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Class of mathematical functions
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex
Subharmonic_function
Type of function in complex analysis
Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Riemannian
Plurisubharmonic_function
Topology in the study of subharmonic functions
natural topology for setting the study of subharmonic functions. In the earliest studies of subharmonic functions, namely those for which Δ u ≥ 0 , {\displaystyle
Fine topology (potential theory)
Fine_topology_(potential_theory)
Type of mathematical functions
condition than a holomorphically convex. The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex domain
Function of several complex variables
Function_of_several_complex_variables
Model in probability theory
potential theory, a subharmonic function f {\displaystyle f} satisfies Δ f ≥ 0 {\displaystyle \Delta f\geq 0} . Any subharmonic function bounded above by
Martingale (probability theory)
Martingale_(probability_theory)
Class of numerical techniques
discrete Laplace operator. Similar to continuous subharmonic functions one can define subharmonic functions for finite-difference approximations u h {\displaystyle
Finite_difference_method
Functions in mathematics
hold, although other properties of harmonic functions may fail. More generally, a function is subharmonic if and only if, in the interior of any ball
Harmonic_function
of the ball (mean value property). Also subharmonic function and superharmonic function. Elementary function: composition of arithmetic operations, exponentials
List_of_types_of_functions
One-dimensional complex manifold
whether function spaces other than the negative subharmonic functions are degenerate, e.g. Riemann surfaces on which all bounded holomorphic functions are
Riemann_surface
Property of functions which is weaker than continuity
used in the proof of the Choquet theorem. Similar ideas applied to subharmonic functions are used in the Perron method for solving the Dirichlet problem
Semi-continuity
Mathematical function that outputs real values
sets), convex functions (on vector and affine spaces), harmonic and subharmonic functions (on Riemannian manifolds), analytic functions (usually of one
Real-valued_function
Harmonic functions as solutions to Laplace's equation
Bloch space, Bergman space and Sobolev space. Subharmonic function – Class of mathematical functions Kellogg's theorem Garabedian, P. R.; Schiffer, M
Potential_theory
Mathematical technique
the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar
Perron_method
Potential in mathematics
interest in potential theory because Iαμ is then a (continuous) subharmonic function off the support of μ, and is lower semicontinuous on all of Rn. Consideration
Riesz_potential
Area of mathematical analysis
function. Maximal functions are used to control pointwise convergence, differentiation of integrals, and boundary limits of harmonic or subharmonic functions
Harmonic_analysis
Chinese-American mathematician (born 1949)
rigidity results for functions on complete Riemannian manifolds. A particularly famous result of his says that a subharmonic function cannot be both positive
Shing-Tung_Yau
Hungarian mathematician (1895–1965)
in which he gave a solution to Plateau's problem, and in 1935, "Subharmonic Functions". His work focused on computer science in the last decade of his
Tibor_Radó
{\displaystyle n\geq 2} ) is a polar set if there is a non-constant subharmonic function u {\displaystyle u} on R n {\displaystyle \mathbb {R} ^{n}} such
Polar_set_(potential_theory)
Russian mathematician (1891–1941)
1934, he studied subharmonic functions, building on the work of Riesz. Samary Aleksandrovich Galpern. I. I. Privalov, Subharmonic Functions, GITTL, Moscow
Ivan_Privalov
Second-order partial differential equation
method, which constructs a candidate solution as the supremum of all subharmonic functions lying below the prescribed boundary data. The resulting Perron solution
Laplace's_equation
Concept within complex analysis
kind of "complex convexity" remains, namely the fact that z → |z|q is subharmonic for every q > 0. As a consequence, if F ( z ) = ∑ n = 0 + ∞ c n z n
Hardy_space
Ukrainian-American mathematician
State University in 1979 (Asymptotic Properties of Meromorphic and Subharmonic Functions), and is currently a distinguished professor at Purdue University
Alexandre_Eremenko
harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and superharmonic
Kelvin_transform
British mathematician (1926–2020)
Research Problems in Function Theory, London: Athlone Press, pp. vii+56. Hayman, W. K.; Kennedy, P. B. (1976), Subharmonic functions. Volume 1, London Mathematical
Walter_Hayman
Ohio State University; published: "On the Problem of Plateau", "Subharmonic Functions", in the Bell System Technical Journal the Busy Beaver problem received
List of University of Szeged people
List_of_University_of_Szeged_people
American mathematician
martingales correspond to harmonic functions, supermartingales to superharmonic functions, and submartingales to subharmonic functions. Quoted from Snell's Obituary
J._Laurie_Snell
Problem of solving a partial differential equation subject to prescribed boundary values
using the Perron method, which relies on the maximum principle for subharmonic functions. This approach is described in many text books. It is not well-suited
Dirichlet_problem
Theorem in quantum information theory
Ortega-Cerda', Joaquim; Tilli, Paolo (2022). "A monotonicity theorem for subharmonic functions on manifolds". arXiv:2212.14008 [math.CA]. Video of a lecture by
Lieb_conjecture
Theorem about entire functions
entire functions that admit a certain estimate from below". Soviet Math. Dokl. 1: 548–552. Kheyfits, A.I. (2013). "Growth of Schrödingerian Subharmonic Functions
Matsaev's_theorem
Mathematics theorem
the function f r ( z ) = f ( r z ) . {\displaystyle f_{r}(z)=f(rz).} The inequalities can also be deduced, following Riesz (1925), using subharmonic functions
Littlewood subordination theorem
Littlewood_subordination_theorem
{R} ^{n}} , the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous. Finer topology If X is a set,
Glossary_of_general_topology
Probability measure on a complex plane
(\log \left|A-\lambda I\right|),\;\lambda \in \mathbb {C} ,} is a subharmonic function and its Laplacian in the distributional sense is a probability measure
Brown_measure
Musical interval
93. Retrieved 14 April 2011. Boswell, George W. "The Neutral Tone as a Function of Folk-Song Text", Yearbook of the International Folk Music Council, vol
Neutral_third
Differential operator in mathematics
continuously differentiable function u {\displaystyle u} is called harmonic if Δ u = 0 {\displaystyle \Delta u=0} , subharmonic if Δ u ≥ 0 {\displaystyle
Laplace_operator
Relates the integral of Gaussian curvature of surfaces to the Euler characteristic
0789.01. MR 1556908. Zbl 0011.22501. Huber, Alfred (1957). "On subharmonic functions and differential geometry in the large". Commentarii Mathematici
Cohn-Vossen's_inequality
Greek mathematician (1908–1974)
important contribution was his work in function theory, in particular Nevanlinna theory and the growth of subharmonic functions. Vorlesungen über Funktionentheorie
Alexander_Dinghas
Class of continuous maps between Riemannian manifolds of the same dimension
shown by David Drasin and Pekka Pankka. If f is an analytic function, then log |f| is subharmonic, and harmonic away from the zeros of f. The corresponding
Quasiregular_map
Basic musical interval
against an E♮ in the bass. Here E♭ was preferred to a D♯ to make the tone's function clear as part of an F dominant seventh chord, and the augmented unison
Semitone
Concept in mathematics
noncompact M by making use of Yau's theorem asserting that nonnegative subharmonic functions which are L2-bounded must be constant. In summary, according to
Harmonic_map
American mathematician
Peter; Schoen, Richard (1984). "Lp and mean value properties of subharmonic functions on Riemannian manifolds". Acta Mathematica. 153 (3–4): 279–301.
Peter_Li_(mathematician)
American mathematician
1090/s0002-9904-1946-08590-0. MR 0016469. ——— (1948). "Entire Functions with Bounded Minimum Modulus; Subharmonic Function Analogues". Annals of Mathematics. 49 (1): 200–213
Maurice_Heins
understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and
Complex_convexity
Wave where gravity is the main restoring force
waves. Alternatively, so-called infragravity waves, which are due to subharmonic nonlinear wave interaction with the wind waves, have periods longer than
Gravity_wave
Green's function for Laplacian
potential is subharmonic on R d {\displaystyle \mathbb {R} ^{d}} . If f {\displaystyle f} is a compactly supported continuous function (or, more generally
Newtonian_potential
Mathematical seminars held in Paris since 1948
des fonctions analytiques et sous-harmoniques (complex analysis, subharmonic functions) Charles Ehresmann, Les connexions infinitésimales dans un espace
Séminaire_Nicolas_Bourbaki
subderivative subderivative. subharmonic A twice continuously differentiable function f {\displaystyle f} is said to be subharmonic if Δ f ≥ 0 {\displaystyle
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Musical interval
one-third of a diatonic semitone and one-fifth of a whole tone, so it may function as a quarter tone, a fifth-tone or a sixth-tone. In just intonation the
Quarter_tone
Gauthier-Villars Paris: 1069–1071. Beckenbach, E. F.; Rado, T. (1933). "Subharmonic Functions and Surfaces of Negative Curvature". Transactions of the American
Cartan–Hadamard_conjecture
Classic entropy of a quantum-mechanical density matrix
Kulikov, A.; Nicola, F.; Ortega-Cerda', J.; Tilli, P. (2022). "A monotonicity theorem for subharmonic functions on manifolds". arXiv:2212.14008 [math.CA].
Wehrl_entropy
American writer
Times, November 3, 2009. Cunzer, Ela-Chaim (1937). On convex and subharmonic functions (Catalog entry for translated manuscript). Translated by Wlordarski
Esther_Hautzig
Convex and subharmonic functions Robert Creighton Buck University of Wisconsin Algebraic and topological properties of linear operators on function spaces
List of Guggenheim Fellowships awarded in 1958
List_of_Guggenheim_Fellowships_awarded_in_1958
Wave with frequency an integer multiple of the fundamental frequency
perfect harmonicPages displaying short descriptions of redirect targets Subharmonic – Having a frequency that is a fraction of a fundamental frequency Xenharmonic
Harmonic
Irish mathematician and chess player
a conjecture of Heins, which concerned a conjecture of Heins on subharmonic functions and gives positive results. That same year he was appointed as an
Patrick_Brendan_Kennedy
Romanian Swedish mathematician
2, 525–568. Borcea, Julius; Bøgvad, Rikard, Piecewise harmonic subharmonic functions and positive Cauchy transforms. Pacific J. Math. 240 (2009), no
Julius_Borcea
Scattering process at the normal-metal-superconductor interface
4515. Octavio, M; Tinkham, M.; Blonder, G. E.; Klapwijk, T. M. (1983). "Subharmonic energy-gap structure in superconducting constrictions". Phys. Rev. B
Andreev_reflection
Electronic circuit
magnetic field. The step recovery diode impulse generator is driven at a subharmonic of the desired output frequency. An electromagnet then tunes the YIG
Frequency_multiplier
Guitar where the bridge extends beyond its usual stop
When played at one part of a string, the opposed part can resonate in a subharmonic of the struck part, depending on a predictable mathematical ratio of
3rd_bridge
Israeli-American mathematician (born 1944)
inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions", Commentarii Mathematici Helvetici 51, no. 1 (1976): 133–161. doi:10
Shmuel_Friedland
Sinusoidal wave whose frequency is an integer multiple
interharmonic with a frequency less than the fundamental is called a subharmonic. The main sources of interharmonics are cycloconverters and arcing loads
Harmonics_(electrical_power)
Israeli professional audio company
Channel Andrew Scheps signature mixing channel strip 2018 Submarine Subharmonic frequency generator 2019 Bass Fingers Virtual instrument plugin 2019
Waves_Audio
Vocal practice
singing. Undertone singing, which involves techniques that comprise subharmonics. It is generated by the combined vibrations of parts of the singing apparatus
Throat_singing
Mathematical technique in complex analysis
fashion to subharmonic and superharmonic functions. To continue the example above, we can impose a growth condition on a holomorphic function f {\displaystyle
Phragmén–Lindelöf_principle
English polymath (1773–1829)
(x) of a body subject to a known load (F), where the constant (k) is a function of both the geometry and material under consideration. Finding k required
Thomas_Young_(scientist)
Audio signal processing technique
guitar, electric bass, or electronic keyboards Spectral band replication Subharmonic enhancer "Aphex 204 User manual – P/n 999–4140, Revision2 Released 09/01/2001"
Exciter_(effect)
Field of electrical engineering
highly complex behaviors including bifurcations, chaos, harmonics, and subharmonics which cannot be produced or analyzed using linear methods. Polynomial
Signal_processing
Class of problems in classical mechanics
inverse of an integer, such as 1⁄3, the second orbit is said to be a subharmonic of the first orbit. The classical central-force problem was solved geometrically
Classical central-force problem
Classical_central-force_problem
Ultrasound imaging method
below. Subharmonic filtering: This works by filtering out all signals but the subharmonic signals. Since tissue generally does not have a subharmonic response
Acoustic_angiography
wave Standing wave ratio Stefan–Boltzmann law Stokes drift Stokes wave Subharmonic Super low frequency Superharmonic Superposition principle Supersonic
Index_of_wave_articles
Instrument that uses electronic circuits to make sound
Greenwood. The Trautonium was invented in 1928. It was based on the subharmonic scale, and the resulting sounds were often used to emulate bell or gong
Electronic_musical_instrument
Identification of nonlinear systems
including systems with exotic behaviours such as chaos, bifurcations, and subharmonics. While NARMAX started as the name of a model it has now developed into
Nonlinear system identification
Nonlinear_system_identification
Theorem in classical mechanics
force depends on the initial velocity of the particle. Harmonic and subharmonic orbits are special types of such closed orbits. A closed trajectory is
Newton's theorem of revolving orbits
Newton's_theorem_of_revolving_orbits
Phenomenon in maths
systems is not known.[citation needed] The circle map also exhibits subharmonic routes to chaos, that is, period doubling of the form 3, 6, 12, 24,.
Arnold_tongue
Theory of the mechanism of hearing
Theory of Hearing. Groups of neurons in the cochlea individually fire at subharmonic frequencies of a sound being heard and collectively phase-lock to match
Volley_theory
In 1957 Karlheinz Stockhausen described this additive series as "a subharmonic proportional series" which, "compared to a scale constructed of chromatic
Duration_series
Non-linear second order differential equation and its attractor
and phase portraits of the Duffing equation, showing the appearance of subharmonics through period-doubling bifurcation – as well chaotic behavior – are
Duffing_equation
Quality of a musical note or sound or tone
×3, ×4, etc. Partials are other overtones. There are also sometimes subharmonics at whole number divisions of the fundamental frequency. Most instruments
Timbre
Non-linear effect in amplitude modulation
using overdriven amplifiers or effects pedals to produce new tones at subharmonics of the tones being played on the instrument. See Power chord#Analysis
Intermodulation
Nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth
scales smaller than the wavelength λ {\displaystyle \lambda } ) and subharmonics (for perturbations at the spatial scales larger than λ {\displaystyle
Stokes_wave
nature and boundary conditions of each mechanical system. Additionally, subharmonics, superharmonics or subsuperharmonics of each mode can also be excited
Mechanical_amplifier
Ability of an imaging system to resolve detail
resulting measure is the contrast transfer function (CTF) and not the MTF. The difference arises from the subharmonics of the square waves and can be easily
Optical_resolution
Former Northwestern University professor, currently a sculptor
1973 monograph: The Auditory Periphery. [2.2] Discovery of fractional subharmonics in cochlear mechanics, including the first report on chaotic behavior
Peter_Dallos
Mathematical conjecture
doi:10.2140/gt.2003.7.713. S2CID 2140632. Hingston, Nancy (2009). "Subharmonic solutions of Hamiltonian equations on tori". Annals of Mathematics. 170
Conley_conjecture
Musical term
Not harmonic Untuned percussion Anharmonicity Dissonance Pseudo-octave Subharmonic How harmonic are harmonics? by Joe Wolfe, accessed 29 June 2008 The Indian
Inharmonicity
Acoustic phenomenon
may benefit from augmented bass harmonics processing. Psychoacoustics Subharmonic Howard, David M.; Angus, J. A. S. (2017). Acoustics and Psychoacoustics
Missing_fundamental
Circuit that creates new frequencies from two signals
they mean switching mixers. Electronics portal Frequency multiplier Subharmonic mixer Product detector Pentagrid converter Beam deflection tube Ring
Frequency_mixer
Analogue modular synthesizer
Positive Slews, were able to function as envelope followers, crude low pass filters, modulating waveforms, subharmonic generators, and audio oscillators
Serge_synthesizer
Study of the propagation of sound in water
As a consequence for a sinusoidal wave input additional harmonic and subharmonic frequencies are generated. When two sinusoidal waves are input, sum and
Underwater_acoustics
Species of amphibian
convey arousal. O. graminea can produce four types of NLP components: subharmonics, deterministic chaos, frequency jumps, or biphonations. Most vocalizations
Odorrana_graminea
introduce the idea of actively perturbing the free shear layer with subharmonics of its Kelvin-Helmholtz instability frequency for increasing the entrainment
Chih-Ming_Ho
American composer (1901–1974)
whose pitch classes are the harmonics or subharmonics of a given fixed tone. These six-tone chords function in Partch's music much the same that the three-tone
Harry_Partch
State of a physical system in phase space
crystals are closely related but different concepts. They both study subharmonic modes that emerge in periodically driven systems. Time crystals focus
Phase_space_crystal
scale Subcircuit board Subcooled liquid Subcooling Subcritical reactor Subharmonic Subhelic arc Subir Sachdev Sublimation (Physics) Sublimation (phase transition)
Index_of_physics_articles_(S)
SUBHARMONIC FUNCTION
SUBHARMONIC FUNCTION
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Biblical
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Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, the son of the functionary Heknofre.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, Functionary of the Interior.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Egyptian
, a great functionary.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
SUBHARMONIC FUNCTION
SUBHARMONIC FUNCTION
Girl/Female
Hindu, Indian, Tamil
Dirty Stunted Grass
Female
English
Variant spelling of English Candy, CANDI means either "candy" the sweet, or "prince of servants."
Boy/Male
Indian
World, Universe
Female
English
 Anglicized form of Scottish Jennet, JANET means "God is gracious."
Boy/Male
Tamil
Daughter
Girl/Female
American, Australian, Christian, French
Lady; Form of Donna; Combination of the Popular Prefix La with Donna; World Ruler
Boy/Male
Indian
The Creater of Vedas Lord Brahma
Surname or Lastname
English, Scottish, and Irish
English, Scottish, and Irish : variant spelling of Hyland.Possibly an Americanized spelling of German Heiland.
Boy/Male
Indian
King of the Earth
Boy/Male
Indian
King, Hope
SUBHARMONIC FUNCTION
SUBHARMONIC FUNCTION
SUBHARMONIC FUNCTION
SUBHARMONIC FUNCTION
SUBHARMONIC FUNCTION
adv.
In a functional manner; as regards normal or appropriate activity.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
pl.
of Functionary
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
v. i.
Alt. of Functionate
a.
Having relation to growth or nutrition; partaking of simple growth and enlargement of the systems of nutrition, apart from the sensorial or distinctively animal functions; vegetal.
n.
One deputed or authorized to perform the functions of another; a substitute in office; a deputy.
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
n.
The doctrine that all the functions of a living organism are due to an unknown vital principle distinct from all chemical and physical forces.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
a.
Belonging or relating to life, either animal or vegetable; as, vital energies; vital functions; vital actions.
a.
Pertaining to the function of an organ or part, or to the functions in general.
a.
Destitute of function, or of an appropriate organ. Darwin.
a.
Of, pertaining to, or designating, certain secret tribunals which flourished in Germany from the end of the 12th century to the middle of the 16th, usurping many of the functions of the government which were too weak to maintain law and order, and inspiring dread in all who came within their jurisdiction.
n.
Fig.: Any cavity, or hollow place, in which any function may be conceived of as operating.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
a.
Producing mathematically perfect harmony or concord; sweetly or perfectly harmonious.
a.
Pertaining to, or connected with, a function or duty; official.
prep.
Acting as a substitute; -- said of abnormal action which replaces a suppressed normal function; as, vicarious hemorrhage replacing menstruation.
v. t.
To assign to some function or office.