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Harmonic Analysis
Harmonic
Harmonic
analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis)
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Harmony
In music, harmony considers the process by which the composition of individual sounds, or superpositions of sounds, is analysed by hearing. Usually, this means simultaneously occurring frequencies, pitches (tones, notes), or chords.[1] The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them.[2] Harmony
Harmony
is often said to refer to the "vertical" aspect of music, as distinguished from melodic line, or the "horizontal" aspect.[3] Counterpoint, which refers to the relationship between melodic lines, and polyphony, which refers to the simultaneous sounding of separate independent voices, are thus sometimes distinguished from harmony. In popular and jazz harmony, chords are named by their root plus various terms and characters indicating their qualities. In many types of music, notably baroque, romantic, modern, and jazz, chords are often augmented with "tensions"
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Data Collection
Data collection
Data collection
is the process of gathering and measuring information on targeted variables in an established systematic fashion, which then enables one to answer relevant questions and evaluate outcomes. Data collection is a component of research in all fields of study including physical and social sciences, humanities, and business. While methods vary by discipline, the emphasis on ensuring accurate and honest collection remains the same. The goal for all data collection is to capture quality evidence that allows analysis to lead to the formulation of convincing and credible answers to the questions that have been posed.Contents1 Importance 2 Impact of faulty data 3 See also 4 References 5 External linksImportance[edit] Regardless of the field of study or preference for defining data (quantitative or qualitative), accurate data collection is essential to maintaining the integrity of research
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Distribution (mathematics)
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist
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Compact Support
In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis.Contents1 Formulation 2 Closed support 3 Compact support 4 Essential support 5 Generalization 6 In probability and measure theory 7 Support of a distribution 8 Singular support 9 Family of supports 10 See also 11 ReferencesFormulation[edit] Suppose that f : X → R is a real-valued function whose domain is an arbitrary set X
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Harmonic (color)
In color theory, color harmony refers to the property that certain aesthetically pleasing color combinations have. These combinations create pleasing contrasts and consonances that are said to be harmonious. These combinations can be of complementary colors, split-complementary colors, color triads, or analogous colors. Color harmony has been a topic of extensive study throughout history, but only since the Renaissance and the Scientific Revolution has it seen extensive codification
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Functional Analysis
Functional analysis
Functional analysis
is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject
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String (music)
A string is the vibrating element that produces sound in string instruments such as the guitar, harp, piano (piano wire), and members of the violin family. Strings are lengths of a flexible material that a musical instrument holds under tension so that they can vibrate freely, but controllably. Strings may be "plain", consisting only of a single material, like steel, nylon, or gut, or wound, having a "core" of one material and an overwinding of another. This is to make the string vibrate at the desired pitch, while maintaining a low profile and sufficient flexibility for playability. The invention of wound strings, such as nylon covered in wound metal, was a crucial step in string instrument technology, because a metal-wound string can produce a lower pitch than a catgut string of similar thickness. This enabled stringed instruments to be made with less thick bass strings
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Differential Equation
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers
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System Of Equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a:System of linear equations, System of nonlinear equations, System of bilinear equations, System of polynomial equations, System of ordinary differential equations, System of partial differential equations, or a System of difference equationsSee also[edit]Simultaneous equations model, a statistical model in the form of simultaneous linear equations Elementary algebra, for elementary methodsThis mathematics-related article is a stub
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Lord Kelvin
William Thomson, 1st Baron Kelvin, OM, GCVO, PC, FRS, FRSE
FRSE
(26 June 1824 – 17 December 1907) was a Scots-Irish[1][2] mathematical physicist and engineer who was born in Belfast
Belfast
in 1824. At the University of Glasgow
Glasgow
he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging discipline of physics in its modern form. He worked closely with mathematics professor Hugh Blackburn
Hugh Blackburn
in his work. He also had a career as an electric telegraph engineer and inventor, which propelled him into the public eye and ensured his wealth, fame and honour. For his work on the transatlantic telegraph project he was knighted in 1866 by Queen Victoria, becoming Sir William Thomson
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Harmonic Series (music)
A harmonic series is the sequence of sounds[1]—pure tones, represented by sinusoidal waves—in which the frequency[2] of each sound is an integer multiple of the fundamental, the lowest frequency.[3] Pitched musical instruments are often based on an acoustic resonator such as a string or a column of air, which oscillates at numerous modes simultaneously. At the frequencies of each vibrating mode, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves. Interaction with the surrounding air causes audible sound waves, which travel away from the instrument
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Abelian Group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers. They are named after early 19th century mathematician Niels Henrik Abel.[1] The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules and vector spaces, are developed. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood
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Lie Group
In mathematics, a Lie group
Lie group
(pronounced /liː/ "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups. Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory
Galois theory
for analysing the discrete symmetries of algebraic equations
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Special Linear Group
In mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant det : GL ⁡ ( n , F ) → F × . displaystyle det colon operatorname GL (n,F)to F^ times
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Dimension (mathematics And Physics)
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.[1][2] Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism
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