārea). For instance, the inverse of the hyperbolic sine function is typically
Inverse_function
Transformations induced by a mathematical group
mathematics. CRC Press. ISBN 978-1-4200-6371-4. Kapovich, Michael (2009), Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser
Group_action
Navigation and surveillance technique
TOAs are multiple and known. When MLAT is used for navigation (as in hyperbolic navigation), the waves are transmitted by the stations and received by
Pseudo-range_multilateration
Model of n-dimensional hyperbolic geometry
Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet
Hyperboloid_model
Three linked but pairwise separated rings
n-colorings. As links, they are Brunnian, alternating, algebraic, and hyperbolic. In arithmetic topology, certain triples of prime numbers have analogous
Borromean_rings
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