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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
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
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
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
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
exponential functions Inverse function Convex function, Concave function Singular function Harmonic function Weakly harmonic function Proper convex function Rational
List_of_real_analysis_topics
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Branch of mathematics
to ordinary calculus which deals with functions. Harmonic analysis deals with the representation of functions or signals as the superposition of basic
Mathematical_analysis
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
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
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
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
Society, second edition, 2002. Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to
Selberg_zeta_function
In mathematics, a quantitative measure of the shape of a set of points
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)
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
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
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)
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
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
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
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
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
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
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
with Harmonic Responses, Mechanical Systems and Signal Processing, 20(8), 1883–1904, (2006) Electrical Engineering Encyclopedia: Describing Functions
Describing_function
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
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)
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
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
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
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
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
\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
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
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
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
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
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
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
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
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
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
Formula for the derivative of a product
formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as ( u ⋅ v ) ′ = u ′
Product_rule
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
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
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
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
Electronic control system
the Nth harmonic of the reference signal. Instead of a simple phase detector, the design uses a harmonic mixer (sampling mixer). The harmonic mixer turns
Phase-locked_loop
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)
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
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
Equation in Fourier analysis
summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined
Poisson_summation_formula
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
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)
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
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
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
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
Formulation of the quantum many-body problem
oscillator. The position and momentum operators of a Harmonic oscillator (or a collection of Harmonic oscillating modes) are given by Hermitian combinations
Second_quantization
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
Description of physical properties at the atomic and subatomic scale
for very few relatively simple model Hamiltonians including the quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the hydrogen
Quantum_mechanics
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
+ U(z). It vanishes on ∂Ω and is harmonic on Ω away from 0. The harmonic conjugate V of U is the unique real function on Ω such that U + iV is holomorphic
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
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
Differentiation under the integral sign formula
< ∞ {\displaystyle -\infty <a(x),b(x)<\infty } and the integrands are functions dependent on x , {\displaystyle x,} the derivative of this integral is
Leibniz_integral_rule
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
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
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
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
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
or functions is called monotone or monotonic if it is either weakly increasing x 1 ≤ x 2 ≤ ⋯ {\displaystyle x_{1}\leq x_{2}\leq \cdots } or weakly decreasing
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
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
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
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
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
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
Microscope imaging technique
structure and function. A second-harmonic microscope obtains contrasts from variations in a specimen's ability to generate second-harmonic light from the
Second-harmonic imaging microscopy
Second-harmonic_imaging_microscopy
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
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
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
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Surname or Lastname
English
English : variant spelling of Weekley.
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
Girl/Female
Greek Latin
Daughter of Ares.
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’.
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.
Male
English
English surname transferred to forename use, from the German personal name Harman, HARMON means "bold/hardy man."
Surname or Lastname
English
English : variant of Weekley.
Girl/Female
Hindu
Pearl Pearly just similar to Pearl
Surname or Lastname
English
English : variant spelling of Weekley.
Girl/Female
American, Australian, British, Christian, English, French, Greek, Latin
A State of Order or Agreement; Unity; Concord; Harmony; Agreement
Surname or Lastname
English
English : variant of Wakeley.
Female
English
Variant spelling of English Harmony, HARMONIE means "concord, harmony."
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Surname or Lastname
English (Somerset)
English (Somerset) : unexplained. Compare Lukey.
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
Girl/Female
American, British, English, Greek, Latin
A State of Order or Agreement; Unity; Concord; Musically in Tune; A Tuneful Sound
Girl/Female
Latin American
Concord.
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.
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
Girl/Female
Australian, Christian, Danish, French, Hebrew, Latin
Supplants; Female Version of Jacob; Supplanter
Girl/Female
Danish German Scandinavian
Girl/Female
Tamil
Name of Goddess Durga
Boy/Male
Tamil
Viranchi | விராஂசீ
Name of Lord Brahma
Girl/Female
Biblical
That bears fruit; or grows.
Boy/Male
Assamese, Gujarati, Hindu, Indian, Malayalam, Marathi, Oriya, Sanskrit, Tamil
Young Man
Girl/Female
Persian
Eagle.
Girl/Female
Muslim
The generous
Female
Egyptian
, the sister of the scribe Thoth.
Male
Swedish
Swedish form of Old Norse Arnviðr, ARVIDH means "eagle tree."
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
WEAKLY HARMONIC-FUNCTION
a.
Of or pertaining to a week, or week days; as, weekly labor.
adv.
Early.
n.
One who understands the principles of harmony or is skillful in applying them in composition; a musical composer.
a.
Not harmonic.
n.
See Harmonic suture, under Harmonic.
adv.
Once a week; by hebdomadal periods; as, each performs service weekly.
a.
Producing mathematically perfect harmony or concord; sweetly or perfectly harmonious.
superl.
Not strong of constitution; infirm; feeble; as, a weakly woman; a man of a weakly constitution.
a.
Alt. of Harmonical
a.
Of, pertaining to, or obtained from, carbon; as, carbonic oxide.
a.
Concordant; musical; consonant; as, harmonic sounds.
v. t.
To accompany with harmony; to provide with parts, as an air, or melody.
adv.
In a weak manner; with little strength or vigor; feebly.
a.
Neatly; dexterously; nimbly.
a.
To make or become weak; to weaken.
a.
Coming, happening, or done once a week; hebdomadary; as, a weekly payment; a weekly gazette.
pl.
of Harmony
n.
A musical note produced by a number of vibrations which is a multiple of the number producing some other; an overtone. See Harmonics.
v. i.
To agree in vocal or musical effect; to form a concord; as, the tones harmonize perfectly.
n.
Alt. of Harmonite