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mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of
Harmonic_measure
Second-order partial differential equation
called the harmonic measure of D {\displaystyle D} . Thus harmonic measure gives a probabilistic solution of the Dirichlet problem: the harmonic extension
Laplace's_equation
Area of mathematical analysis
their boundary behavior. The methods of harmonic analysis decompose functions and related objects, such as measures, into components based on symmetries
Harmonic_analysis
Method for reconstructing a harmonic function in a domain
reconstructing an harmonic function in a domain from its values on the boundary of the domain. In modern terms, the balayage operator maps a measure μ {\displaystyle
Balayage
Solution to a specific type of stochastic differential equation
the harmonic measure of B on ∂D is invariant under all rotations of D about x and coincides with the normalized surface measure on ∂D. The harmonic measure
Itô_diffusion
Measurement of the harmonic distortion present in a signal
The total harmonic distortion (THD or THDi) is a measurement of the harmonic distortion present in a signal and is defined as the ratio of the sum of the
Total_harmonic_distortion
Russian-American mathematician (born 1955)
Mathematics in 1986 under Nikolai Nikolski with thesis Metric properties of harmonic measure (title translated from Russian). He was an academic at the Steklov
Nikolai_Georgievich_Makarov
Left-invariant (or right-invariant) measure on locally compact topological group
used in harmonic analysis on locally compact groups, particularly in the theory of Pontryagin duality. To prove the existence of a Haar measure on a locally
Haar_measure
Special mathematical functions defined on the surface of a sphere
fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal
Spherical_harmonics
Rate at which chords change (or progress) in a musical composition
and chord changes twice a measure has a fast harmonic rhythm and a slow surface rhythm (1 note per chord change). Harmonic rhythm may be described as
Harmonic_rhythm
American mathematician
computational geometry; and in particular for topics such as fractals, harmonic measure, conformal and quasiconformal mappings and Julia sets. Along with Peter
Christopher_J._Bishop
Theorem of Fourier transforms of Borel measures
Fourier-Stieltjes transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier
Bochner's_theorem
Statistical measure of a test's accuracy
Understanding Conference (MUC-4, 1992). The traditional F-measure or balanced F-score (F1 score) is the harmonic mean of precision and recall: F 1 = 2 r e c a l
F-score
Inverse of the average of the inverses of a set of numbers
In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is sometimes used for ratios and rates such as speeds, and is
Harmonic_mean
Sequence of frequencies
The harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental
Harmonic_series_(music)
Measure of a transformer's ability to withstand harmonic distortion
power transformer is a measure of how well it can handle harmonic distortion. Transformers which are designed to handle harmonic distortion are referred
K-factor (electrical engineering)
K-factor_(electrical_engineering)
for any z0 in Ω, the harmonic measure w( ⋅ ) = w(z0, Ω, ⋅) is absolutely continuous with respect to one-dimensional Hausdorff measure and its Radon–Nikodym
Muckenhoupt_weights
Functions in mathematics
mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle
Harmonic_function
Chinese mathematician (born 1991)
Mathematics, an endowed professorship. Wang's research is in harmonic analysis and geometric measure theory, particularly the family of Kakeya-type problems
Hong_Wang
Application of Fourier analysis to non-abelian topological groups
In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative
Noncommutative harmonic analysis
Noncommutative_harmonic_analysis
Study of geometric properties of sets through measure theory
Connections to singular integrals, Fourier transform, Frostman measures, harmonic measures, etc Currents, a generalization of the concept of oriented manifolds
Geometric_measure_theory
Divergent sum of positive unit fractions
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: ∑ i = 1 ∞ 1 i = 1 1 + 1 2 + 1 3 + 1 4 + 1 5 +
Harmonic_series_(mathematics)
In Euclidean space, a measure of that set's "size"
capacity of a set in Euclidean space is a measure of the "size" of that set. Unlike, say, Lebesgue measure, which measures a set's volume or physical extent,
Capacity_of_a_set
Process forming a path from many random steps
used to calculate solutions to Laplace's equation, to estimate the harmonic measure, and for various constructions in analysis and combinatorics. In psychology
Random_walk
Generalization of volume to non-integer number of dimensions
are fundamental in geometric measure theory. They appear naturally in harmonic analysis or potential theory. Let ( X , ρ ) {\displaystyle (X,\rho )} be
Hausdorff_measure
Mathematical algorithm
knowledge of Green's functions for the specific domains. (see also Harmonic measure) When it is possible to use it, the Green's function first-passage
Walk-on-spheres_method
American mathematician
Academic Press. 1981. ISBN 0-122-76150-2. with Donald E. Marshall: Harmonic Measures. Cambridge University Press. 2005. ISBN 0-521-47018-8. "2003 Steele
John_B._Garnett
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
a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle
Positive_harmonic_function
Mathematical measure space associated to a random walk
harmonic functions on the group, and to the space of trajectories of the random walk on the group (both with respect to a given probability measure)
Poisson_boundary
Number taken as representative of a list of numbers
overly influenced by the much higher incomes of the few rich people. The harmonic mean, defined as the reciprocal of the mean of the reciprocals, is used
Average
Branch of mathematics
topological spaces, measure spaces, and function spaces. Its major areas include complex analysis, functional analysis, measure theory, harmonic analysis, and
Mathematical_analysis
Repetitive variation of some measure about a central value
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or
Oscillation
Theorem in complex analysis
ISBN 978-0-8218-5270-5 Garnett, John B.; Marshall, Donald E. (2005), Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, ISBN 0-521-47018-8
Carathéodory's theorem (conformal mapping)
Carathéodory's_theorem_(conformal_mapping)
Aspect of music
effects created by distinct pitches or tones coinciding with one another; harmonic objects such as chords, textures and tonalities are identified, defined
Harmony
Alexander (2017). "On Tsirelson's theorem about triple points for harmonic measure". International Mathematics Research Notices. 2018 (12): 3671–3683
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
American music theorist
371. Springer Science & Business Media. ISBN 9781852337971. "Harmonic entropy is a measure of the uncertainty in pitch perception, and it provides a physical
Paul_Erlich
Polish-Canadian mathematician
University of British Columbia. Her main research specialties are harmonic analysis, geometric measure theory, and additive combinatorics. Łaba earned a master's
Izabella_Łaba
mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular
Zonal_spherical_harmonics
Concept in mathematics
differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential
Harmonic_map
Shape containing unit line segments in all directions
sets in the plane of measure zero and in 1928 that there are Kakeya needle sets in the plane of arbitrarily small positive measure. There are no Kakeya
Kakeya_set
N-th root of the product of n numbers
arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least
Geometric_mean
of Ω when compared to the surface measure on the boundary of Ω. Carleson measures have many applications in harmonic analysis and the theory of partial
Carleson_measure
Audio compression standard
measure of linear dependence is identified as the pitch period. The spectral envelope is represented by a set of amplitude values, one per harmonic.
Harmonic Vector Excitation Coding
Harmonic_Vector_Excitation_Coding
American award for mathematical analysis
Wolff for "his work in harmonic analysis," "harmonic measure, and unique continuation," including Counterexamples with harmonic gradients in ℝ3. Essays
Bôcher_Memorial_Prize
be a finite non-negative measure on X {\displaystyle X} . The generalized Poisson distribution associated with the measure F {\displaystyle F} is defined
Generalized Poisson distribution on a locally compact Abelian group
Generalized_Poisson_distribution_on_a_locally_compact_Abelian_group
Polish mathematician
committee that includes Zdunik. Zdunik, Anna (1991). "Harmonic Measure Versus Hausdorff Measures on Repellers for Holomorphic Maps". Transactions of the
Anna_Zdunik
Sinusoidal wave whose frequency is an integer multiple
system, a harmonic of a voltage or current waveform is a sinusoidal wave whose frequency is an integer multiple of the fundamental frequency. Harmonic frequencies
Harmonics_(electrical_power)
In mathematics, in the field of harmonic analysis, the van der Corput lemma is an estimate for oscillatory integrals named after the Dutch mathematician
Van der Corput lemma (harmonic analysis)
Van_der_Corput_lemma_(harmonic_analysis)
American physicist
has been cited over 4,000 times. Sander's latest work involves the harmonic measure of fractals, cell motility and malignant brain tumors, statistical
Leonard_Sander
Iterative method in conformal mapping
Alternating Procedures Garnett, John B.; Marshall, Donald E. (2005), Harmonic Measure, Cambridge University Press, ISBN 1139443097 Freitag, Eberhard (2011)
Schwarz_alternating_method
Measure that is 1 if and only if a specified element is in the set
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element x or not. It is one way of formalizing the
Dirac_measure
Quality of a musical note or sound or tone
tristimulus measures the relative weight of the first harmonic; the second tristimulus measures the relative weight of the second, third, and fourth harmonics taken
Timbre
Russian mathematician (born 1989)
Hausdorff measures on the zero sets of Laplace eigenfunctions defined on compact smooth manifolds and an estimate (from below) in harmonic analysis and
Aleksandr Logunov (mathematician)
Aleksandr_Logunov_(mathematician)
2 points about which a triangle can be inverted into an equilateral triangle
that point. That is, the isodynamic point is the point for which the harmonic measure of the three arcs is equal. Given a univariate polynomial P ( z ) =
Isodynamic_point
Dao-Xing; Brody, Elmer J. (1972). Measure and Integration Theory on Infinite-Dimensional Spaces: Abstract Harmonic Analysis. Ukraine: Academic Press.
Cylinder_set_measure
This is a list of harmonic analysis topics. See also list of Fourier analysis topics and list of Fourier-related transforms, which are more directed towards
List of harmonic analysis topics
List_of_harmonic_analysis_topics
term is found by multiplying the previous one by a fixed non-zero number Harmonic progression – a sequence formed by taking the reciprocals of the terms
List_of_real_analysis_topics
Finnish mathematician (1895–1980)
accomplished violinist with a deep interest in the music of Jean Sibelius. Harmonic measure Nevanlinna theory Nevanlinna class (functions of bounded type) Nevanlinna
Rolf_Nevanlinna
Canadian philosopher and mathematician (born 1973)
Columbia with a dissertation on Symmetrization, Green’s Functions, Harmonic Measures and Difference Equations, under John J. F. Fournier in 1996, and published
Alexander_Pruss
Ratio of active power to apparent power
of the power system frequency. Distortion power factor is a measure of how much the harmonic distortion of a load current decreases the average power transferred
Power_factor
log-concavity, harmonic convexity, and so on. The mathematician Christer Borell was a pioneer of the detailed study of convex measures on locally convex
Convex_measure
Topological index in graph theory
The harmonic index is a topological index based on the degrees of vertices in a graph. Introduced by Fajtlowicz in 1987, it has become an important descriptor
Harmonic_index
University of Oregon, Jorge M. López co-wrote a book on Sidon sequences in harmonic analysis with Kenneth Ross. These sequences were first introduced by Simon
Jorge_M._López
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
Alteration of the original shape of a signal
noise. Quality measures that reflect both noise and distortion include the signal-to-noise and distortion (SINAD) ratio and total harmonic distortion plus
Distortion
Method of mathematical integration
integral can be generalized in a straightforward way to more general spaces, measure spaces, such as those that arise in probability theory. The term Lebesgue
Lebesgue_integral
Association football club in Glasgow City, Scotland
Athletic, which its opponents won 5–0. A measure of the strength of Glasgow football at the time is shown by Harmonic, one of the minor clubs in the city,
Harmonic_F.C.
Notion of boundary associated with a group
probability measure on G {\displaystyle G} . A function f {\displaystyle f} on G {\displaystyle G} is called μ {\displaystyle \mu } -harmonic if f ( g )
Furstenberg_boundary
unique harmonic function ωi on G equal to 1 on ∂Di and 0 on ∂Dj for j ≠ i (the so-called harmonic measure of ∂Di). The ωi's sum to 1. The harmonic function
Planar_Riemann_surface
Harmonic pitch class profiles (HPCP) is a group of features that a computer program extracts from an audio signal, based on a pitch class profile—a descriptor
Harmonic_pitch_class_profiles
Microscope imaging technique
Second-harmonic imaging microscopy (SHIM) is based on a nonlinear optical effect known as second-harmonic generation (SHG). SHIM has been established as
Second-harmonic imaging microscopy
Second-harmonic_imaging_microscopy
Pattern-recognition performance metrics
considered class. A measure that combines precision and recall is the harmonic mean of precision and recall, the traditional F-measure or balanced F-score:
Precision_and_recall
for every function u harmonic and integrable over D with respect to area measure, the integral of u with respect to this measure is given by a "quadrature
Quadrature_domains
Subject in mathematics
In mathematics, measure theory in topological vector spaces refers to the extension of measure theory to topological vector spaces. Such spaces are often
Measure theory in topological vector spaces
Measure_theory_in_topological_vector_spaces
Musical chord
respect to G. The harmonic seventh chord is a dominant seventh chord formed by a major triad plus a harmonic seventh interval. The harmonic seventh interval
Seventh_chord
Green's function for Laplacian
named for Isaac Newton, who first discovered it and proved that it was a harmonic function in the special case of three variables, where it served as the
Newtonian_potential
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
The quasi-harmonic approximation is a phonon-based model of solid-state physics used to describe volume-dependent thermal effects, such as the thermal
Quasi-harmonic_approximation
Mathematical measure of a function's variability
mathematics Total variation – Measure of local oscillation behavior Bounded mean oscillation – Real-valued function Harmonic map – Concept in mathematics
Dirichlet_energy
1997 promotional single by Tool
written in D Phrygian dominant scale, also known as the fifth mode of the G harmonic minor scale. In March 2023, Rolling Stone ranked "Forty Six & 2" at No
Forty_Six_&_2
Harmonic device in Western music
chords for C major. Each chord is accompanied by its standard number in harmonic notation. In this notation, a secondary dominant is usually labeled with
Secondary_chord
American mathematician (1931–2018)
was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus
Elias_M._Stein
Canadian–American mathematician
1090/S0002-9939-1982-0652448-7. Jenkins, James A. (1991). "Some estimates for harmonic measures. II". Proceedings of the American Mathematical Society. 111 (2): 441
James_Allister_Jenkins
Measure of change in a periodic variable
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal
Amplitude
Relationship between harmonic force and velocity
Mechanical impedance is a measure of how much a structure resists motion when subjected to a harmonic force. It relates forces with velocities acting
Mechanical_impedance
American mathematician (born 1931)
kernel measure. This decomposition then led to many other developments in the study of harmonic analysis on Lie groups in which the Gaussian measure on Euclidean
Leonard_Gross
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)
Keyboard composition by Johann Sebastian Bach
(second inversion) minor chord with C as its root borrowed from the parallel harmonic minor. The A♭ may be considered as its enharmonic, G♯, which creates a
Prelude and Fugue in C major, BWV 846
Prelude_and_Fugue_in_C_major,_BWV_846
Measuring device for harmonic noise
A total harmonic distortion analyzer calculates the total harmonic content of a sinewave with some distortion, expressed as total harmonic distortion (THD)
Total harmonic distortion analyzer
Total_harmonic_distortion_analyzer
Any musical scale used in jazz
displaying the influence of African music. The E♭ and B♭ are blue notes. The harmonic minor scale is also of value to many improvisors, as it provides an alternative
Jazz_scale
Musical interval
semitones, or 600 cents. In traditional functional harmony, the tritone is a harmonic and melodic dissonance and tritones in chords push toward resolution. For
Tritone
Device designed to reduce vibrations in structures
A tuned mass damper (TMD), also known as a harmonic absorber or seismic damper, is a device mounted in structures to reduce mechanical vibrations, consisting
Tuned_mass_damper
Metric space with double measurement
Lebesgue measure. The definition of a doubling measure may seem arbitrary, or purely of geometric interest. However, many results from classical harmonic analysis
Doubling_space
Branch of mathematics concerning probability
of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample
Probability_theory
Branch of mathematics
and volume are extended by measure theory, which studies methods of assigning a size or measure to sets, where the measures follow rules similar to those
Geometry
Model in probability theory
function bounded above by a harmonic function for all points on the boundary of a ball is bounded above by the harmonic function inside the ball. Similarly
Martingale (probability theory)
Martingale_(probability_theory)
Musical structure of salsa
accordance to the harmonic sequence, as Rick Davies observes in his detailed analysis of the first moña: The moña consists of a two-measure module and its
Salsa_(musical_structure)
Musical chord
together gives the dominant seventh chord its dissonant quality (i.e. its harmonic instability). Dominant seventh chords are often built on the fifth scale
Dominant_seventh_chord
Music term
element, that is to say human,...and "it will evolve depending on the harmonic context, the place in the song, the sound of the musician's instrument
Groove_(music)
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