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Hyperbolic

''Hyperbolic'' is an adjective with several distinct meanings across mathematics, physics, rhetoric, and other fields. In rhetoric and language, it relates to hyperbole, a figure of speech involving deliberate exaggeration for emphasis or effect. In mathematics, the term appears in contexts such as hyperbolic functions (analogous to trigonometric functions but based on the hyperbola), hyperbolic geometry (a non-Euclidean geometry of constant negative curvature), hyperbolic space, and hyperbolic distributions (probability models with heavy tails). In physics, it describes hyperbolic trajectories (unbound orbital paths), hyperbolic motion (in special relativity), and hyperbolic partial differential equations (wave-like behaviors). Other applications include hyperbolic growth (exponential-like increase) and various structures in topology and computer science. For detailed discussions, see the sections below.

Mathematics

Hyperbolic Functions

Hyperbolic functions, often denoted as sinh, cosh, and tanh, serve as analogs to the trigonometric functions sine, cosine, and tangent, but are defined in terms of exponential functions rather than circular motion. These functions arise naturally from the geometry of the hyperbola and play a key role in various mathematical and physical contexts, particularly in solving differential equations involving exponential growth or decay. Unlike trigonometric functions, which are periodic and bounded, hyperbolic functions are non-periodic and unbounded, reflecting the hyperbolic geometry's contrasting properties. The primary hyperbolic functions are defined as follows:

\sinh z = \frac{e^z - e^{-z}}{2}, \quad \cosh z = \frac{e^z + e^{-z}}{2}, \quad \tanh z = \frac{\sinh z}{\cosh z}.

^[1] These definitions stem from the parametric representation of the unit hyperbola x^2 - y^2 = 1, where points on the right branch can be parameterized as x = \cosh u and y = \sinh u, with u representing the hyperbolic angle or rapidity; this parameterization satisfies the hyperbola's equation directly due to the fundamental identity \cosh^2 u - \sinh^2 u = 1.^[2] Key identities mirror those of trigonometric functions but with sign changes. The Pythagorean identity is \cosh^2 z - \sinh^2 z = 1. Addition formulas include \cosh(u + v) = \cosh u \cosh v + \sinh u \sinh v and \sinh(u + v) = \sinh u \cosh v + \cosh u \sinh v, which facilitate computations similar to trigonometric expansions. The inverse hyperbolic functions reverse these mappings. For example, \operatorname{arcsinh} z = \ln\left(z + \sqrt{z^2 + 1}\right), defined for all real z with range (-\infty, \infty). Similarly, \operatorname{arccosh} z = \ln\left(z + \sqrt{z^2 - 1}\right) for z \geq 1, with range [0, \infty), and \operatorname{arctanh} z = \frac{1}{2} \ln\left(\frac{1 + z}{1 - z}\right) for |z| < 1, with range (-\infty, \infty). These logarithmic forms highlight the deep connection to exponentials. Hyperbolic functions are intimately linked to exponential functions and are fundamental solutions to linear differential equations of the form y'' - k^2 y = 0, where \sinh(kx) and \cosh(kx) provide the general solution basis. A prominent application is the catenary curve, describing the shape of a hanging chain under uniform gravity, given by y = a \cosh(x/a), where a is a constant related to the chain's linear density and tension; this equation emerges from balancing gravitational and tensile forces in the governing differential equation. The development of hyperbolic functions traces back to the 18th century. Vincenzo Riccati introduced them in 1757 as part of his work on solving cubic equations and differential equations, using notations like Sh and Ch in his Opuscula ad res physicas et mathematicas pertinentia. Johann Heinrich Lambert formalized and systematized the theory in 1768, employing sinh and cosh notations and integrating them into broader trigonometric frameworks.^[3]^[4]

Hyperbolic Geometry

Hyperbolic geometry is a non-Euclidean geometry characterized by the failure of Euclid's parallel postulate, allowing for infinitely many lines through a point not on a given line that do not intersect it.^[5] Unlike Euclidean geometry, where the sum of angles in a triangle equals 180 degrees, in hyperbolic geometry this sum is always less than 180 degrees, leading to unique properties such as exponential growth of area with radius.^[6] This geometry provides a consistent mathematical framework alternative to Euclidean plane geometry, with applications in areas like topology and cosmology.^[7] The development of hyperbolic geometry traces back to early 19th-century efforts to rigorously examine Euclid's axioms. Carl Friedrich Gauss explored ideas of non-Euclidean geometry privately around 1800 but did not publish them, influencing his correspondents.^[7] Independently, Nikolai Ivanovich Lobachevsky published his work in 1829, presenting a complete axiomatic system, while János Bolyai included his discoveries in a 1832 appendix to his father's book, both demonstrating the consistency of a geometry without the parallel postulate.^[8] These contributions resolved long-standing debates about the necessity of Euclid's fifth postulate and paved the way for modern differential geometry.^[7] The core axioms of hyperbolic geometry consist of Euclid's first four postulates—concerning points, lines, circles, and right angles—along with the hyperbolic parallel postulate: given a line and a point not on it, there exist at least two (in fact, infinitely many) lines through the point parallel to the given line.^[9] This replacement implies that the sum of the interior angles in any triangle is strictly less than 180 degrees, with the difference, known as the angular defect, being positive and dependent on the triangle's size.^[10] Additional axioms, such as those for continuity and betweenness, are typically assumed to ensure a complete ordered geometry.^[6] Several models embed hyperbolic geometry within Euclidean space for visualization and computation. The Poincaré disk model represents the hyperbolic plane as the interior of a unit disk, with geodesics as circular arcs orthogonal to the boundary; the metric is given by ds = \frac{2 |dz|}{1 - |z|^2}, where distances involve inverse hyperbolic functions, such as the distance from the center to a point at Euclidean distance r being \mathrm{acosh}\left( \frac{1 + r^2}{1 - r^2} \right).^[11] The upper half-plane model uses the region above the real axis with a similar conformal metric, while the Klein-Beltrami model projects geodesics as straight lines in a disk, preserving straight-line properties but distorting angles.^[12] These models confirm the consistency of hyperbolic geometry by satisfying all axioms within Euclidean subsets.^[5] Key theorems highlight the distinctive properties of hyperbolic geometry. The area of a triangle is proportional to its angular defect \Delta = \pi - (\alpha + \beta + \gamma), where \alpha, \beta, \gamma are the angles in radians; specifically, area A = k \Delta for some constant k > 0 depending on the curvature (often normalized to k = 1).^[13] This implies an upper bound on triangle areas, as \Delta < \pi.^[14] Regarding parallels, through a point not on a line, there are two limiting parallels that approach the line asymptotically at infinity without intersecting, while other parallels diverge; this asymptotic behavior distinguishes hyperbolic from Euclidean parallelism.^[5] Hyperbolic geometry finds applications in tiling problems, where regular polygons with interior angle sums allowing more than six around a vertex can tile the plane without gaps or overlaps. For instance, regular pentagons (angle ≈108°) or heptagons enable infinite tessellations satisfying \frac{1}{p} + \frac{1}{q} < \frac{1}{2} for Schläfli symbol {p, q}, unlike the Euclidean constraint of exactly six equilateral triangles or four squares.^[15] Such tilings illustrate the excess space in hyperbolic planes, with patterns growing exponentially. Hyperbolic functions appear in metric calculations across models, providing algebraic tools for distances and angles. This two-dimensional framework extends briefly to higher-dimensional hyperbolic spaces for broader structures.^[11] Hyperbolic n-space, denoted \mathbb{H}^n, is defined as the unique (up to isometry) simply connected complete Riemannian manifold with constant sectional curvature -1.^[16] This uniqueness follows from the Killing-Hopf theorem, which classifies complete simply connected Riemannian manifolds of constant curvature.^[16] Building on the foundational two-dimensional hyperbolic plane, \mathbb{H}^n generalizes these properties to higher dimensions while preserving the negative curvature that distinguishes hyperbolic geometry from Euclidean or spherical spaces.^[17] Several models realize \mathbb{H}^n, each embedding it into a familiar ambient space while inducing the hyperbolic metric. The Poincaré ball model extends the two-dimensional Poincaré disk to an open unit ball in \mathbb{R}^n, where the metric is given by ds^2 = \frac{4 \sum dx_i^2}{(1 - \|x\|^2)^2}, ensuring constant curvature -1.^[17] This conformal model facilitates visualization and computations, particularly in higher dimensions. Another prominent realization is the hyperboloid model, which embeds \mathbb{H}^n as the upper sheet of the hyperboloid \{x \in \mathbb{R}^{n+1} : \langle x, x \rangle = -1, x_0 > 0\} in Minkowski space \mathbb{R}^{n,1} equipped with the Lorentzian metric \langle x, y \rangle = -x_0 y_0 + \sum_{i=1}^n x_i y_i.^[18] The induced Riemannian metric on this sheet yields the desired curvature, and geodesics correspond to intersections with planes through the origin.^[18] Coordinates in these models often involve hyperbolic functions, such as sinh and cosh, to parametrize distances and angles.^[18] Hyperbolic groups provide an algebraic counterpart to the geometric structure of hyperbolic spaces. Introduced by Mikhail Gromov, a finitely generated group \Gamma is hyperbolic if its Cayley graph with respect to any finite generating set is a \delta-hyperbolic metric space for some \delta \geq 0, meaning geodesic triangles are \delta-thin: each side lies within \delta of the union of the other two.^[19] This condition captures a coarse notion of negative curvature in the word metric.^[19] Prominent examples include free groups of finite rank, whose Cayley graphs are trees and thus \delta = 0-hyperbolic, and fundamental groups of closed orientable surfaces of genus g \geq 2, which act properly and cocompactly on \mathbb{H}^2.^[19] In topology, hyperbolic groups arise as fundamental groups of hyperbolic manifolds and play a key role in classifying spaces. For instance, the boundary at infinity of a hyperbolic space on which a hyperbolic group acts properly and cocompactly serves as a model for the classifying space for proper actions, \underline{E}\Gamma, facilitating the study of manifold bordism and cohomology.^[20] In computer science, the negative curvature of hyperbolic groups enables efficient graph algorithms, such as linear-time solutions for the word problem and isometry testing in Cayley graphs.^[21] Recent applications extend to machine learning, where Poincaré embeddings leverage the exponential volume growth of hyperbolic space to represent hierarchical data, such as word taxonomies, outperforming Euclidean embeddings in tasks like link prediction on knowledge graphs.^[22] These embeddings, introduced by Nickel and Kiela, project symbolic data into the Poincaré ball, capturing tree-like structures with lower distortion than in flat spaces.^[22] As of 2025, hyperbolic deep learning has advanced further, with applications in foundation models, computer vision, and biological sequence modeling, as surveyed in recent literature.^[23]^[24]

Physics

Hyperbolic Trajectories

In astrodynamics, a hyperbolic trajectory describes the path of an object moving around a central gravitational body with sufficient velocity to escape its influence, characterized by an orbital eccentricity e > 1 and positive total specific energy \varepsilon > 0. This unbound orbit forms one branch of a hyperbola, with the central body at the external focus, allowing the object to approach from infinity, reach a point of closest approach (periapsis), and then recede back to infinity. Unlike bound elliptical orbits, hyperbolic paths do not close and are relevant for scenarios involving escape velocities, such as interstellar objects or spacecraft departing a planet's sphere of influence.^[25] The geometry of a hyperbolic trajectory is governed by the conic section equation in polar coordinates, expressed as

r = \frac{h^2 / \mu}{1 + e \cos \theta},

where r is the radial distance from the focus, \theta is the true anomaly, h is the specific angular momentum, and \mu is the standard gravitational parameter of the central body. The semi-major axis a, which quantifies the scale of the hyperbola and is negative by convention, relates to the specific energy via

a = -\frac{\mu}{2\varepsilon}.

This negative value distinguishes hyperbolas from ellipses, where a > 0, and underscores the positive energy enabling escape. Parametric representations may involve hyperbolic functions for position and time, linking briefly to hyperbolic motion concepts in relativity, though the Newtonian form suffices for most gravitational analyses.^[25] Key parameters include the impact parameter b, defined as the perpendicular distance from the central body to the incoming asymptote, given by

b = \frac{h^2}{\mu \sqrt{e^2 - 1}},

which determines the closeness of the encounter and influences the overall deflection. The deflection angle \delta, representing the change in the object's velocity direction due to gravity, is

\delta = 2 \arcsin\left( \frac{1}{e} \right).

For example, with e = 1.272, \delta \approx 103.6^\circ, illustrating how higher eccentricities result in smaller deflections approaching straight-line paths. These parameters are essential for predicting encounter outcomes in scattering problems.^[25]^[26] Hyperbolic trajectories are prominently applied in comet paths and spacecraft missions. Many long-period comets, such as interstellar visitors like 2I/Borisov, follow hyperbolic orbits with e > 1, escaping the Solar System after perihelion passage, though historical approximations for near-parabolic comets like Halley's (with e \approx 0.967) sometimes treat segments as hyperbolic for simplified velocity calculations near the Sun. In astrodynamics, they enable gravity-assist maneuvers, where spacecraft exploit planetary flybys to alter speed and direction; for instance, NASA's Voyager 2 mission utilized successive hyperbolic escapes around Jupiter (e \approx 1.33), Saturn, Uranus, and Neptune to achieve interstellar velocities exceeding 15 km/s relative to the Sun, reducing travel times dramatically without additional propulsion. Recent analyses of Voyager data, incorporating multi-body perturbations, refine these trajectories for long-term interstellar predictions, highlighting evolving computational methods in post-2020 studies.^[27]^[28]

Hyperbolic Motion

In special relativity, hyperbolic motion describes the trajectory of an object undergoing constant proper acceleration \alpha, where the proper acceleration is the acceleration measured in the instantaneous rest frame of the object. This results in a worldline that traces a hyperbola in Minkowski spacetime, satisfying the equation (ct)^2 - x^2 = (c^2 / \alpha)^2, with c as the speed of light and the origin chosen such that the object starts from rest at x = c^2 / \alpha.^[29] The hyperbola has asymptotes at x = \pm ct, reflecting the approach to the speed of light as acceleration persists. This motion contrasts with uniform acceleration in classical mechanics, as relativistic effects prevent the velocity from exceeding c. The kinematics of hyperbolic motion are parameterized using rapidity \phi, a hyperbolic angle that relates to the object's velocity v = c \tanh(\phi) and Lorentz factor \gamma = \cosh(\phi). The rapidity evolves linearly with proper time \tau as \phi = (\alpha \tau)/c, providing a natural way to handle the non-linear velocity increase. The position as a function of rapidity is given by x = (c^2 / \alpha) (\cosh(\phi) - 1) and ct = (c^2 / \alpha) \sinh(\phi), ensuring the worldline adheres to the hyperbolic relation. These expressions utilize hyperbolic functions to parameterize the path, maintaining consistency with the Lorentz-invariant proper acceleration.^[29] Hyperbolic motion forms the basis for Rindler coordinates, which describe the spacetime experienced by uniformly accelerating observers, transforming the Minkowski metric into a form suitable for analyzing acceleration effects. In these coordinates, the accelerating observer's worldline corresponds to a hyperbola, and the coordinate system reveals a Rindler horizon—an event horizon analogous to that in black hole physics—beyond which signals cannot reach the observer due to the finite speed of light. This horizon arises because, from the accelerated frame, distant inertial observers appear to freeze and redshift indefinitely. The concept of hyperbolic motion was first discussed by Max Born in 1909 in the context of rigid body dynamics in relativity, where he analyzed the motion of an electron under constant proper acceleration, terming it "hyperbolic motion" due to the curvature hyperbola in spacetime diagrams. Born's work laid foundational insights into relativistic kinematics for accelerated systems, later formalized in standard relativity textbooks.^[30]

Hyperbolic Partial Differential Equations

Hyperbolic partial differential equations (PDEs) are a class of PDEs characterized by wave-like propagation of information, distinct from elliptic and parabolic types. For a general second-order linear PDE in two variables of the form a u_{xx} + 2b u_{xy} + c u_{yy} + d u_x + e u_y + f u = g, the type is determined by the discriminant b^2 - ac. The equation is hyperbolic if b^2 - ac > 0, indicating two distinct real characteristics along which solutions propagate. A canonical example is the one-dimensional wave equation \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, where a = -c^2, b = 0, c = 1, yielding a positive discriminant c^2 > 0.^[31] Characteristics play a central role in hyperbolic PDEs, representing curves or surfaces along which discontinuities or information propagate without alteration. For first-order hyperbolic PDEs, such as the linear advection equation u_t + a u_x = 0, the method of characteristics reduces the PDE to ordinary differential equations (ODEs) along these curves, enabling explicit solutions like u(t, x) = u_0(x - a t) for initial data u(0, x) = u_0(x). In higher-order cases, characteristics are derived from the principal symbol of the PDE operator, guiding the solution structure and domain of dependence. This method extends to systems of first-order hyperbolic PDEs, ensuring well-posedness in appropriate function spaces.^[32]^[33] Analytical solution techniques for hyperbolic PDEs often exploit characteristic coordinates. For the one-dimensional wave equation on the infinite line with initial conditions u(0, x) = \phi(x) and u_t(0, x) = \psi(x), D'Alembert's formula provides the exact solution:

u(x, t) = \frac{\phi(x + c t) + \phi(x - c t)}{2} + \frac{1}{2c} \int_{x - c t}^{x + c t} \psi(s) \, ds.

This formula illustrates the finite propagation speed, where the solution at (x, t) depends only on initial data within [x - c t, x + c t]. For nonlinear hyperbolic systems, characteristics may intersect, leading to shocks, but the method remains foundational for understanding solution behavior.^[34] Hyperbolic PDEs arise in numerous physical applications modeling wave phenomena. In acoustics, the wave equation describes sound propagation in media, capturing pressure variations and reflections. Electromagnetism is governed by Maxwell's equations, which form a symmetric hyperbolic system in first-order form, ensuring causality and wave speeds at the speed of light. Nonlinear hyperbolic conservation laws, such as the Lighthill-Whitham-Richards (LWR) model \partial_t \rho + \partial_x (\rho v(\rho)) = 0 for traffic density \rho, simulate flow dynamics including shock formation for traffic jams.^[35]^[36]^[37] Recent advances in numerical methods for hyperbolic conservation laws address challenges like shock capturing and high accuracy. Finite volume schemes, such as high-order well-balanced reconstructions, preserve steady states and handle source terms effectively in balance laws, improving simulations of discontinuous solutions. Data-driven physics-informed finite volume methods integrate machine learning to approximate small-scale shocks, enhancing resolution without excessive computational cost since 2021. These developments, building on Godunov-type schemes, enable robust simulations in complex geometries post-2010.^[38]^[39]

Rhetoric and Language

Hyperbole

Hyperbole is a figure of speech characterized by deliberate and obvious exaggeration, employed to emphasize a point, evoke strong emotions, or create a vivid effect, without the intention of being taken literally.^[40] The term derives from the Greek word hyperbolē, meaning "a throwing beyond" or "excess," reflecting its origins in classical rhetoric as a tool for amplification.^[41] In ancient Greek rhetoric, hyperbole emerged as a key device for persuasion and stylistic enhancement, with early uses attributed to orators like Isocrates in the 4th century BCE, who employed it in encomia to exaggerate virtues for emphatic praise.^[42] Aristotle formalized its role in his Rhetoric (Book 3, Chapter 11), describing hyperbole as a form of excess that enhances emotional appeal and wit, particularly when used playfully to counter exaggeration in opponents' arguments.^[43] Unlike understatement, known as meiosis, which diminishes for ironic or modest effect, hyperbole amplifies to intensify impact, such as in the common expression "I'm so hungry I could eat a horse," where the impossibility underscores genuine appetite.^[44] Hyperbole serves multiple rhetorical functions, including evoking humor through absurdity, stirring emotions like passion or frustration, and generating memorable imagery that aids persuasion.^[45] In literature, it heightens dramatic tension, as seen in Shakespeare's Romeo and Juliet, where Romeo exclaims that Juliet's eyes are "Two of the fairest stars in all the heaven," exaggerating her beauty to convey overwhelming infatuation.^[46] This device extends into hyperbolic language in broader literary contexts, amplifying narrative themes.^[45] Culturally, hyperbole permeates modern discourse for persuasive ends. In advertising, visual and verbal exaggerations, such as claims of products being "the best ever," boost ad liking and engagement by creating incongruity that sparks humor and attention.^[47] Political rhetoric often deploys it to rally support, with statements like characterizing policy threats as existential crises protected under doctrines like rhetorical hyperbole in legal contexts.^[48] On social media, hyperbole fuels memes and posts, such as "I'm dying of laughter," to express exaggerated enthusiasm or sarcasm, enhancing virality through emotional amplification.^[49]

Hyperbolic Language in Literature

Hyperbolic language in literature extends beyond mere exaggeration to serve as a stylistic tool that amplifies emotional depth, character portrayal, and thematic resonance, often blending vivid imagery with narrative intent to engage readers on multiple levels. In poetry, particularly ancient epics, hyperbole manifests through extended similes that elevate mundane comparisons to grand, almost cosmic scales, creating a sense of awe and universality in the storytelling. For instance, Homer's Iliad employs epic similes, such as likening the Trojan warriors' flight to leaves scattered by the wind or chaff winnowed by farmers, to hyperbolic proportions that underscore the chaos and inevitability of battle, transforming individual actions into emblematic forces of nature.^[50]^[51] In novels, this technique exaggerates character traits to highlight social critiques or psychological states; Charles Dickens, for example, uses hyperbole in Great Expectations to depict Mr. Wopsle's theatrical pomposity through his exaggerated declamation of a simple prayer, blending elements "like a religious cross of the Ghost in Hamlet with Richard the Third," thereby satirizing pretension while making the character's flaws memorably grotesque.^[52] The evolution of hyperbolic language traces a path from classical restraint to Romantic intensification and modernist surrealism, adapting to cultural shifts in expression. In the Romantic era, Lord Byron's "She Walks in Beauty" intensifies natural imagery through hyperbole, portraying the subject's grace as harmonizing "all that's best of dark and bright" in a flawless, almost divine equilibrium that elevates human beauty to ethereal perfection.^[53]^[54] By the 20th century, this evolved into surrealist forms, as seen in Gabriel García Márquez's magical realism, where hyperbolic distortions of reality—such as rains of flowers or ascensions to heaven—blend the fantastical with the everyday to critique historical traumas in Latin America, amplifying the absurdities of colonialism and isolation.^[55]^[56] In literary analysis, hyperbolic language plays a pivotal role in satire and irony by distorting proportions to expose societal follies or human contradictions, often through overstatement that invites readers to recognize underlying truths. For example, in satirical works, hyperbole enlarges vices to absurd levels, as in Jonathan Swift's A Modest Proposal, where suggesting the consumption of infants mocks economic exploitation, using exaggeration to provoke moral outrage.^[45] This device also fosters irony by contrasting stated extremes with implied realities, heightening the comedic or critical bite. Psychologically, hyperbolic language intensifies readers' emotional responses, increasing perceived impact and believability of narratives; studies show it heightens arousal and empathy, making victims' testimonies in fictional accounts seem more credible and emotionally resonant, thus deepening reader immersion and ethical engagement.^[57]^[58] In modern contexts, hyperbolic language permeates graphic novels and film scripts, where visual and verbal exaggeration amplifies dramatic tension or humor for multimedia impact. In graphic novels like Alan Moore's Watchmen, hyperbolic depictions of superhuman feats—such as characters surviving nuclear blasts—exaggerate moral dilemmas to satirize heroism, blending textual overstatements with illustrative extremes for heightened visual rhetoric.^[59] In film scripts, dialogue often employs hyperbole to inflate threats and underscore power dynamics, enhancing cinematic intensity; a classic example is the line "I'm gonna make him an offer he can't refuse" from The Godfather (screenplay by Mario Puzo and Francis Ford Coppola, adapted from Puzo's novel). Post-2015 digital literature, particularly fanfiction on platforms like Archive of Our Own, incorporates hyperbolic tropes—such as amplified romantic entanglements or character god-like abilities—to subvert canon and explore identity, reflecting participatory reading trends where exaggeration fosters communal creativity and emotional catharsis in online communities.^[60]^[61] A notable case study is William Shakespeare's deployment of hyperbole, which varies distinctly between tragedies and comedies to modulate tone and effect. In tragedies like Romeo and Juliet, hyperbole intensifies passion and foreshadows doom, as Romeo's declaration that Juliet's beauty "teaches the torches to burn bright" elevates love to cosmic proportions, underscoring its tragic volatility.^[46] Conversely, in comedies such as The Comedy of Errors, it fuels farce and mistaken identities, with Dromio's hyperbolic rants comparing a character's girth to global landmarks—"She is spherical, like a globe"—to generate slapstick humor and relieve tension through absurdity. This contrast illustrates hyperbole's versatility: tragic amplification builds pathos, while comic excess diffuses conflict, tailoring emotional arcs to genre demands.^[62]

Other Applications

Hyperbolic Growth

Hyperbolic growth describes a form of accelerating increase in a quantity that approaches infinity in finite time, known as a finite-time singularity. This model is governed by the differential equation

\frac{dy}{dt} = k y^{1+\delta},

where k > 0 and \delta > 0 are constants, and initial condition y(0) = y_0 > 0. To solve, separate variables: y^{-1-\delta} dy = k \, dt. Integrating both sides yields \int y^{-1-\delta} dy = k \int dt, or -\frac{1}{\delta} y^{-\delta} = k t + C. Applying the initial condition gives C = -\frac{1}{\delta y_0^\delta}, so y^{-\delta} = y_0^{-\delta} - \delta k t. Thus, y(t) = y_0 \left( 1 - \delta k t y_0^\delta \right)^{-1/\delta}, which diverges as t \to t_s^-, where the singularity time is t_s = \frac{1}{k \delta y_0^\delta}.^[63] Unlike exponential growth, where \frac{dy}{dt} = k y leads to y(t) = y_0 e^{kt} and unbounded growth over infinite time with constant relative rate, hyperbolic growth features a relative rate \frac{1}{y} \frac{dy}{dt} = k y^\delta that itself accelerates, causing the quantity to explode in finite time. This distinction highlights hyperbolic models' suitability for systems with positive feedback loops amplifying growth.^[63] In population dynamics, hyperbolic growth has approximated human population trends from antiquity through the 20th century, where social and technological factors created self-reinforcing expansion. Heinz von Foerster and colleagues fitted historical data (1 AD to 1960) to a specific case with \delta = 1, predicting a singularity around 2026 if trends persisted, though later demographic transitions shifted patterns toward logistic limits. Similar models apply in ecology to describe rapid population booms in resource-rich environments before density-dependent factors intervene, such as algal blooms or insect outbreaks.^[64]^[65] Technological progress, including extensions of Moore's Law on transistor density doubling, has been analyzed through hyperbolic lenses to capture accelerating innovation rates driven by cumulative knowledge. Ray Kurzweil's Singularity Hypothesis posits that computational paradigms evolve hyperbolically, merging human and machine intelligence by mid-century. Economic applications extend this to GDP trajectories, where von Foerster-inspired models forecast potential singularities from sustained feedback, though empirical fits weaken post-1960 due to resource constraints.^[66] Recent debates on AI-driven hyperbolic growth, updated in the 2020s, center on whether exponential compute scaling (e.g., via AI training) could trigger a technological singularity akin to Kurzweil's 2045 timeline, with critics arguing physical limits preclude sustained acceleration. These discussions remain unresolved, emphasizing risks of unchecked feedback in intelligent systems.^[67]

Hyperbolic Distributions

Hyperbolic distributions form a class of continuous probability distributions characterized by their density functions, which incorporate modified Bessel functions of the third kind and exhibit symmetry around zero in their standard parameterization. The generalized hyperbolic (GH) distribution, the foundational member of this class, arises as a normal variance-mean mixture in which the mixing distribution follows a generalized inverse Gaussian law, enabling flexible modeling of skewness and tail behavior. It is parameterized by five quantities: λ (a shape parameter governing the form of the mixing distribution), α (a positive scale-like parameter controlling tail decay), β (a skewness parameter with |β| < α), δ (a non-negative scale parameter), and μ (a location parameter representing the mean in symmetric cases).^[68] Specific instances of the GH distribution include the hyperbolic distribution, obtained when λ = 1. Another prominent special case is the normal inverse Gaussian (NIG) distribution, arising when λ = -1/2. The variance-gamma distribution arises when δ = 0, which simplifies to a difference of two gamma distributions and accommodates infinite activity in Lévy process representations without a Brownian motion component.^[69] These subtypes inherit the GH framework's versatility, with the NIG often used for its flexibility in modeling skewness and the variance-gamma prized for its closed-form characteristic function. Key properties of hyperbolic distributions include their heavy-tailed nature, where the tails decay exponentially or polynomially depending on α and λ, making them suitable for capturing extreme events in data with excess kurtosis.^[70] Moments exist up to order r if r < λ for the GH case, with explicit recursive formulas available for mean, variance, and higher moments as functions of the parameters, though finite moments require α > |β|.^[71] The density function for the GH distribution is given by

f(x; \lambda, \alpha, \beta, \delta, \mu) = \frac{(\delta / \alpha)^\lambda}{ \sqrt{2\pi} K_\lambda(\delta \gamma) } \exp\left( \beta (x - \mu) \right) \alpha^{\lambda - 1/2} \frac{ K_{\lambda - 1/2} \left( \alpha \sqrt{\delta^2 + (x - \mu)^2} \right) }{ \left( \delta^2 + (x - \mu)^2 \right)^{(\lambda - 1/2)/2} },

where γ = √(α² - β²) and K_ν denotes the modified Bessel function of the second kind of order ν, highlighting the distribution's analytic tractability despite its complexity. In applications, hyperbolic distributions are extensively employed in finance to model asset returns exhibiting leptokurtosis and asymmetry, such as daily stock log-returns, where they outperform normal or Student's t distributions in fitting empirical tail risks.^[72] They serve as marginal distributions for Lévy processes in option pricing and risk management, with the GH Lévy model capturing both jumps and diffusion for improved volatility forecasting.^[73] Recent advances in Bayesian inference for these models, particularly post-2018, have leveraged Markov chain Monte Carlo methods to incorporate prior information on tail parameters, enhancing parameter uncertainty quantification in high-dimensional financial datasets.^[74] Parameter estimation typically relies on maximum likelihood methods, often implemented via expectation-maximization (EM) algorithms to handle the non-closed-form likelihood, with simulations confirming convergence for sample sizes above 100 under varied initializations.^[75]

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Hyperbolic Functions -- from Wolfram MathWorld
Hyperbolic Functions ; sinhz, = (e^z-e^(-z))/2 ; = -sinh(-z) ; coshz, = (e^z+e^(-z))/2 ; = cosh(-z) ; tanhz, = (e^z-e^(-z))/(e^z+.
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Vincenzo Riccati (1707 - 1775) - Biography - MacTutor
17 January 1775. Treviso (now Italy). Summary: Vincenzo Riccati was an Italian Jesuit who worked on hyperbolic functions. Thumbnail of Vincenzo Riccati View ...
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Earliest Uses of Symbols for Trigonometric and Hyperbolic Functions
Vincenzo Riccati (1707-1775), who introduced the hyperbolic functions, used Sh. and Ch. for hyperbolic sine and cosine.Missing: definitions | Show results with:definitions
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[PDF] 4 Hyperbolic Geometry
A non-Euclidean geometry is (typically) a model satisfying most of Hilbert's axioms but for which parallels might not exist or are non-unique: There exists a ...<|control11|><|separator|>
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[PDF] Hyperbolic Geometry
It is reasonable that Gauss had an influence on Lobachevsky's thought (although Gauss had not solidified his own ideas by 1806). Lobachevsky is given some ...
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Nineteenth Century Geometry - Stanford Encyclopedia of Philosophy
Jul 26, 1999 · Gauss, Lobachevsky and Bolyai—unbeknownst to each other—coincided in calling b1 and b2 the parallels to a through Q. μ is called the angle ...
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Foundations of Geometry - KSU Math
Hyperbolic geometry is the geometry where we accept the first four axioms of Euclidean geometry but negate the fifth postulate.
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[PDF] Chapter 4: Introduction to Hyperbolic Geometry - Mathematics
Axiom 2: Any line segment may be continued indefinitely. Axiom 3: A circle of any radius and any center can be drawn. Axiom 4: Any two right angles are ...
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[PDF] Chapter 9 - Poincaré's Disk Model for Hyperbolic Geometry
means that our standard distance formula will not work. We introduce a distance metric by dρ = 2dr. 1 − r2 where ρ represents the hyperbolic distance and r ...
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The Poincaré model for the hyperbolic plane, Section 7
The second model that we use to represent the hyperbolic plane is called the Poincaré disk model, named after the great French mathematician, Henri Poincaré.
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https://www.ias.edu/sites/default/files/math/csdm/05-06/rkrauthgamer_algorithms_on_negatively_curved_spaces.pdf
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Hyperbolic Tessellations - Clark University
A regular tessellation, or tiling, is a covering of the plane by regular polygons so that the same number of polygons meet at each vertex.
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[PDF] Extremal problems and probabilistic methods in hyperbolic geometry
Aug 19, 2020 · One big difference with lower dimensional manifold is Wang's theorem, which states: Theorem 1.5.1 ([Wan72]). The number of hyperbolic n- ...
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[PDF] Geometry of hyperbolic space - Matrix Editions
The most natural – if not the easiest to visualize – is the hyperboloid model, which imitates the definition of the unit sphere. We use the symbol Hn to denote ...
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[PDF] HYPERBOLIC GROUPS - M. Gromov - IHES
Non-definition: A group r is called semi hyperbolic if it looks as if it admits a discrete cocompact isometric action on a space of non-positive curvature. ( ...
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[PDF] Relative hyperbolicity, classifying spaces, and lower algebraic K ...
For Γ a relatively hyperbolic group, we construct a model for the universal space among Γ-spaces with isotropy on the family VC of virtually cyclic ...
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[PDF] Algorithms on negatively curved spaces - Institute for Advanced Study
In this section, we give the basic definitions and proper- ties of spaces that one might consider "negatively curved.” Such spaces include classical examples ...
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Poincaré Embeddings for Learning Hierarchical Representations
May 22, 2017 · We introduce a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space -- or more precisely into an n- ...
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[PDF] ORBITAL MECHANICS FOR ENGINEERING STUDENTS - hlevkin
This textbook evolved from a formal set of notes developed over nearly ten years of teaching an introductory course in orbital mechanics for aerospace ...
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[PDF] Classical Mechanics LECTURE 20: OPEN ORBITS
20.3 Hyperbolic orbit: the angle of deflection, φ. 20.3.1 Method 1 : using ... ▻ Orbit equation : r(θ) = r0. 1+e cos θ. ▻ Ellipse : e < 1 x a. 2. + y b. 2.
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Hyperbolic Orbit - an overview | ScienceDirect Topics
A hyperbola is an open curve with a negative semimajor axis (see Fig. 3); therefore, the total energy ξ is positive and eccentricity e is greater than unity.
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Planetary Voyage - NASA Science
The following are hyperbolic elements for NASA's Voyager 1 and 2 during various legs of their travels to the outer planets of the solar system. Voyager 1 was ...Missing: escape astrodynamics
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[PDF] Relativistic hyperbolic motion and its higher order kinematic quantities
Nov 2, 2022 · The relativistic hyperbolic motion happens when a particle moves in a Minkowski spacetime with constant proper acceleration. It has been studied ...
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[PDF] The theory of the rigid electron in the kinematics of the principle of ...
(Leipzig) 30. (1909), 1-56. The theory of the rigid electron in the kinematics of the principle of relativity. By Max Born. Dedicated ...
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2. Partial Differential Equations
In this chapter, procedures will be developed for classifying partial differential equations as elliptic, parabolic or hyperbolic. The different types of ...
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[PDF] 2 First-Order Equations: Method of Characteristics
We can use ODE theory to solve the characteristic equations, then piece together these characteristic curves to form a surface. Such a surface will provide us ...
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1. Hyperbolic Partial Differential Equations - SIAM.org
Mar 23, 2012 · We begin this chapter by considering the simplest hyperbolic equation and then extend our discussion to include hyperbolic systems of equations ...
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[PDF] The one dimensional Wave Equation: D'Alembert's Solution
This decomposition is used to derive the classical D'Alembert Solution to the wave equation on the domain (−∞, ∞) with prescribed initial displacements and.
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[PDF] Partial Differential Equations with Applications to Wave Theory - KTH
This equation has a wide range of use in acoustics, elastomechanics and electro- dynamics. The field quantity (x; y; z; t) depends on the problem studied.
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On the hyperbolicity of Maxwell's equations with a local constitutive ...
Apr 25, 2011 · After splitting Maxwell's equations into evolution equations and constraints, we derive the characteristic equation and we discuss its ...
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Analytical and grid-free solutions to the Lighthill–Whitham–Richards ...
The LWR partial differential equation (PDE) is a first order scalar hyperbolic conservation law that computes the evolution of a density function (the density ...
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A Very Easy High-Order Well-Balanced Reconstruction for ...
Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact ...
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A data-driven physics-informed finite-volume scheme for ...
Jul 15, 2021 · We propose a data-driven physics-informed finite-volume scheme for the approximation of small-scale dependent shocks.
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What is Hyperbole? || Oregon State Guide to Literary Terms
Hyperbole describes the sense of over-reaching, or grasping beyond what is necessary in order to describe a certain feeling, an experience, or response.
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Hyperbole - Etymology, Origin & Meaning
Hyperbole, from Greek hyperbolē meaning "exaggeration" (literally "a throwing beyond"), originates in rhetoric as obvious exaggeration, noted by Aristotle ...
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Chapter 38: Hyperbole – Longinus, On the Sublime
Aristotle uses “hyperbole” both in a general sense meaning “excess” (Rhetoric I 463a2) and as a technical term in rhetoric (14l3a22): colloquial hyperboles ...
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Rhetoric by Aristotle - The Internet Classics Archive
The best way to counteract any exaggeration is the well-worn device by which the speaker puts in some criticism of himself; for then people feel it must be all ...
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Hyperbole ~ Definition, Examples & Meaning - BachelorPrint
Nov 10, 2023 · Hyperbole vs. meiosis vs. litotes ; Rhetorical device, Definition, Example ; Hyperbole, Deliberate and obvious exaggeration used for emphasis, I'm ...
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Examples and Definition of Hyperbole as Literary Device
Hyperbole is a figure of speech and literary device that creates heightened effect through deliberate exaggeration.
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Hyperbole in Romeo and Juliet | Analysis & Examples - Study.com
She says 'If he be married, my grave is like to be my wedding bed. ' In other words, 'I'll die if he is married. ' Not only is this hyperbole, meant to show ...
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The Role of Visual Hyperbole in Advertising Effectiveness
This study makes important distinctions among these terms and shows that hyperbolic ads produce more ad liking than nonhyperbolic ads. Subjects measuring high ...
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Rhetorical Hyperbole | The First Amendment Encyclopedia
Apr 14, 2020 · Rhetorical hyperbole is a First Amendment-based doctrine that often provides protection to exaggerated, over-the-top speech in defamation cases.
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“SO FUCKING CUTE IM DYING”: The use of hyperbole on X
Exaggeration in language. Hyperbole. 7Figurative language has received a lot of attention from scholars in different areas, such as cognitive linguistics, ...
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Epic Similes in The Iliad by Homer | Purpose & Examples - Study.com
In "The Iliad," Homer uses epic similes to make the work more engaging. They combine to create the impression of a truly universal conflict.
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12. Mapping the Hypertext: Similes in Iliad XXII
Only the extended or long similes in Iliad XXII will be examined in this study. I will not deal with 'short' similes characterized as such by a single point of ...
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Great Expectations: Hyperbole 3 key examples - LitCharts
When characters act pompously in a Dickens novel, the author is most often indicating that they shouldn't be taken seriously by the reader. Mr. Wopsle is a ...
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How Does Charles Dickens Use Of Hyperbole - 561 Words | Bartleby
Dickens uses hyperbole and imagery to convey how Pip feels about what is going on and the effect of his surroundings on him. The usage of hyperbole creates ...
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The Meaning of Figures of Speech Used in “She Walks in Beauty” by ...
The study identifies simile, metaphor, personification, alliteration, synecdoche, and hyperbole as key figures of speech utilized by Byron in 'She Walks in ...
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Lord Byron – She Walks in Beauty | Genius
sky and stars etc — in stanza one. Stanza two describes her grace and beauty in hyperbolic ...
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Magical Realism in Gabriel Garcia Marquez' s "One Hundred Years ...
Nov 2, 2020 · This dissertation attempts to conduct an empirical analysis to examine, on the one hand, the extent to which this genre has been incorporated in Gabriel Garcia ...<|separator|>
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García Márquez and Magical Realism - Oxford Academic
This article focuses on why Gabriel García Márquez's novel One Hundred Years of Solitude is the ur-magical realist text, which put magical realism on the world ...
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The role of hyperbole in conveying emotionality: the case of victim ...
Oct 10, 2024 · The presence of hyperbole significantly increased the intensity of the emotion as felt by participants themselves, and as perceived from the ...
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The impact of hyperbole on perception of victim testimony
Results from this experiment demonstrated that the use of hyperbole made testimonies more believable and made the victim seem more impacted.<|separator|>
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What is a Hyperbole? Definition & Examples - StudioBinder
Jul 30, 2024 · Hyperbole is a literary device used to draw emphasis through extreme exaggeration, with examples in film and literature.Missing: graphic | Show results with:graphic
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The Role of Hyperbole and Metaphor in Film - Ejournal Undiksha
Feb 25, 2025 · Overall, hyperbole and metaphor contribute significantly to enhancing emotional depth, humor, and audience engagement.
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[PDF] The 'Fanfic Lens': Fan Writing's Impact on Media Consumption
Mar 2, 2023 · The 'Fanfic Lens' is a set of literacy skills gained from fan communities, shaping how fanfiction authors consume media, and is a fandom- ...
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The Comedy of Errors: Hyperbole 3 key examples - LitCharts
Get everything you need to know about Hyperbole in The Comedy of Errors. Analysis, related characters, quotes, themes, and symbols.
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Modeling Population Growth: Exponential and Hyperbolic Modeling
Learn how to solve for the model's parameters in a very simple way. In this paper, we use initial values from 1960 and 2009;
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Doomsday: Friday, 13 November, A.D. 2026 | Science
Doomsday: Friday, 13 November, AD 2026: At this date human population will approach infinity if it grows as it has grown in the last two millenia.Missing: hyperbolic | Show results with:hyperbolic
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The end of hyperbolic growth in human population and CO2 emissions
The growth rate has increased from 1/(2800 years) to 1/(64 years) now. We predict a population peak in 2030 at 8.23 billion people and subsequent decline.
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[PDF] Against the singularity hypothesis | Global Priorities Institute
Thesingularity hypothesisis a radical hypothesis about the future of artificial intelligence on which self-improving artificial agents will quickly become ...
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Against the singularity hypothesis | Philosophical Studies
May 10, 2024 · One famous model from this period held that continued hyperbolic growth would lead to an infinite population by 2026 (Von Foerster et al., 1960) ...
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[PDF] GeneralizedHyperbolic: The Generalized Hyperbolic Distribution
param. Parameter vector of the hyperbolic distribution. Details. The formulae used for the mean, variance and mode are as given in Barndorff-Nielsen and Blæsild.
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[PDF] The Variance-Gamma Distribution: A Review - arXiv
Mar 9, 2023 · The VG distribution is a special case of the generalized hyperbolic (GH) distribution that is also popular in financial modelling; see, for.
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Moments of the generalized hyperbolic distribution
Dec 14, 2010 · In this paper we demonstrate a recursive method for obtaining the moments of the generalized hyperbolic distribution. The method is readily ...
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[PDF] Moments of the Generalized Hyperbolic Distribution
Dec 9, 2009 · Recently, Barndorff-Nielsen and Stelzer (2005) developed expressions for the moments (and absolute moments) of the GH distribution as the sum of ...
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Hyperbolic distributions in finance - Project Euclid
Distributional assumptions for the returns on the underlying assets play a key role in valuation theories for derivative securities.
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Application of Generalized Hyperbolic Lévy Motions to Finance
Generalized hyperbolic Lévy motions are processes which allow for an almost perfect fit to financial data. We discuss in detail what the consequences for asset ...
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Full article: Computational aspects of likelihood-based inference for ...
Dec 8, 2024 · Our interest focuses in fact on maximum likelihood estimation of the Generalized hyperbolic distribution, working in the univariate case. This ...
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[PDF] an em algorithm for maximum likelihood estimation of barndorff ...
We have derived an EM algorithm to compute Maximum Likeli- hood parameter estimates for Barndorff-Nielsen's flexible, multi- variate Generalized Hyperbolic ...