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Functions in mathematics
distributions). A weakly harmonic function coincides almost everywhere with a strongly harmonic function, and is in particular smooth. A weakly harmonic distribution
Harmonic_function
that a function is weakly harmonic if and only if it is harmonic. Thus weakly harmonic is actually equivalent to the seemingly stronger harmonic condition
Weakly_harmonic_function
Second-order partial differential equation
interpreted in a weak sense. A function u ∈ H l o c 1 ( Ω ) {\displaystyle u\in H_{\mathrm {loc} }^{1}(\Omega )} is called weakly harmonic if ∫ Ω ∇ u ⋅ ∇
Laplace's_equation
Area of mathematical analysis
Harmonic analysis is an area of mathematical analysis that emerged from the study of harmonic functions, and especially their boundary behavior. The methods
Harmonic_analysis
In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure
Positive_harmonic_function
Concept in mathematics
the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions. Informally, the
Harmonic_map
Mathematical function
the M-Whittaker function are weak Maass forms. When the spectral parameter is specialized to the harmonic point they lead to harmonic Maass forms. The
Harmonic_Maass_form
Complex-differentiable part of a Maass wave function
a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were
Mock_modular_form
Mathematical method in electrical engineering
Harmonic balance is a method used to calculate the steady-state response of nonlinear differential equations, and is mostly applied to nonlinear electrical
Harmonic_balance
Function that "converges" to periodicity
finite-dimensional vector space. A function on a locally compact group is called weakly almost periodic if its orbit is weakly relatively compact in L ∞ {\displaystyle
Almost_periodic_function
exponential functions Inverse function Convex function, Concave function Singular function Harmonic function Weakly harmonic function Proper convex function Rational
List_of_real_analysis_topics
defined on an open subset U of M, is harmonic if each individual coordinate function xi is a harmonic function on U. That is, one requires that Δ g x
Harmonic_coordinates
real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps, namely
Harmonic_morphism
Theorem in complex analysis
maximum principle if they achieve their maxima at the boundary of D. Harmonic functions and, more generally, solutions of elliptic partial differential equations
Maximum_principle
Maximal functions appear in many forms in harmonic analysis (an area of mathematics). One of the most important of these is the Hardy–Littlewood maximal
Maximal_function
End of a musical phrase with resolution
partial resolution, especially in music of the 16th century onwards. A harmonic cadence is a progression of two or more chords that concludes a phrase
Cadence
Inequality for Harmonic Functions
Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). Harnack's inequality
Harnack's_inequality
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
Type of vector space in math
any orthonormal sequence {fn} converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence {xn} is bounded, by the
Hilbert_space
Branch of mathematics
spaces, measure spaces, and function spaces. Its major areas include complex analysis, functional analysis, measure theory, harmonic analysis, and the theory
Mathematical_analysis
Nonlinear optical process
Second-harmonic generation (SHG), also known as frequency doubling, is the lowest-order wave-wave nonlinear interaction that occurs in various systems
Second-harmonic_generation
Society, second edition, 2002. Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to
Selberg_zeta_function
Mathematical transform that expresses a function of time as a function of frequency
still a continuous function of frequency ( ξ {\displaystyle \xi } or ω {\displaystyle \omega } ). When the sinusoids are harmonically related (i.e. when
Fourier_transform
Mathematical operator in real and harmonic analysis
non-linear operator used in real analysis and harmonic analysis. The operator takes a locally integrable function f : R d → C {\displaystyle f:\mathbb {R}
Hardy–Littlewood maximal function
Hardy–Littlewood_maximal_function
Fish that can generate electric fields
species and by function. Electric fish have evolved many specialised behaviours. The predatory African sharptooth catfish eavesdrops on its weakly electric
Electric_fish
Mathematical form
In potential theory (the study of harmonic functions) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar
Dirichlet_form
Complex complementary error function
permittivity functions of amorphous oxides have resonances (due to phonons) that are sometimes too complicated to fit using simple harmonic oscillators
Faddeeva_function
Function spaces generalizing finite-dimensional p norm spaces
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes
Lp_space
Functions in harmonic analysis mathematics
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly
Singular_integral
Measure of the shape of a function
is a sequence μ n ′ {\displaystyle {\mu _{n}}'} that weakly converges to a distribution function μ {\displaystyle \mu } having α k {\displaystyle \alpha
Moment_(mathematics)
Musical interval
The harmonic seventh interval (also known as the septimal minor seventh, or subminor seventh) is one with an exact 7:4 ratio (about 969 cents). This is
Harmonic_seventh
Mathematical function with no sudden changes
a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies
Continuous_function
Type of topological space in mathematics
with weaker notions of locally compact. Every closed set in a weakly locally compact space (= condition (1) in the definitions above) is weakly locally
Locally_compact_space
Characteristic property of holomorphic functions
That is, u is a harmonic function. This means that the divergence of the gradient is zero, and so the fluid is incompressible. The function v also satisfies
Cauchy–Riemann_equations
Decomposition of periodic functions
Fourier series for a square wave. As more harmonics are added, the partial sums converge to the square wave. Function s 6 ( x ) {\displaystyle s_{6}(x)} (in
Fourier_series
Method for constructing existence proofs and calculating solutions in variational calculus
J} is bounded, and J {\displaystyle J} is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence u n → u 0 {\displaystyle
Direct method in the calculus of variations
Direct_method_in_the_calculus_of_variations
Integral transform and linear operator
{y}{\pi \,\left(x^{2}+y^{2}\right)}}} Furthermore, there is a unique harmonic function v defined in the upper half-plane such that F(z) = u(z) + i v(z) is
Hilbert_transform
with Harmonic Responses, Mechanical Systems and Signal Processing, 20(8), 1883–1904, (2006) Electrical Engineering Encyclopedia: Describing Functions
Describing_function
Theorem in mathematics
mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if
Inverse_function_theorem
Model in probability theory
subharmonic function f {\displaystyle f} satisfies Δ f ≥ 0 {\displaystyle \Delta f\geq 0} . Any subharmonic function bounded above by a harmonic function for
Martingale (probability theory)
Martingale_(probability_theory)
Type of spring
torsional harmonic oscillators that can oscillate with a rotational motion about the axis of the torsion spring, clockwise and counterclockwise, in harmonic motion
Torsion_spring
Electrical resistance attributed to contacting interfaces
and adsorbed water molecules, which lead to capacitor-type junctions at weakly contacting asperities and resistor type contacts at strongly contacting
Contact_resistance
Theorem of Fourier transforms of Borel measures
More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact
Bochner's_theorem
Laser science process
generation in the perturbative (weak field) regime is characterised by rapidly decreasing efficiency with increasing harmonic order. This behaviour can be
High_harmonic_generation
Fourier transform of the probability density function
and characteristic functions is sequentially continuous. That is, whenever a sequence of distribution functions Fj(x) converges (weakly) to some distribution
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Instantaneous rate of change (mathematics)
quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input
Derivative
available earlier. Non-tangential boundary values of an analytic or harmonic function exist at almost all boundary points of non-tangential boundedness
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
American mathematician (1943–2024)
Eells, who with Joseph Sampson had recently published a paper introducing harmonic map heat flow. Hamilton was inspired to formulate a version of Eells and
Richard_S._Hamilton
\Phi } is weakly(resp. strongly) continuous, then clearly so is F {\displaystyle F} . On the other hand, consider now a positive-definite function F {\displaystyle
Positive-definite function on a group
Positive-definite_function_on_a_group
Conditions for switching order of integration in calculus
integral (however, it is true if the function is continuous on the rectangle; in multivariable calculus, this weaker result is sometimes also called Fubini's
Fubini's_theorem
Description of a quantum-mechanical system
its energy is called the zero-point energy, and the wave function is a Gaussian. The harmonic oscillator, like the particle in a box, illustrates the generic
Schrödinger_equation
Electronic circuit
an even function to generate even harmonics or an odd function for odd harmonics. See Even and odd functions#Harmonics. A full wave rectifier, for example
Frequency_multiplier
real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point
Hopf_lemma
Concept within complex analysis
around. Given a function f ~ ∈ L p ( T ) {\displaystyle {\tilde {f}}\in L^{p}(\mathbf {T} )} , with p ≥ 1, one can regain a (harmonic) function f on the unit
Hardy_space
Normed vector space that is complete
the weak*-topology of the bidual. The Banach space X {\displaystyle X} is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent
Banach_space
Integral expressing the amount of overlap of one function as it is shifted over another
a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle f*g} , as the
Convolution
curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform
Singular integral operators on closed curves
Singular_integral_operators_on_closed_curves
Differential calculus on function spaces
J\geq 0} there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema, depending on
Calculus_of_variations
Certain vector fields are the sum of an irrotational and a solenoidal vector field
133–140. Sheldon Axler, Paul Bourdon, Wade Ramey "Bounded Harmonic FunctionsHarmonic Function Theory (= Graduate Texts in Mathematics 137). Springer, New
Helmholtz_decomposition
Branch of mathematics
of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the
Fourier_analysis
harmonic analysis, where one studies harmonic functions: the kernel of the Laplace operator. This leads to such functions as the spherical harmonics,
Maps_of_manifolds
Study of rates of change
are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input equals
Differential_calculus
Mathematics of real numbers and real functions
(how well a function is represented by a series or partial sum), harmonic analysis (what can be inferred about the regularity of the function from its representations)
Real_analysis
Difference in pitch between two notes
sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord. In
Interval_(music)
Type of differential equation
solid is a harmonic function. It is usually a matter of straightforward computation to check whether or not a given function is harmonic. For instance
Partial_differential_equation
Foundational principle in quantum physics
the context of harmonic analysis the uncertainty principle implies that one cannot at the same time localize the value of a function and its Fourier
Uncertainty_principle
Relationship between derivatives and integrals
differentiating a function (calculating its slopes, or rate of change at every point on its domain) with the concept of integrating a function (calculating
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Conjecture on zeros of the zeta function
48 (5): 89–155, MR 0020594 Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to
Riemann_hypothesis
Number taken as representative of a list of numbers
+f(x_{n})\right]\right)} where f is any invertible function. The harmonic mean is an example of this using f(x) = 1/x, and the geometric mean
Average
Specific figure of merit in electronics
assumption of a weakly nonlinear system, meaning that higher-order nonlinear terms are small enough to be negligible. In practice, the weakly nonlinear assumption
Third-order_intercept_point
Branch of mathematics that studies dynamical systems
applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups)
Ergodic_theory
Mathematical measure space associated to a random walk
(X_{t})_{*}\nu } almost surely weakly converges to a Dirac mass. Let f {\displaystyle f} be a μ {\displaystyle \mu } -harmonic function on G {\displaystyle G}
Poisson_boundary
Electronic control system
in 1921 in the Shortt-Synchronome clock. Spontaneous synchronization of weakly coupled pendulum clocks was noted by the Dutch physicist Christiaan Huygens
Phase-locked_loop
Type of musical note
song that is not part of the implied or expressed chord set out by the harmonic framework. In contrast, a chord tone is a note that is a part of the functional
Nonchord_tone
Mathematical equation
{\displaystyle \Omega } . This result implies the interior regularity of harmonic functions in Ω {\displaystyle \Omega } , but it does not say anything about
Weyl's lemma (Laplace equation)
Weyl's_lemma_(Laplace_equation)
Theoretical framework in physics
the promotion of a classical harmonic oscillator to a quantum harmonic oscillator. The displacement of a classical harmonic oscillator is described by x
Quantum_field_theory
Mathematical theorem
fact that exchanging the order of partial derivatives of a multivariate function f ( x 1 , x 2 , … , x n ) {\displaystyle f\left(x_{1},\,x_{2},\,\ldots
Symmetry of second derivatives
Symmetry_of_second_derivatives
Fundamental theorem in probability theory and statistics
random variables X1, X2, ... ∈ L2(Ω) be such that Xn → 0 weakly in L2(Ω) and X n → 1 weakly in L1(Ω). Then there exist integers n1 < n2 < ⋯ such that
Central_limit_theorem
Harmonic device in Western music
functions are the secondary mediant, the secondary submediant, and the secondary subtonic. Barbershop seventh chord – Major triad plus the harmonic seventh
Secondary_chord
Subgenre of jazz music developed in the U.S. in mid-1940s
substitute chords—along with virtuosic improvisation based on a combination of harmonic structure, scales, and occasional references to the melody. Bebop developed
Bebop
mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Theorem in differential geometry
each Busemann function is in fact (weakly) a harmonic function. Weyl's lemma implies the infinite differentiability of the Busemann functions. Then, the
Splitting_theorem
Mathematical function for the probability a given outcome occurs in an experiment
probability distributions ( P n ) {\displaystyle (P_{n})} is said to converge weakly (or in distribution) to a probability distribution P {\displaystyle P} if
Probability_distribution
Australian and American mathematician (born 1975)
for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory. He is a professor of mathematics
Terence_Tao
Bimodal function
condition for n = 1, 2.) A function is negative semi-definite if the inequality is reversed. A function is definite if the weak inequality is replaced with
Positive-definite_function
Type of curve in geometry
"Selberg zeta function". Zeta Functions of Graphs: A Stroll through the Garden. Cambridge: Cambridge University Press. Selberg, Atle (1956). "Harmonic analysis
Prime_geodesic
Representation theory
JSTOR 2041084 Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
(pseudo-)Riemannian manifold whose geodesics are reversible
definition of symmetric space to that of weakly symmetric Riemannian space, or in current terminology weakly symmetric space. These are defined as Riemannian
Symmetric_space
Set of statistical processes for estimating the relationships among variables
What is multiple regression used for? – Multiple regression Regression of Weakly Correlated Data – how linear regression mistakes can appear when Y-range
Regression_analysis
Stock phrases in 18th century musical style
tonic function to dominant function "opening" of the harmonic progression in the first pair of events, and then a dominant function to tonic function "closing"
Galant_Schemata
German mathematician
Ono Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms, Arxiv Preprint 2011 Adriana Salerno, "Road to Partition:
Jan_Hendrik_Bruinier
Geometry notion in mathematics
spherical functions in harmonic analysis, known for symmetric spaces, has not yet been developed. Akhiezer, D. N.; Vinberg, E. B. (1999), "Weakly symmetric
Weakly_symmetric_space
Norwegian mathematician (1917–2007)
MR 0067143. Zbl 0057.28502. Selberg, A. (1956). "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to
Atle_Selberg
On when a family of real, continuous functions has a uniformly convergent subsequence
equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators
Arzelà–Ascoli_theorem
Some remarkable congruences for the partition function
(2013). "Algebraic Formulas for the Coefficients of Half-Integral Weight Harmonic Weak Maas Forms" (PDF). Advances in Mathematics. 246: 198–219. arXiv:1104
Ramanujan's_congruences
Probability distribution
{\displaystyle H(N,\alpha -1)} is the generalized harmonic number. This makes Zipf's probability density function derivable from Pareto's. f ( x ) = α x m α
Pareto_distribution
Mathematical function conceived as a crude model
An artificial neuron is a mathematical function conceived as a model of a biological neuron in a neural network. The artificial neuron is the elementary
Artificial_neuron
Set of points linked through the evolution function of a dynamical system
quadratic polynomial. It tends to weakly attracting fixed point with multiplier=0.99993612384259 Critical orbit tends to weakly attracting point. One can see
Orbit_(dynamics)
Concept in the philosophy of mind
processes weakly embodied and bodily processes weakly embodied and extrabodily processes The first and third claims signify a strong and a weak reading
4E_cognition
Spectroscopic technique
_{zx}^{2})\right]}}} In the double-harmonic approximations, the potential energy is expanded to the second order near equilibrium (harmonic force fields), while polarizability
Raman_spectroscopy
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
Girl/Female
American, Australian, British, Christian, English, French, Greek, Latin
A State of Order or Agreement; Unity; Concord; Harmony; Agreement
Female
English
English name derived from the vocabulary word harmony, from Greek Harmonia, HARMONY means "concord, harmony."
Girl/Female
Christian & English(British/American/Australian)
Harmony
Surname or Lastname
English
English : variant spelling of Weekley.
Male
English
English surname transferred to forename use, from the German personal name Harman, HARMON means "bold/hardy man."
Girl/Female
Hindu
Pearl Pearly just similar to Pearl
Girl/Female
American, Australian, British, Chinese, Christian, English, French, Greek, Latin
A State of Order or Agreement; A Beautiful Blending; Agreement; Concord; Musical Combination of Chords; Harmony; Joining
Surname or Lastname
English
English : variant of Weekley.
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Surname or Lastname
English
English : variant spelling of Weekley.
Girl/Female
Latin American
Concord.
Female
English
Variant spelling of English Harmony, HARMONIE means "concord, harmony."
Surname or Lastname
English (Somerset)
English (Somerset) : unexplained. Compare Lukey.
Female
Greek
(ΑÏμονία) Greek name HARMONIA means "concord, harmony." In mythology, this is the name of the daughter of Ares and Aphrodite. Her Latin name is Concordia.
Surname or Lastname
Irish (mainly County Louth)
Irish (mainly County Louth) : generally of English origin (see 1); but sometimes also used as a variant of Harman or Hardiman, i.e. an Anglicized form of Gaelic Ó hArgadáin (see Hargadon).English : variant spelling of Harman 1.
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Surname or Lastname
English
English : variant of Wakeley.
Girl/Female
American, British, English, Greek, Latin
A State of Order or Agreement; Unity; Concord; Musically in Tune; A Tuneful Sound
Surname or Lastname
English
English : habitational name from a place in Northamptonshire called Weekley, from Old English wīc ‘settlement’, perhaps in this case a Roman settlement, Latin vicus + lēah ‘wood’, ‘clearing’.
Girl/Female
Greek Latin
Daughter of Ares.
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
Girl/Female
Hindu
A tree
Boy/Male
Indian, Punjabi, Sikh
Exalted Religion
Boy/Male
American, Australian, Chinese, Dutch, French, German, Hindu, Indian, Jamaican, Japanese, Latin, Shakespearean, Spanish, Swiss
Pilgrim to Rome; Citizen of Rome; Of the Romans; From Rome
Girl/Female
Spanish American
Good.
Girl/Female
Biblical
My peace, my happiness, my recompense.
Girl/Female
Indian
She who fulfills
Girl/Female
Tamil
Sapling, Newborn
Surname or Lastname
English
English : habitational name from Winslow, a place in Buckinghamshire named from the genitive case of the Old English personal name or byname Wine (meaning ‘friend’) + Old English hlÄw ‘hill’, ‘mound’, ‘barrow’.Edward Winslow (1595–1655), one of the founders of the Plymouth Colony who sailed on the Mayflower in 1620, was born in Droitwich, Worcestershire, England. He was a governor of the colony and also served as agent of the Massachusetts Bay Company in France. In 1621 he married Susanna, the widow of William White, the first marriage in New England. Their son Josiah (c.1629–80) was governor of Plymouth Colony from 1673 to 1680, the first native-born governor in North America. He had numerous prominent descendents.
Male
Arthurian
, a formidable boar hunted by Arthur.
Boy/Male
Muslim
Sacrifice
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
n.
See Harmonic suture, under Harmonic.
pl.
of Harmony
a.
Not harmonic.
a.
Neatly; dexterously; nimbly.
v. t.
To accompany with harmony; to provide with parts, as an air, or melody.
n.
A musical note produced by a number of vibrations which is a multiple of the number producing some other; an overtone. See Harmonics.
n.
Alt. of Harmonite
v. i.
To agree in vocal or musical effect; to form a concord; as, the tones harmonize perfectly.
a.
Alt. of Harmonical
adv.
In a weak manner; with little strength or vigor; feebly.
a.
Of, pertaining to, or obtained from, carbon; as, carbonic oxide.
a.
Coming, happening, or done once a week; hebdomadary; as, a weekly payment; a weekly gazette.
a.
Of or pertaining to a week, or week days; as, weekly labor.
a.
To make or become weak; to weaken.
a.
Concordant; musical; consonant; as, harmonic sounds.
adv.
Once a week; by hebdomadal periods; as, each performs service weekly.
n.
One who understands the principles of harmony or is skillful in applying them in composition; a musical composer.
adv.
Early.
superl.
Not strong of constitution; infirm; feeble; as, a weakly woman; a man of a weakly constitution.
a.
Producing mathematically perfect harmony or concord; sweetly or perfectly harmonious.