In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicate ...

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Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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Trivially, the zero module {0} is injective. Every vector space Q is injective. Reason: if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V. The new extending basis vectors span a subspace K of V and V is the internal direct sum of Q and K. Note that the direct complement K of Q is not uniquely determined by Q, and likewise the extending map g in ...

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Injective module, Injective module - Definition, Injective module - Examples, Injective module - Facts, Injective module - Generalization

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Any product of (even infinitely many) injective modules is injective. Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules or infinite direct sums of injective modules need not be injective. In Baer's original paper, he proved a useful result, usually known as Baer's Criterion, for checking whether a module is injective: a left R-module Q is injective if and only if any homomorphism g : I → Q defined on a left ideal I of See also:

Injective module, Injective module - Definition, Injective module - Examples, Injective module - Facts, Injective module - Generalization

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Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups. Right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C; these functors form a functor category C-Mod which is the natural gene ...

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Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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Suppose we wish to modulate a simple sine wave on a carrier wave. The equation for the carrier wave of frequency ωc, taking its phase to be a reference phase of zero, is c(t) = Csin(ωct). The equation for the simple sine wave of frequency ωm (the signal we wish to broadcast) is m(t) = Msin(ω ...

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Amplitude modulation, Amplitude modulation - Applications in radio, Amplitude modulation - AM vs. FM, Amplitude modulation - Forms of AM, Amplitude modulation - Example, Amplitude modulation - A more general example, Amplitude modulation - Modulation index, Amplitude modulation - Amplitude modulator designs, Amplitude modulation - Circuits, Amplitude modulation - Low level, Amplitude modulation - High level

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Each format builds on concepts introduced in its predecessors. Sound/Pro/Noisetracker module (file extension: .mod) (in fact, originally on the Amiga files were called MOD.<something>, for example mod.MySong1. Later the file name format was changed so that all file names ended in ".mod", like MySong1.mod) The format that started it all. Uses inverse-frequency note numbers. 4 voices, up to 32 in later variations of the format. Pattern data is not packed. Instruments are simple volume levels; samples and instruments c ...

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Module file, Module file - Modules, Module file - Popular formats, Module file - Software module file players and converters, Module file - Hardware module file players

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The Lunar Module was the portion of the Apollo spacecraft that landed on the moon and returned to lunar orbit. It is divided into two major parts, the Descent Module and the Ascent Module. The Descent Modules contains the landing gear, landing radar antenna, descent rocket engine, and fuel to land on the moon. It also had several cargo compartments used to carry among other things, the Apollo Lunar Surface Experiment Packages ALSEP, Mobile Equipment Cart (a hand pulled equipment cart—Apollo 14), the Lunar Rover (moon car)—Apollo 1 ...

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Apollo Lunar Module, Apollo Lunar Module - History, Apollo Lunar Module - Lunar Module LM specifications, Apollo Lunar Module - Lunar Modules produced, Apollo Lunar Module - LM Truck, Apollo Lunar Module - In fiction

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With the proper identifications, we can again say that every element x of the direct sum can be written in one and only one way as a sum of finitely many elements of the Mi. If the Mi are actually vector spaces, then the dimension of the direct sum is equal to the sum of the dimensions of the Mi. The same is true for t ...

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Direct sum of modules, Direct sum of modules - Construction for vector spaces and abelian groups, Direct sum of modules - Construction for two vector spaces, Direct sum of modules - Construction for two abelian groups, Direct sum of modules - Construction for an arbitrary family of modules, Direct sum of modules - Properties, Direct sum of modules - Internal direct sum, Direct sum of modules - Categorical interpretation, Direct sum of modules - Direct sum of modules with additional structure, Direct sum of modules - Direct sum of Banach spaces, Direct sum of modules - Direct sum of Hilbert spaces

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A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of vector bundles. This can be made precise for the ring of continuous real-valued functions on a compact Hausdorff space, as well as for the ring of smooth functions on a compact smooth manifold (see Swan's theorem). Vector bundles are locally free. If there is some notion of "localization" which can be carried over to modules, such as is given at localization of a ring, one can define locally free modules, and t ...

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Projective module, Projective module - Definitions, Projective module - Direct summands of free modules, Projective module - Lifting property, Projective module - Vector bundles and locally free modules, Projective module - Facts

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Given a set E, we can construct a free R-module over E, denoted by C(E), as follows: As a set, C(E) contains the functions f : E → R such that f(x) = 0 for all but finitely many x in E. Addition: for two elements f, g ∈ C(E), we define f + g ∈ C(E) by (f + g)(x) = f(x) + g(x) for all x ∈ ...

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Free module, Free module - Construction, Free module - Universal property

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Edwin Armstrong presented his paper: "A Method of Reducing Disturbances in Radio Signaling by a System of Frequency Modulation", which first described FM radio, before the New York section of the Institute of Radio Engineers on November 6, 1935. Frequency modulation requires a wider bandwidth than amplitude modulation by an equivalent modulating signal, but this also makes the signal more robust against interference. Frequency modulation is also more robust against simple signal amplitude fading phenomena. As a result, FM was chosen a ...

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Frequency modulation, Frequency modulation - Applications in radio, Frequency modulation - Theory, Frequency modulation - Modulation Index

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A basic AM radio transmitter works by first DC-shifting the modulating signal, then multiplying it with the carrier wave using a frequency mixer. The output of this process is a signal with the same frequency as the carrier but with peaks and troughs that vary in proportion to the strength of the modulating signal. This is amplified and fed to an antenna. Amplitude modulation - AM vs. FM. AM radio's main limitation is its susceptibility to atmospheric interference, which is heard as static from the receive ...

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Amplitude modulation, Amplitude modulation - Applications in radio, Amplitude modulation - AM vs. FM, Amplitude modulation - Forms of AM, Amplitude modulation - Example, Amplitude modulation - A more general example, Amplitude modulation - Modulation index, Amplitude modulation - Amplitude modulator designs, Amplitude modulation - Circuits, Amplitude modulation - Low level, Amplitude modulation - High level

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In telecommunications, the width of the pulses correspond with specific data values encoded at one end and decoded at the other. Pulses of various lengths (the information itself) will be sent at regular intervals (the carrier frequency of the modulation). _ _ _ _ _ _ _ _ | | | | | | | | | | | | | | | | Clock | | | | | | | | | | | | | | | | __| |____| |__ ...

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Pulse-width modulation, Pulse-width modulation - Telecommunications, Pulse-width modulation - Power delivery, Pulse-width modulation - Voltage regulation, Pulse-width modulation - Audio effects and amplifcation

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In its basic form, amplitude modulation produces a signal with power concentrated at the carrier frequency and in two adjacent sidebands. Each sideband is equal in bandwidth to that of the modulating signal and is a mirror image of the other. Thus, most of the power output by an AM transmitter is effectively wasted: half the power is concentrated at the carrier frequency, which carries no useful information (beyond the fact that a signal is present); the remaining power is split between t ...

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Amplitude modulation, Amplitude modulation - Applications in radio, Amplitude modulation - AM vs. FM, Amplitude modulation - Forms of AM, Amplitude modulation - Example, Amplitude modulation - A more general example, Amplitude modulation - Modulation index, Amplitude modulation - Amplitude modulator designs, Amplitude modulation - Circuits, Amplitude modulation - Low level, Amplitude modulation - High level

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The Apollo Lunar Module came into being because NASA chose to reach the moon via a lunar orbit rendezvous (LOR) instead of a direct ascent or Earth orbit rendezvous (EOR) (see Choosing a mission mode for more information on the available rendezvous types). Both a direct ascent and an EOR would have involved the entire Apollo spacecraft landing on the moon; once the decision had been made to proceed using LOR, it became necessary to produce a separa ...

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Apollo Lunar Module, Apollo Lunar Module - History, Apollo Lunar Module - Lunar Module LM specifications, Apollo Lunar Module - Lunar Modules produced, Apollo Lunar Module - LM Truck, Apollo Lunar Module - In fiction

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One should notice a clear similarity between the definitions of the direct sum of two vector spaces and of two abelian groups. In fact, each is a special case of the construction of the direct sum of two modules. Additionally, by modifying the definition one can accommodate the direct sum of an infinite family of modules. The precise definition is as follows. Assume R is some ring, and {Mi : i in I} is a family of left R-modules indexed by the set I. The direct sumSee also:

Direct sum of modules, Direct sum of modules - Construction for vector spaces and abelian groups, Direct sum of modules - Construction for two vector spaces, Direct sum of modules - Construction for two abelian groups, Direct sum of modules - Construction for an arbitrary family of modules, Direct sum of modules - Properties, Direct sum of modules - Internal direct sum, Direct sum of modules - Categorical interpretation, Direct sum of modules - Direct sum of modules with additional structure, Direct sum of modules - Direct sum of Banach spaces, Direct sum of modules - Direct sum of Hilbert spaces

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If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M,+). The set of all group endomorphisms of M is denoted EndZ(M) and forms a ring under addition and composition, and sending a ring element r of R to its action actually define ...

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Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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Main article on oversampling Let's consider a signal at frequence f0 and a sampling frequence of fs much higher than Nyquist rate (see Fig. 3). ΔΣ modulation is based on oversampling technique to reduce the noise in the band of interest (green), which also avoid the using of high-precision analog circuits for the anti-aliasing filter. The quantization noise spectrum is the same as in Nyquist (not oversampling) con ...

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Delta-sigma modulation, Delta-sigma modulation - Principle, Delta-sigma modulation - Quantization theory formulas, Delta-sigma modulation - Oversampling, Delta-sigma modulation - Example of decimation, Delta-sigma modulation - Changes from Δ-modulation, Delta-sigma modulation - Naming

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J. Chowning, "The Synthesis of Complex Audio Spectra by Means of Frequency Modulation," Journal of the Audio Engineering Society 21(7), 1973 Dodge, Charles and Jerse, Thomas A. (1997). Computer Music: Synthesis, Composition and Performance. New York: Schirmer Books. ISBN 0-02-864682-7. ...

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Frequency modulation synthesis, Frequency modulation synthesis - Reference

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Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and any r in R, the product rn is in N (or nr for a right module). If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if, for any m, n in M and r, s in R, f(rmSee also:

Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations

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As with all modulation schemes, QAM conveys data by changing some aspect of a base signal, the carrier wave, (usually a sinusoid) in response to a data signal. In the case of QAM, the amplitude of two quadrature waves is changed (modulated or keyed) to represent the data signal. Phase modulation (analogue PM) and phase-shift keying (digital PSK) can be regarded as a special case of QAM, where the amplitude of the modulating signal is constant, with only the phase varying. This can also be extended to frequency modulation (FM) and frequency-shift keying (FSK), as ...

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Quadrature amplitude modulation, Quadrature amplitude modulation - Overview, Quadrature amplitude modulation - Ideal structure, Quadrature amplitude modulation - Transmitter, Quadrature amplitude modulation - Receiver, Quadrature amplitude modulation - Performance, Quadrature amplitude modulation - Definitions, Quadrature amplitude modulation - Rectangular QAM, Quadrature amplitude modulation - Odd-k QAM, Quadrature amplitude modulation - Non-rectangular QAM

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