text
stringlengths 105
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if
$$
L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y)
$$
for every
$$
x,y\in X
$$
and some constant
$$
L>0
$$
.
### Normed spaces
An important special case is that of normed spaces; in this case it is natural to consider linear embeddings.
One of the basic questions that can be asked about a finite-dimensional normed space
$$
(X, \| \cdot \|)
$$
is, what is the maximal dimension such that the Hilbert space can be linearly embedded into with constant distortion?
The answer is given by Dvoretzky's theorem.
## Category theory
In category theory, there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable under pullbacks.
Ideally the class of all embedded subobjects of a given object, up to isomorphism, should also be small, and thus an ordered set.
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Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable under pullbacks.
Ideally the class of all embedded subobjects of a given object, up to isomorphism, should also be small, and thus an ordered set. In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a closure operator).
In a concrete category, an embedding is a morphism
$$
f:A\rightarrow B
$$
that is an injective function from the underlying set of
$$
A
$$
to the underlying set of
$$
B
$$
and is also an initial morphism in the following sense:
If
$$
g
$$
is a function from the underlying set of an object
$$
C
$$
to the underlying set of
$$
A
$$
, and if its composition with
$$
f
$$
is a morphism
$$
fg:C\rightarrow B
$$
, then _ BLOCK8_ itself is a morphism.
A factorization system for a category also gives rise to a notion of embedding. If
$$
(E,M)
$$
is a factorization system, then the morphisms in
$$
M
$$
may be regarded as the embeddings, especially when the category is well powered with respect to
$$
M
$$
.
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A factorization system for a category also gives rise to a notion of embedding. If
$$
(E,M)
$$
is a factorization system, then the morphisms in
$$
M
$$
may be regarded as the embeddings, especially when the category is well powered with respect to
$$
M
$$
. Concrete theories often have a factorization system in which
$$
M
$$
consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article.
As usual in category theory, there is a dual concept, known as quotient. All the preceding properties can be dualized.
An embedding can also refer to an embedding functor.
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In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers.
## Overview
The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity.
The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fields, étale cohomology provides topological invariants associated to algebraic varieties. p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields.
## History
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p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields.
## History
### 19th century: early arithmetic geometry
In the early 19th century, Carl Friedrich Gauss observed that non-zero integer solutions to homogeneous polynomial equations with rational coefficients exist if non-zero rational solutions exist.
In the 1850s, Leopold Kronecker formulated the Kronecker–Weber theorem, introduced the theory of divisors, and made numerous other connections between number theory and algebra. He then conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his twelfth problem, which outlines a goal to have number theory operate only with rings that are quotients of polynomial rings over the integers.
### Early-to-mid 20th century: algebraic developments and the Weil conjectures
In the late 1920s, André Weil demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to the Mordell–Weil theorem which demonstrates that the set of rational points of an abelian variety is a finitely generated abelian group.
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He then conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his twelfth problem, which outlines a goal to have number theory operate only with rings that are quotients of polynomial rings over the integers.
### Early-to-mid 20th century: algebraic developments and the Weil conjectures
In the late 1920s, André Weil demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to the Mordell–Weil theorem which demonstrates that the set of rational points of an abelian variety is a finitely generated abelian group.
Modern foundations of algebraic geometry were developed based on contemporary commutative algebra, including valuation theory and the theory of ideals by Oscar Zariski and others in the 1930s and 1940s.
In 1949, André Weil posed the landmark Weil conjectures about the local zeta-functions of algebraic varieties over finite fields. These conjectures offered a framework between algebraic geometry and number theory that propelled Alexander Grothendieck to recast the foundations making use of sheaf theory (together with Jean-Pierre Serre), and later scheme theory, in the 1950s and 1960s. Bernard Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960.
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These conjectures offered a framework between algebraic geometry and number theory that propelled Alexander Grothendieck to recast the foundations making use of sheaf theory (together with Jean-Pierre Serre), and later scheme theory, in the 1950s and 1960s. Bernard Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960. Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with Michael Artin and Jean-Louis Verdier) by 1965. The last of the Weil conjectures (an analogue of the Riemann hypothesis) would be finally proven in 1974 by Pierre Deligne.
### Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond
Between 1956 and 1957, Yutaka Taniyama and Goro Shimura posed the Taniyama–Shimura conjecture (now known as the modularity theorem) relating elliptic curves to modular forms. This connection would ultimately lead to the first proof of Fermat's Last Theorem in number theory through algebraic geometry techniques of modularity lifting developed by Andrew Wiles in 1995.
In the 1960s, Goro Shimura introduced Shimura varieties as generalizations of modular curves. Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures.
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In the 1960s, Goro Shimura introduced Shimura varieties as generalizations of modular curves. Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures.
In papers in 1977 and 1978, Barry Mazur proved the torsion conjecture giving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. In 1996, the proof of the torsion conjecture was extended to all number fields by Loïc Merel.
In 1983, Gerd Faltings proved the Mordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstrates finite generation of the set of rational points as opposed to finiteness).
In 2001, the proof of the local Langlands conjectures for GLn was based on the geometry of certain Shimura varieties.
In the 2010s, Peter Scholze developed perfectoid spaces and new cohomology theories in arithmetic geometry over p-adic fields with application to Galois representations and certain cases of the weight-monodromy conjecture.
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In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame. The two differ only by a multiplicative constant and the units of measurement. The principle is described by the physicist Albert Einstein's formula:
$$
E = mc^2
$$
. In a reference frame where the system is moving, its relativistic energy and relativistic mass (instead of rest mass) obey the same formula.
The formula defines the energy () of a particle in its rest frame as the product of mass () with the speed of light squared (). Because the speed of light is a large number in everyday units (approximately ), the formula implies that a small amount of mass corresponds to an enormous amount of energy.
Rest mass, also called invariant mass, is a fundamental physical property of matter, independent of velocity.
### Massless particles
such as photons have zero invariant mass, but massless free particles have both momentum and energy.
The equivalence principle implies that when mass is lost in chemical reactions or nuclear reactions, a corresponding amount of energy will be released. The energy can be released to the environment (outside of the system being considered) as radiant energy, such as light, or as thermal energy. The principle is fundamental to many fields of physics, including nuclear and particle physics.
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The energy can be released to the environment (outside of the system being considered) as radiant energy, such as light, or as thermal energy. The principle is fundamental to many fields of physics, including nuclear and particle physics.
Mass–energy equivalence arose from special relativity as a paradox described by the French polymath Henri Poincaré (1854–1912). Einstein was the first to propose the equivalence of mass and energy as a general principle and a consequence of the symmetries of space and time. The principle first appeared in "Does the inertia of a body depend upon its energy-content?", one of his annus mirabilis papers, published on 21 November 1905. The formula and its relationship to momentum, as described by the energy–momentum relation, were later developed by other physicists.
## Description
Mass–energy equivalence states that all objects having mass, or massive objects, have a corresponding intrinsic energy, even when they are stationary. In the rest frame of an object, where by definition it is motionless and so has no momentum, the mass and energy are equal or they differ only by a constant factor, the speed of light squared ().
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## Description
Mass–energy equivalence states that all objects having mass, or massive objects, have a corresponding intrinsic energy, even when they are stationary. In the rest frame of an object, where by definition it is motionless and so has no momentum, the mass and energy are equal or they differ only by a constant factor, the speed of light squared (). In Newtonian mechanics, a motionless body has no kinetic energy, and it may or may not have other amounts of internal stored energy, like chemical energy or thermal energy, in addition to any potential energy it may have from its position in a field of force. These energies tend to be much smaller than the mass of the object multiplied by , which is on the order of 1017 joules for a mass of one kilogram. Due to this principle, the mass of the atoms that come out of a nuclear reaction is less than the mass of the atoms that go in, and the difference in mass shows up as heat and light with the same equivalent energy as the difference. In analyzing these extreme events, Einstein's formula can be used with as the energy released (removed), and as the change in mass.
In relativity, all the energy that moves with an object (i.e., the energy as measured in the object's rest frame) contributes to the total mass of the body, which measures how much it resists acceleration.
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In analyzing these extreme events, Einstein's formula can be used with as the energy released (removed), and as the change in mass.
In relativity, all the energy that moves with an object (i.e., the energy as measured in the object's rest frame) contributes to the total mass of the body, which measures how much it resists acceleration. If an isolated box of ideal mirrors could contain light, the individually massless photons would contribute to the total mass of the box by the amount equal to their energy divided by . For an observer in the rest frame, removing energy is the same as removing mass and the formula indicates how much mass is lost when energy is removed. In the same way, when any energy is added to an isolated system, the increase in the mass is equal to the added energy divided by .
## Mass in special relativity
An object moves at different speeds in different frames of reference, depending on the motion of the observer. This implies the kinetic energy, in both Newtonian mechanics and relativity, is 'frame dependent', so that the amount of relativistic energy that an object is measured to have depends on the observer. The relativistic mass of an object is given by the relativistic energy divided by . Because the relativistic mass is exactly proportional to the relativistic energy, relativistic mass and relativistic energy are nearly synonymous; the only difference between them is the units.
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The relativistic mass of an object is given by the relativistic energy divided by . Because the relativistic mass is exactly proportional to the relativistic energy, relativistic mass and relativistic energy are nearly synonymous; the only difference between them is the units. The rest mass or invariant mass of an object is defined as the mass an object has in its rest frame, when it is not moving with respect to the observer. The rest mass is the same for all inertial frames, as it is independent of the motion of the observer, it is the smallest possible value of the relativistic mass of the object. Because of the attraction between components of a system, which results in potential energy, the rest mass is almost never additive; in general, the mass of an object is not the sum of the masses of its parts. The rest mass of an object is the total energy of all the parts, including kinetic energy, as observed from the center of momentum frame, and potential energy. The masses add up only if the constituents are at rest (as observed from the center of momentum frame) and do not attract or repel, so that they do not have any extra kinetic or potential energy. Massless particles are particles with no rest mass, and therefore have no intrinsic energy; their energy is due only to their momentum.
### Relativistic mass
Relativistic mass depends on the motion of the object, so that different observers in relative motion see different values for it.
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Massless particles are particles with no rest mass, and therefore have no intrinsic energy; their energy is due only to their momentum.
### Relativistic mass
Relativistic mass depends on the motion of the object, so that different observers in relative motion see different values for it. The relativistic mass of a moving object is larger than the relativistic mass of an object at rest, because a moving object has kinetic energy. If the object moves slowly, the relativistic mass is nearly equal to the rest mass and both are nearly equal to the classical inertial mass (as it appears in Newton's laws of motion). If the object moves quickly, the relativistic mass is greater than the rest mass by an amount equal to the mass associated with the kinetic energy of the object. Massless particles also have relativistic mass derived from their kinetic energy, equal to their relativistic energy divided by , or . The speed of light is one in a system where length and time are measured in natural units and the relativistic mass and energy would be equal in value and dimension. As it is just another name for the energy, the use of the term relativistic mass is redundant and physicists generally reserve mass to refer to rest mass, or invariant mass, as opposed to relativistic mass. A consequence of this terminology is that the mass is not conserved in special relativity, whereas the conservation of momentum and conservation of energy are both fundamental laws.
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As it is just another name for the energy, the use of the term relativistic mass is redundant and physicists generally reserve mass to refer to rest mass, or invariant mass, as opposed to relativistic mass. A consequence of this terminology is that the mass is not conserved in special relativity, whereas the conservation of momentum and conservation of energy are both fundamental laws.
### Conservation of mass and energy
Conservation of energy is a universal principle in physics and holds for any interaction, along with the conservation of momentum. The classical conservation of mass, in contrast, is violated in certain relativistic settings. This concept has been experimentally proven in a number of ways, including the conversion of mass into kinetic energy in nuclear reactions and other interactions between elementary particles. While modern physics has discarded the expression 'conservation of mass', in older terminology a relativistic mass can also be defined to be equivalent to the energy of a moving system, allowing for a conservation of relativistic mass. Mass conservation breaks down when the energy associated with the mass of a particle is converted into other forms of energy, such as kinetic energy, thermal energy, or radiant energy.
Massless particles
Massless particles have zero rest mass.
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Mass conservation breaks down when the energy associated with the mass of a particle is converted into other forms of energy, such as kinetic energy, thermal energy, or radiant energy.
Massless particles
Massless particles have zero rest mass. The Planck–Einstein relation for the energy for photons is given by the equation , where is the Planck constant and is the photon frequency. This frequency and thus the relativistic energy are frame-dependent. If an observer runs away from a photon in the direction the photon travels from a source, and it catches up with the observer, the observer sees it as having less energy than it had at the source. The faster the observer is traveling with regard to the source when the photon catches up, the less energy the photon would be seen to have. As an observer approaches the speed of light with regard to the source, the redshift of the photon increases, according to the relativistic Doppler effect. The energy of the photon is reduced and as the wavelength becomes arbitrarily large, the photon's energy approaches zero, because of the massless nature of photons, which does not permit any intrinsic energy.
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As an observer approaches the speed of light with regard to the source, the redshift of the photon increases, according to the relativistic Doppler effect. The energy of the photon is reduced and as the wavelength becomes arbitrarily large, the photon's energy approaches zero, because of the massless nature of photons, which does not permit any intrinsic energy.
### Composite systems
For closed systems made up of many parts, like an atomic nucleus, planet, or star, the relativistic energy is given by the sum of the relativistic energies of each of the parts, because energies are additive in these systems. If a system is bound by attractive forces, and the energy gained in excess of the work done is removed from the system, then mass is lost with this removed energy. The mass of an atomic nucleus is less than the total mass of the protons and neutrons that make it up. This mass decrease is also equivalent to the energy required to break up the nucleus into individual protons and neutrons. This effect can be understood by looking at the potential energy of the individual components. The individual particles have a force attracting them together, and forcing them apart increases the potential energy of the particles in the same way that lifting an object up on earth does. This energy is equal to the work required to split the particles apart. The mass of the Solar System is slightly less than the sum of its individual masses.
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This energy is equal to the work required to split the particles apart. The mass of the Solar System is slightly less than the sum of its individual masses.
For an isolated system of particles moving in different directions, the invariant mass of the system is the analog of the rest mass, and is the same for all observers, even those in relative motion. It is defined as the total energy (divided by ) in the center of momentum frame. The center of momentum frame is defined so that the system has zero total momentum; the term center of mass frame is also sometimes used, where the center of mass frame is a special case of the center of momentum frame where the center of mass is put at the origin. A simple example of an object with moving parts but zero total momentum is a container of gas. In this case, the mass of the container is given by its total energy (including the kinetic energy of the gas molecules), since the system's total energy and invariant mass are the same in any reference frame where the momentum is zero, and such a reference frame is also the only frame in which the object can be weighed.
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A simple example of an object with moving parts but zero total momentum is a container of gas. In this case, the mass of the container is given by its total energy (including the kinetic energy of the gas molecules), since the system's total energy and invariant mass are the same in any reference frame where the momentum is zero, and such a reference frame is also the only frame in which the object can be weighed. In a similar way, the theory of special relativity posits that the thermal energy in all objects, including solids, contributes to their total masses, even though this energy is present as the kinetic and potential energies of the atoms in the object, and it (in a similar way to the gas) is not seen in the rest masses of the atoms that make up the object. Similarly, even photons, if trapped in an isolated container, would contribute their energy to the mass of the container. Such extra mass, in theory, could be weighed in the same way as any other type of rest mass, even though individually photons have no rest mass. The property that trapped energy in any form adds weighable mass to systems that have no net momentum is one of the consequences of relativity. It has no counterpart in classical Newtonian physics, where energy never exhibits weighable mass.
### Relation to gravity
Physics has two concepts of mass, the gravitational mass and the inertial mass.
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It has no counterpart in classical Newtonian physics, where energy never exhibits weighable mass.
### Relation to gravity
Physics has two concepts of mass, the gravitational mass and the inertial mass. The gravitational mass is the quantity that determines the strength of the gravitational field generated by an object, as well as the gravitational force acting on the object when it is immersed in a gravitational field produced by other bodies. The inertial mass, on the other hand, quantifies how much an object accelerates if a given force is applied to it. The mass–energy equivalence in special relativity refers to the inertial mass. However, already in the context of Newtonian gravity, the weak equivalence principle is postulated: the gravitational and the inertial mass of every object are the same. Thus, the mass–energy equivalence, combined with the weak equivalence principle, results in the prediction that all forms of energy contribute to the gravitational field generated by an object. This observation is one of the pillars of the general theory of relativity.
The prediction that all forms of energy interact gravitationally has been subject to experimental tests. One of the first observations testing this prediction, called the Eddington experiment, was made during the solar eclipse of May 29, 1919.
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The prediction that all forms of energy interact gravitationally has been subject to experimental tests. One of the first observations testing this prediction, called the Eddington experiment, was made during the solar eclipse of May 29, 1919. During the eclipse, the English astronomer and physicist Arthur Eddington observed that the light from stars passing close to the Sun was bent. The effect is due to the gravitational attraction of light by the Sun. The observation confirmed that the energy carried by light indeed is equivalent to a gravitational mass. Another seminal experiment, the Pound–Rebka experiment, was performed in 1960. In this test a beam of light was emitted from the top of a tower and detected at the bottom. The frequency of the light detected was higher than the light emitted. This result confirms that the energy of photons increases when they fall in the gravitational field of the Earth. The energy, and therefore the gravitational mass, of photons is proportional to their frequency as stated by the Planck's relation.
## Efficiency
In some reactions, matter particles can be destroyed and their associated energy released to the environment as other forms of energy, such as light and heat. One example of such a conversion takes place in elementary particle interactions, where the rest energy is transformed into kinetic energy.
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## Efficiency
In some reactions, matter particles can be destroyed and their associated energy released to the environment as other forms of energy, such as light and heat. One example of such a conversion takes place in elementary particle interactions, where the rest energy is transformed into kinetic energy. Such conversions between types of energy happen in nuclear weapons, in which the protons and neutrons in atomic nuclei lose a small fraction of their original mass, though the mass lost is not due to the destruction of any smaller constituents. Nuclear fission allows a tiny fraction of the energy associated with the mass to be converted into usable energy such as radiation; in the decay of the uranium, for instance, about 0.1% of the mass of the original atom is lost. In theory, it should be possible to destroy matter and convert all of the rest-energy associated with matter into heat and light, but none of the theoretically known methods are practical. One way to harness all the energy associated with mass is to annihilate matter with antimatter. Antimatter is rare in the universe, however, and the known mechanisms of production require more usable energy than would be released in annihilation. CERN estimated in 2011 that over a billion times more energy is required to make and store antimatter than could be released in its annihilation.
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Antimatter is rare in the universe, however, and the known mechanisms of production require more usable energy than would be released in annihilation. CERN estimated in 2011 that over a billion times more energy is required to make and store antimatter than could be released in its annihilation.
As most of the mass which comprises ordinary objects resides in protons and neutrons, converting all the energy of ordinary matter into more useful forms requires that the protons and neutrons be converted to lighter particles, or particles with no mass at all. In the Standard Model of particle physics, the number of protons plus neutrons is nearly exactly conserved. Despite this, Gerard 't Hooft showed that there is a process that converts protons and neutrons to antielectrons and neutrinos. This is the weak SU(2) instanton proposed by the physicists Alexander Belavin, Alexander Markovich Polyakov, Albert Schwarz, and Yu. S. Tyupkin. This process, can in principle destroy matter and convert all the energy of matter into neutrinos and usable energy, but it is normally extraordinarily slow. It was later shown that the process occurs rapidly at extremely high temperatures that would only have been reached shortly after the Big Bang.
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This process, can in principle destroy matter and convert all the energy of matter into neutrinos and usable energy, but it is normally extraordinarily slow. It was later shown that the process occurs rapidly at extremely high temperatures that would only have been reached shortly after the Big Bang.
Many extensions of the standard model contain magnetic monopoles, and in some models of grand unification, these monopoles catalyze proton decay, a process known as the Callan–Rubakov effect. This process would be an efficient mass–energy conversion at ordinary temperatures, but it requires making monopoles and anti-monopoles, whose production is expected to be inefficient. Another method of completely annihilating matter uses the gravitational field of black holes. The British theoretical physicist Stephen Hawking theorized it is possible to throw matter into a black hole and use the emitted heat to generate power. According to the theory of Hawking radiation, however, larger black holes radiate less than smaller ones, so that usable power can only be produced by small black holes.
## Extension for systems in motion
Unlike a system's energy in an inertial frame, the relativistic energy (
$$
E_{\rm rel}
$$
) of a system depends on both the rest mass (
$$
m_0
$$
) and the total momentum of the system.
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According to the theory of Hawking radiation, however, larger black holes radiate less than smaller ones, so that usable power can only be produced by small black holes.
## Extension for systems in motion
Unlike a system's energy in an inertial frame, the relativistic energy (
$$
E_{\rm rel}
$$
) of a system depends on both the rest mass (
$$
m_0
$$
) and the total momentum of the system. The extension of Einstein's equation to these systems is given by:
$$
\begin{align}
E_{\rm rel}^2 - |\mathbf{p} |^2 c^2 &= m_0^2 c^4 \\
\end{align}
$$
or
$$
\begin{align}
E_{\rm rel}^2 - (pc)^2 &= (m_0 c^2)^2 \\
\end{align}
$$
or
$$
\begin{align}
E_{\rm rel} = \sqrt{ (m_0 c^2)^2 + (pc)^2 } \,\!
\end{align}
$$
where the
$$
(pc)^2
$$
term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered.
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## Extension for systems in motion
Unlike a system's energy in an inertial frame, the relativistic energy (
$$
E_{\rm rel}
$$
) of a system depends on both the rest mass (
$$
m_0
$$
) and the total momentum of the system. The extension of Einstein's equation to these systems is given by:
$$
\begin{align}
E_{\rm rel}^2 - |\mathbf{p} |^2 c^2 &= m_0^2 c^4 \\
\end{align}
$$
or
$$
\begin{align}
E_{\rm rel}^2 - (pc)^2 &= (m_0 c^2)^2 \\
\end{align}
$$
or
$$
\begin{align}
E_{\rm rel} = \sqrt{ (m_0 c^2)^2 + (pc)^2 } \,\!
\end{align}
$$
where the
$$
(pc)^2
$$
term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. This equation is called the energy–momentum relation and reduces to
$$
E_{\rm rel} = mc^2
$$
when the momentum term is zero.
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The extension of Einstein's equation to these systems is given by:
$$
\begin{align}
E_{\rm rel}^2 - |\mathbf{p} |^2 c^2 &= m_0^2 c^4 \\
\end{align}
$$
or
$$
\begin{align}
E_{\rm rel}^2 - (pc)^2 &= (m_0 c^2)^2 \\
\end{align}
$$
or
$$
\begin{align}
E_{\rm rel} = \sqrt{ (m_0 c^2)^2 + (pc)^2 } \,\!
\end{align}
$$
where the
$$
(pc)^2
$$
term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. This equation is called the energy–momentum relation and reduces to
$$
E_{\rm rel} = mc^2
$$
when the momentum term is zero. For photons where
$$
m_0 = 0
$$
, the equation reduces to
$$
E_{\rm rel} = pc
$$
.
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This equation is called the energy–momentum relation and reduces to
$$
E_{\rm rel} = mc^2
$$
when the momentum term is zero. For photons where
$$
m_0 = 0
$$
, the equation reduces to
$$
E_{\rm rel} = pc
$$
.
## Low-speed approximation
Using the Lorentz factor, , the energy–momentum can be rewritten as and expanded as a power series:
$$
E = m_0 c^2 \left[1 + \frac{1}{2} \left(\frac{v}{c}\right)^2 + \frac{3}{8} \left(\frac{v}{c}\right)^4 + \frac{5}{16} \left(\frac{v}{c}\right)^6 + \ldots \right].
$$
For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because is small. For low speeds, all but the first two terms can be ignored:
$$
E \approx m_0 c^2 + \frac{1}{2} m_0 v^2.
$$
In classical mechanics, both the term and the high-speed corrections are ignored.
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## Low-speed approximation
Using the Lorentz factor, , the energy–momentum can be rewritten as and expanded as a power series:
$$
E = m_0 c^2 \left[1 + \frac{1}{2} \left(\frac{v}{c}\right)^2 + \frac{3}{8} \left(\frac{v}{c}\right)^4 + \frac{5}{16} \left(\frac{v}{c}\right)^6 + \ldots \right].
$$
For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because is small. For low speeds, all but the first two terms can be ignored:
$$
E \approx m_0 c^2 + \frac{1}{2} m_0 v^2.
$$
In classical mechanics, both the term and the high-speed corrections are ignored. The initial value of the energy is arbitrary, as only the change in energy can be measured and so the term is ignored in classical physics.
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For low speeds, all but the first two terms can be ignored:
$$
E \approx m_0 c^2 + \frac{1}{2} m_0 v^2.
$$
In classical mechanics, both the term and the high-speed corrections are ignored. The initial value of the energy is arbitrary, as only the change in energy can be measured and so the term is ignored in classical physics. While the higher-order terms become important at higher speeds, the Newtonian equation is a highly accurate low-speed approximation; adding in the third term yields:
$$
E \approx m_0 c^2 + \frac{1}{2}m_0 v^2 \left(1 + \frac{3v^2}{4c^2}\right)
$$
.
The difference between the two approximations is given by
$$
\tfrac{3v^2}{4c^2}
$$
, a number very small for everyday objects. In 2018 NASA announced the Parker Solar Probe was the fastest ever, with a speed of . The difference between the approximations for the Parker Solar Probe in 2018 is
$$
\tfrac{3v^2}{4c^2} \approx 3.9 \times 10^{-8}
$$
, which accounts for an energy correction of four parts per hundred million.
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In 2018 NASA announced the Parker Solar Probe was the fastest ever, with a speed of . The difference between the approximations for the Parker Solar Probe in 2018 is
$$
\tfrac{3v^2}{4c^2} \approx 3.9 \times 10^{-8}
$$
, which accounts for an energy correction of four parts per hundred million. The gravitational constant, in contrast, has a standard relative uncertainty of about
$$
2.2 \times 10^{-5}
$$
.
## Applications
### Application to nuclear physics
The nuclear binding energy is the minimum energy that is required to disassemble the nucleus of an atom into its component parts. The mass of an atom is less than the sum of the masses of its constituents due to the attraction of the strong nuclear force. The difference between the two masses is called the mass defect and is related to the binding energy through Einstein's formula. The principle is used in modeling nuclear fission reactions, and it implies that a great amount of energy can be released by the nuclear fission chain reactions used in both nuclear weapons and nuclear power.
A water molecule weighs a little less than two free hydrogen atoms and an oxygen atom. The minuscule mass difference is the energy needed to split the molecule into three individual atoms (divided by ), which was given off as heat when the molecule formed (this heat had mass).
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A water molecule weighs a little less than two free hydrogen atoms and an oxygen atom. The minuscule mass difference is the energy needed to split the molecule into three individual atoms (divided by ), which was given off as heat when the molecule formed (this heat had mass). Similarly, a stick of dynamite in theory weighs a little bit more than the fragments after the explosion; in this case the mass difference is the energy and heat that is released when the dynamite explodes. Such a change in mass may only happen when the system is open, and the energy and mass are allowed to escape. Thus, if a stick of dynamite is detonated in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light would still be equal to the original mass of the chamber and dynamite. If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation. Thus, a 21.5 kiloton () nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents of the box would be heated to millions of degrees without changing total mass and weight.
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This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation. Thus, a 21.5 kiloton () nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents of the box would be heated to millions of degrees without changing total mass and weight. If a transparent window passing only electromagnetic radiation were opened in such an ideal box after the explosion, and a beam of X-rays and other lower-energy light allowed to escape the box, it would eventually be found to weigh one gram less than it had before the explosion. This weight loss and mass loss would happen as the box was cooled by this process, to room temperature. However, any surrounding mass that absorbed the X-rays (and other "heat") would gain this gram of mass from the resulting heating, thus, in this case, the mass "loss" would represent merely its relocation.
### Practical examples
Einstein used the centimetre–gram–second system of units (cgs), but the formula is independent of the system of units. In natural units, the numerical value of the speed of light is set to equal 1, and the formula expresses an equality of numerical values: .
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### Practical examples
Einstein used the centimetre–gram–second system of units (cgs), but the formula is independent of the system of units. In natural units, the numerical value of the speed of light is set to equal 1, and the formula expresses an equality of numerical values: . In the SI system (expressing the ratio in joules per kilogram using the value of in metres per second):
(≈ 9.0 × 1016 joules per kilogram).
So the energy equivalent of one kilogram of mass is
- 89.9 petajoules
- 25.0 billion kilowatt-hours (or 25,000 GW·h)
- 21.5 trillion kilocalories (or 21.5 Pcal)
- 85.2 trillion BTUs (or 0.0852 quads)
or the energy released by combustion of any of the following:
- 21 500 kilotons of TNT-equivalent energy (or 21.5 Mt)
- litres or US gallons of automotive gasoline
Any time energy is released, the process can be evaluated from an perspective. For instance, the "gadget"-style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling.
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For instance, the "gadget"-style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling. The electromagnetic radiation and kinetic energy (thermal and blast energy) released in this explosion carried the missing gram of mass.
Whenever energy is added to a system, the system gains mass, as shown when the equation is rearranged:
- A spring's mass increases whenever it is put into compression or tension. Its mass increase arises from the increased potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring.
- Raising the temperature of an object (increasing its thermal energy) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of platinum and iridium. If its temperature is allowed to change by 1 °C, its mass changes by 1.5 picograms (1 pg = ).
- A spinning ball has greater mass than when it is not spinning. Its increase of mass is exactly the equivalent of the mass of energy of rotation, which is itself the sum of the kinetic energies of all the moving parts of the ball.
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If its temperature is allowed to change by 1 °C, its mass changes by 1.5 picograms (1 pg = ).
- A spinning ball has greater mass than when it is not spinning. Its increase of mass is exactly the equivalent of the mass of energy of rotation, which is itself the sum of the kinetic energies of all the moving parts of the ball. For example, the Earth itself is more massive due to its rotation, than it would be with no rotation. The rotational energy of the Earth is greater than 1024 Joules, which is over 107 kg.
## History
While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass, though nearly all previous authors thought that the energy that contributes to mass comes only from electromagnetic fields. Once discovered, Einstein's formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.
### Developments prior to Einstein
Eighteenth century theories on the correlation of mass and energy included that devised by the English scientist Isaac Newton in 1717, who speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks, where he asks: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?"
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Once discovered, Einstein's formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.
### Developments prior to Einstein
Eighteenth century theories on the correlation of mass and energy included that devised by the English scientist Isaac Newton in 1717, who speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks, where he asks: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?" Swedish scientist and theologian Emanuel Swedenborg, in his Principia of 1734 theorized that all matter is ultimately composed of dimensionless points of "pure and total motion". He described this motion as being without force, direction or speed, but having the potential for force, direction and speed everywhere within it.
During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various ether theories. In 1873 the Russian physicist and mathematician Nikolay Umov pointed out a relation between mass and energy for ether in the form of , where . English engineer Samuel Tolver Preston in 1875 and the Italian industrialist and geologist Olinto De Pretto in 1903, following physicist Georges-Louis Le Sage, imagined that the universe was filled with an ether of tiny particles that always move at speed .
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In 1873 the Russian physicist and mathematician Nikolay Umov pointed out a relation between mass and energy for ether in the form of , where . English engineer Samuel Tolver Preston in 1875 and the Italian industrialist and geologist Olinto De Pretto in 1903, following physicist Georges-Louis Le Sage, imagined that the universe was filled with an ether of tiny particles that always move at speed . Each of these particles has a kinetic energy of up to a small numerical factor, giving a mass–energy relation.
In 1905, independently of Einstein, French polymath Gustave Le Bon speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative philosophy of physics.
#### Electromagnetic mass
There were many attempts in the 19th and the beginning of the 20th century—like those of British physicists J. J. Thomson in 1881 and Oliver Heaviside in 1889, and George Frederick Charles Searle in 1897, German physicists Wilhelm Wien in 1900 and Max Abraham in 1902, and the Dutch physicist Hendrik Antoon Lorentz in 1904—to understand how the mass of a charged object depends on the electrostatic field. This concept was called electromagnetic mass, and was considered as being dependent on velocity and direction as well.
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#### Electromagnetic mass
There were many attempts in the 19th and the beginning of the 20th century—like those of British physicists J. J. Thomson in 1881 and Oliver Heaviside in 1889, and George Frederick Charles Searle in 1897, German physicists Wilhelm Wien in 1900 and Max Abraham in 1902, and the Dutch physicist Hendrik Antoon Lorentz in 1904—to understand how the mass of a charged object depends on the electrostatic field. This concept was called electromagnetic mass, and was considered as being dependent on velocity and direction as well. Lorentz in 1904 gave the following expressions for longitudinal and transverse electromagnetic mass:
$$
m_{L}=\frac{m_{0}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
$$
,
where
$$
m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}
$$
Another way of deriving a type of electromagnetic mass was based on the concept of radiation pressure.
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This concept was called electromagnetic mass, and was considered as being dependent on velocity and direction as well. Lorentz in 1904 gave the following expressions for longitudinal and transverse electromagnetic mass:
$$
m_{L}=\frac{m_{0}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
$$
,
where
$$
m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}
$$
Another way of deriving a type of electromagnetic mass was based on the concept of radiation pressure. In 1900, French polymath Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass
$$
m_{em}=\frac{E_{em}}{c^2}\,.
$$
By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes.
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Lorentz in 1904 gave the following expressions for longitudinal and transverse electromagnetic mass:
$$
m_{L}=\frac{m_{0}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
$$
,
where
$$
m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}
$$
Another way of deriving a type of electromagnetic mass was based on the concept of radiation pressure. In 1900, French polymath Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass
$$
m_{em}=\frac{E_{em}}{c^2}\,.
$$
By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes.
Austrian physicist Friedrich Hasenöhrl showed in 1904 that electromagnetic cavity radiation contributes the "apparent mass"
$$
m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}
$$
to the cavity's mass.
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In 1900, French polymath Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass
$$
m_{em}=\frac{E_{em}}{c^2}\,.
$$
By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes.
Austrian physicist Friedrich Hasenöhrl showed in 1904 that electromagnetic cavity radiation contributes the "apparent mass"
$$
m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}
$$
to the cavity's mass. He argued that this implies mass dependence on temperature as well.
### Einstein: mass–energy equivalence
Einstein did not write the exact formula in his 1905 Annus Mirabilis paper "Does the Inertia of an object Depend Upon Its Energy Content?"; rather, the paper states that if a body gives off the energy by emitting light, its mass diminishes by . This formulation relates only a change in mass to a change in energy without requiring the absolute relationship. The relationship convinced him that mass and energy can be seen as two names for the same underlying, conserved physical quantity. He has stated that the laws of conservation of energy and conservation of mass are "one and the same".
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The relationship convinced him that mass and energy can be seen as two names for the same underlying, conserved physical quantity. He has stated that the laws of conservation of energy and conservation of mass are "one and the same". Einstein elaborated in a 1946 essay that "the principle of the conservation of mass… proved inadequate in the face of the special theory of relativity. It was therefore merged with the energy conservation principle—just as, about 60 years before, the principle of the conservation of mechanical energy had been combined with the principle of the conservation of heat [thermal energy]. We might say that the principle of the conservation of energy, having previously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass—and holds the field alone."
#### Mass–velocity relationship
In developing special relativity, Einstein found that the kinetic energy of a moving body is
$$
E_k = m_0 c^2( \gamma -1 ) = m_0 c^2\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - 1\right),
$$
with the velocity, the rest mass, and the Lorentz factor.
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We might say that the principle of the conservation of energy, having previously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass—and holds the field alone."
#### Mass–velocity relationship
In developing special relativity, Einstein found that the kinetic energy of a moving body is
$$
E_k = m_0 c^2( \gamma -1 ) = m_0 c^2\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - 1\right),
$$
with the velocity, the rest mass, and the Lorentz factor.
He included the second term on the right to make sure that for small velocities the energy would be the same as in classical mechanics, thus satisfying the correspondence principle:
$$
E_k = \frac{1}{2}m_0 v^2 + \cdots
$$
Without this second term, there would be an additional contribution in the energy when the particle is not moving.
#### Einstein's view on mass
Einstein, following Lorentz and Abraham, used velocity- and direction-dependent mass concepts in his 1905 electrodynamics paper and in another paper in 1906. In Einstein's first 1905 paper on , he treated as what would now be called the rest mass, and it has been noted that in his later years he did not like the idea of "relativistic mass".
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#### Einstein's view on mass
Einstein, following Lorentz and Abraham, used velocity- and direction-dependent mass concepts in his 1905 electrodynamics paper and in another paper in 1906. In Einstein's first 1905 paper on , he treated as what would now be called the rest mass, and it has been noted that in his later years he did not like the idea of "relativistic mass".
In modern physics terminology, relativistic energy is used in lieu of relativistic mass and the term "mass" is reserved for the rest mass. Historically, there has been considerable debate over the use of the concept of "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. One view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. Another view, attributed to Norwegian physicist Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection.
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One view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. Another view, attributed to Norwegian physicist Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection.
#### Einstein's 1905 derivation
Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct expression for the kinetic energy of particles:
$$
E_{k}=mc^{2}\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)
$$
.
Now the question remained open as to which formulation applies to bodies at rest. This was tackled by Einstein in his paper "Does the inertia of a body depend upon its energy content?", one of his Annus Mirabilis papers. Here, Einstein used to represent the speed of light in vacuum and to represent the energy lost by a body in the form of radiation. Consequently, the equation was not originally written as a formula but as a sentence in German saying that "if a body gives off the energy in the form of radiation, its mass diminishes by ."
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Here, Einstein used to represent the speed of light in vacuum and to represent the energy lost by a body in the form of radiation. Consequently, the equation was not originally written as a formula but as a sentence in German saying that "if a body gives off the energy in the form of radiation, its mass diminishes by ." A remark placed above it informed that the equation was approximated by neglecting "magnitudes of fourth and higher orders" of a series expansion. Einstein used a body emitting two light pulses in opposite directions, having energies of before and after the emission as seen in its rest frame. As seen from a moving frame, becomes and becomes . Einstein obtained, in modern notation:
$$
\left(H_{0}-E_{0}\right)-\left(H_{1}-E_{1}\right)=E\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)
$$
.
He then argued that can only differ from the kinetic energy by an additive constant, which gives
$$
K_{0}-K_{1}=E\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)
$$
.
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Einstein obtained, in modern notation:
$$
\left(H_{0}-E_{0}\right)-\left(H_{1}-E_{1}\right)=E\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)
$$
.
He then argued that can only differ from the kinetic energy by an additive constant, which gives
$$
K_{0}-K_{1}=E\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)
$$
.
Neglecting effects higher than third order in after a Taylor series expansion of the right side of this yields:
$$
K_{0}-K_{1}=\frac{E}{c^{2}}\frac{v^{2}}{2}.
$$
Einstein concluded that the emission reduces the body's mass by , and that the mass of a body is a measure of its energy content.
The correctness of Einstein's 1905 derivation of was criticized by German theoretical physicist Max Planck in 1907, who argued that it is only valid to first approximation.
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Neglecting effects higher than third order in after a Taylor series expansion of the right side of this yields:
$$
K_{0}-K_{1}=\frac{E}{c^{2}}\frac{v^{2}}{2}.
$$
Einstein concluded that the emission reduces the body's mass by , and that the mass of a body is a measure of its energy content.
The correctness of Einstein's 1905 derivation of was criticized by German theoretical physicist Max Planck in 1907, who argued that it is only valid to first approximation. Another criticism was formulated by American physicist Herbert Ives in 1952 and the Israeli physicist Max Jammer in 1961, asserting that Einstein's derivation is based on begging the question. Other scholars, such as American and Chilean philosophers John Stachel and Roberto Torretti, have argued that Ives' criticism was wrong, and that Einstein's derivation was correct. American physics writer Hans Ohanian, in 2008, agreed with Stachel/Torretti's criticism of Ives, though he argued that Einstein's derivation was wrong for other reasons.
#### Relativistic center-of-mass theorem of 1906
Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for the center-of-mass theorem to hold.
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American physics writer Hans Ohanian, in 2008, agreed with Stachel/Torretti's criticism of Ives, though he argued that Einstein's derivation was wrong for other reasons.
#### Relativistic center-of-mass theorem of 1906
Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for the center-of-mass theorem to hold. On this occasion, Einstein referred to Poincaré's 1900 paper and wrote: "Although the merely formal considerations, which we will need for the proof, are already mostly contained in a work by H. Poincaré2, for the sake of clarity I will not rely on that work." In Einstein's more physical, as opposed to formal or mathematical, point of view, there was no need for fictitious masses. He could avoid the perpetual motion problem because, on the basis of the mass–energy equivalence, he could show that the transport of inertia that accompanies the emission and absorption of radiation solves the problem. Poincaré's rejection of the principle of action–reaction can be avoided through Einstein's , because mass conservation appears as a special case of the energy conservation law.
#### Further developments
There were several further developments in the first decade of the twentieth century.
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Poincaré's rejection of the principle of action–reaction can be avoided through Einstein's , because mass conservation appears as a special case of the energy conservation law.
#### Further developments
There were several further developments in the first decade of the twentieth century. In May 1907, Einstein explained that the expression for energy of a moving mass point assumes the simplest form when its expression for the state of rest is chosen to be (where is the mass), which is in agreement with the "principle of the equivalence of mass and energy". In addition, Einstein used the formula , with being the energy of a system of mass points, to describe the energy and mass increase of that system when the velocity of the differently moving mass points is increased. Max Planck rewrote Einstein's mass–energy relationship as in June 1907, where is the pressure and the volume to express the relation between mass, its latent energy, and thermodynamic energy within the body. Subsequently, in October 1907, this was rewritten as and given a quantum interpretation by German physicist Johannes Stark, who assumed its validity and correctness. In December 1907, Einstein expressed the equivalence in the form and concluded: "A mass is equivalent, as regards inertia, to a quantity of energy . […] It appears far more natural to consider every inertial mass as a store of energy."
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In December 1907, Einstein expressed the equivalence in the form and concluded: "A mass is equivalent, as regards inertia, to a quantity of energy . […] It appears far more natural to consider every inertial mass as a store of energy." American physical chemists Gilbert N. Lewis and Richard C. Tolman used two variations of the formula in 1909: and , with being the relativistic energy (the energy of an object when the object is moving), is the rest energy (the energy when not moving), is the relativistic mass (the rest mass and the extra mass gained when moving), and is the rest mass. The same relations in different notation were used by Lorentz in 1913 and 1914, though he placed the energy on the left-hand side: and , with being the total energy (rest energy plus kinetic energy) of a moving material point, its rest energy, the relativistic mass, and the invariant mass.
In 1911, German physicist Max von Laue gave a more comprehensive proof of from the stress–energy tensor, which was later generalized by German mathematician Felix Klein in 1918.
Einstein returned to the topic once again after World War II and this time he wrote in the title of his article intended as an explanation for a general reader by analogy.
#### Alternative version
An alternative version of Einstein's thought experiment was proposed by American theoretical physicist Fritz Rohrlich in 1990, who based his reasoning on the Doppler effect.
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Einstein returned to the topic once again after World War II and this time he wrote in the title of his article intended as an explanation for a general reader by analogy.
#### Alternative version
An alternative version of Einstein's thought experiment was proposed by American theoretical physicist Fritz Rohrlich in 1990, who based his reasoning on the Doppler effect. Like Einstein, he considered a body at rest with mass . If the body is examined in a frame moving with nonrelativistic velocity , it is no longer at rest and in the moving frame it has momentum . Then he supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy . In its rest frame, the object remains at rest after the emission since the two beams are equal in strength and carry opposite momentum. However, if the same process is considered in a frame that moves with velocity to the left, the pulse moving to the left is redshifted, while the pulse moving to the right is blue shifted. The blue light carries more momentum than the red light, so that the momentum of the light in the moving frame is not balanced: the light is carrying some net momentum to the right. The object has not changed its velocity before or after the emission. Yet in this frame it has lost some right-momentum to the light. The only way it could have lost momentum is by losing mass.
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Yet in this frame it has lost some right-momentum to the light. The only way it could have lost momentum is by losing mass. This also solves Poincaré's radiation paradox. The velocity is small, so the right-moving light is blueshifted by an amount equal to the nonrelativistic Doppler shift factor . The momentum of the light is its energy divided by , and it is increased by a factor of . So the right-moving light is carrying an extra momentum given by:
$$
\Delta P = {v \over c}{E \over 2c} .
$$
The left-moving light carries a little less momentum, by the same amount . So the total right-momentum in both light pulses is twice . This is the right-momentum that the object lost.
$$
2\Delta P = v {E\over c^2} .
$$
The momentum of the object in the moving frame after the emission is reduced to this amount:
$$
P' = Mv - 2\Delta P = \left(M - {E\over c^2}\right)v .
$$
So the change in the object's mass is equal to the total energy lost divided by .
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So the total right-momentum in both light pulses is twice . This is the right-momentum that the object lost.
$$
2\Delta P = v {E\over c^2} .
$$
The momentum of the object in the moving frame after the emission is reduced to this amount:
$$
P' = Mv - 2\Delta P = \left(M - {E\over c^2}\right)v .
$$
So the change in the object's mass is equal to the total energy lost divided by . Since any emission of energy can be carried out by a two-step process, where first the energy is emitted as light and then the light is converted to some other form of energy, any emission of energy is accompanied by a loss of mass. Similarly, by considering absorption, a gain in energy is accompanied by a gain in mass.
### Radioactivity and nuclear energy
It was quickly noted after the discovery of radioactivity in 1897 that the total energy due to radioactive processes is about one million times greater than that involved in any known molecular change, raising the question of where the energy comes from.
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### Radioactivity and nuclear energy
It was quickly noted after the discovery of radioactivity in 1897 that the total energy due to radioactive processes is about one million times greater than that involved in any known molecular change, raising the question of where the energy comes from. After eliminating the idea of absorption and emission of some sort of Lesagian ether particles, the existence of a huge amount of latent energy, stored within matter, was proposed by New Zealand physicist Ernest Rutherford and British radiochemist Frederick Soddy in 1903. Rutherford also suggested that this internal energy is stored within normal matter as well. He went on to speculate in 1904: "If it were ever found possible to control at will the rate of disintegration of the radio-elements, an enormous amount of energy could be obtained from a small quantity of matter. "
Einstein's equation does not explain the large energies released in radioactive decay, but can be used to quantify them. The theoretical explanation for radioactive decay is given by the nuclear forces responsible for holding atoms together, though these forces were still unknown in 1905. The enormous energy released from radioactive decay had previously been measured by Rutherford and was much more easily measured than the small change in the gross mass of materials as a result.
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The theoretical explanation for radioactive decay is given by the nuclear forces responsible for holding atoms together, though these forces were still unknown in 1905. The enormous energy released from radioactive decay had previously been measured by Rutherford and was much more easily measured than the small change in the gross mass of materials as a result. Einstein's equation, by theory, can give these energies by measuring mass differences before and after reactions, but in practice, these mass differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive decay energies with a calorimeter was thought possibly likely to allow measurement of changes in mass difference, as a check on Einstein's equation itself. Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which was known by then to release enough energy to possibly be "weighed," when missing from the system. However, radioactivity seemed to proceed at its own unalterable pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. Rutherford was reported in 1933 to have declared that this energy could not be exploited efficiently: "Anyone who expects a source of power from the transformation of the atom is talking moonshine.
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However, radioactivity seemed to proceed at its own unalterable pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. Rutherford was reported in 1933 to have declared that this energy could not be exploited efficiently: "Anyone who expects a source of power from the transformation of the atom is talking moonshine. "
This outlook changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single nuclides and their reactions to be calculated directly, and compared with the sum of masses for the particles that made up their composition. In 1933, the energy released from the reaction of lithium-7 plus protons giving rise to two alpha particles, allowed Einstein's equation to be tested to an error of ±0.5%. However, scientists still did not see such reactions as a practical source of power, due to the energy cost of accelerating reaction particles. After the very public demonstration of huge energies released from nuclear fission after the atomic bombings of Hiroshima and Nagasaki in 1945, the equation became directly linked in the public eye with the power and peril of nuclear weapons.
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However, scientists still did not see such reactions as a practical source of power, due to the energy cost of accelerating reaction particles. After the very public demonstration of huge energies released from nuclear fission after the atomic bombings of Hiroshima and Nagasaki in 1945, the equation became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured on page 2 of the Smyth Report, the official 1945 release by the US government on the development of the atomic bomb, and by 1946 the equation was linked closely enough with Einstein's work that the cover of Time magazine prominently featured a picture of Einstein next to an image of a mushroom cloud emblazoned with the equation. Einstein himself had only a minor role in the Manhattan Project: he had cosigned a letter to the U.S. president in 1939 urging funding for research into atomic energy, warning that an atomic bomb was theoretically possible. The letter persuaded Roosevelt to devote a significant portion of the wartime budget to atomic research. Without a security clearance, Einstein's only scientific contribution was an analysis of an isotope separation method in theoretical terms. It was inconsequential, on account of Einstein not being given sufficient information to fully work on the problem.
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Without a security clearance, Einstein's only scientific contribution was an analysis of an isotope separation method in theoretical terms. It was inconsequential, on account of Einstein not being given sufficient information to fully work on the problem.
While is useful for understanding the amount of energy potentially released in a fission reaction, it was not strictly necessary to develop the weapon, once the fission process was known, and its energy measured at 200 MeV (which was directly possible, using a quantitative Geiger counter, at that time). The physicist and Manhattan Project participant Robert Serber noted that somehow "the popular notion took hold long ago that Einstein's theory of relativity, in particular his equation , plays some essential role in the theory of fission. Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of relativity is not required in discussing fission. The theory of fission is what physicists call a non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly." There are other views on the equation's importance to nuclear reactions.
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The theory of fission is what physicists call a non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly." There are other views on the equation's importance to nuclear reactions. In late 1938, the Austrian-Swedish and British physicists Lise Meitner and Otto Robert Frisch—while on a winter walk during which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission—directly used Einstein's equation to help them understand the quantitative energetics of the reaction that overcame the "surface tension-like" forces that hold the nucleus together, and allowed the fission fragments to separate to a configuration from which their charges could force them into an energetic fission. To do this, they used packing fraction, or nuclear binding energy values for elements. These, together with use of allowed them to realize on the spot that the basic fission process was energetically possible.
### Einstein's equation written
According to the Einstein Papers Project at the California Institute of Technology and Hebrew University of Jerusalem, there remain only four known copies of this equation as written by Einstein. One of these is a letter written in German to Ludwik Silberstein, which was in Silberstein's archives, and sold at auction for $1.2 million, RR Auction of Boston, Massachusetts said on May 21, 2021.
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In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries above the main diagonal are zero. Similarly, a square matrix is called if all the entries below the main diagonal are zero.
Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.
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By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.
## Description
A matrix of the form
$$
L = \begin{bmatrix}
\ell_{1,1} & & & & 0 \\
\ell_{2,1} & \ell_{2,2} & & & \\
\ell_{3,1} & \ell_{3,2} & \ddots & & \\
_BLOCK0_ \ell_{n,1} & \ell_{n,2} & \ldots & \ell_{n,n-1} & \ell_{n,n}
\end{bmatrix}
$$
is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form
$$
U = \begin{bmatrix}
u_{1,1} & u_{1,2} & u_{1,3} & \ldots & u_{1,n} \\
_BLOCK1_\end{bmatrix}
$$
is called an upper triangular matrix or right triangular matrix.
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U if and only if all its leading principal minors are non-zero.
## Description
A matrix of the form
$$
L = \begin{bmatrix}
\ell_{1,1} & & & & 0 \\
\ell_{2,1} & \ell_{2,2} & & & \\
\ell_{3,1} & \ell_{3,2} & \ddots & & \\
_BLOCK0_ \ell_{n,1} & \ell_{n,2} & \ldots & \ell_{n,n-1} & \ell_{n,n}
\end{bmatrix}
$$
is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form
$$
U = \begin{bmatrix}
u_{1,1} & u_{1,2} & u_{1,3} & \ldots & u_{1,n} \\
_BLOCK1_\end{bmatrix}
$$
is called an upper triangular matrix or right triangular matrix. A lower or left triangular matrix is commonly denoted with the variable L, and an upper or right triangular matrix is commonly denoted with the variable U or R.
A matrix that is both upper and lower triangular is diagonal.
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## Description
A matrix of the form
$$
L = \begin{bmatrix}
\ell_{1,1} & & & & 0 \\
\ell_{2,1} & \ell_{2,2} & & & \\
\ell_{3,1} & \ell_{3,2} & \ddots & & \\
_BLOCK0_ \ell_{n,1} & \ell_{n,2} & \ldots & \ell_{n,n-1} & \ell_{n,n}
\end{bmatrix}
$$
is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form
$$
U = \begin{bmatrix}
u_{1,1} & u_{1,2} & u_{1,3} & \ldots & u_{1,n} \\
_BLOCK1_\end{bmatrix}
$$
is called an upper triangular matrix or right triangular matrix. A lower or left triangular matrix is commonly denoted with the variable L, and an upper or right triangular matrix is commonly denoted with the variable U or R.
A matrix that is both upper and lower triangular is diagonal. Matrices that are similar to triangular matrices are called triangularisable.
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A lower or left triangular matrix is commonly denoted with the variable L, and an upper or right triangular matrix is commonly denoted with the variable U or R.
A matrix that is both upper and lower triangular is diagonal. Matrices that are similar to triangular matrices are called triangularisable.
A non-square (or sometimes any) matrix with zeros above (below) the diagonal is called a lower (upper) trapezoidal matrix. The non-zero entries form the shape of a trapezoid.
###
### Examples
The matrix
$$
\begin{bmatrix}
1 & 0 & 0 \\
2 & 96 & 0 \\
4 & 9 & 69
\end{bmatrix}
$$
is lower triangular, and
$$
\begin{bmatrix}
1 & 4 & 1 \\
0 & 6 & 9 \\
0 & 0 & 1
\end{bmatrix}
$$
is upper triangular.
## Forward and back substitution
A matrix equation in the form
$$
L\mathbf{x} = \mathbf{b}
$$
or
$$
U\mathbf{x} = \mathbf{b}
$$
is very easy to solve by an iterative process called forward substitution for lower triangular matrices and analogously back substitution for upper triangular matrices.
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### Examples
The matrix
$$
\begin{bmatrix}
1 & 0 & 0 \\
2 & 96 & 0 \\
4 & 9 & 69
\end{bmatrix}
$$
is lower triangular, and
$$
\begin{bmatrix}
1 & 4 & 1 \\
0 & 6 & 9 \\
0 & 0 & 1
\end{bmatrix}
$$
is upper triangular.
## Forward and back substitution
A matrix equation in the form
$$
L\mathbf{x} = \mathbf{b}
$$
or
$$
U\mathbf{x} = \mathbf{b}
$$
is very easy to solve by an iterative process called forward substitution for lower triangular matrices and analogously back substitution for upper triangular matrices. The process is so called because for lower triangular matrices, one first computes
$$
x_1
$$
, then substitutes that forward into the next equation to solve for
$$
x_2
$$
, and repeats through to
$$
x_n
$$
. In an upper triangular matrix, one works backwards, first computing
$$
x_n
$$
, then substituting that back into the previous equation to solve for
$$
x_{n-1}
$$
, and repeating through
$$
x_1
$$
.
Notice that this does not require inverting the matrix.
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In an upper triangular matrix, one works backwards, first computing
$$
x_n
$$
, then substituting that back into the previous equation to solve for
$$
x_{n-1}
$$
, and repeating through
$$
x_1
$$
.
Notice that this does not require inverting the matrix.
### Forward substitution
The matrix equation Lx = b can be written as a system of linear equations
$$
\begin{matrix}
\ell_{1,1} x_1 & & & & & & & = & b_1 \\
\ell_{2,1} x_1 & + & \ell_{2,2} x_2 & & & & & = & b_2 \\
_BLOCK0_ \ell_{m,1} x_1 & + & \ell_{m,2} x_2 & + & \dotsb & + & \ell_{m,m} x_m & = & b_m \\
\end{matrix}
$$
Observe that the first equation (
$$
\ell_{1,1} x_1 = b_1
$$
) only involves
$$
x_1
$$
, and thus one can solve for
$$
x_1
$$
directly.
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Notice that this does not require inverting the matrix.
### Forward substitution
The matrix equation Lx = b can be written as a system of linear equations
$$
\begin{matrix}
\ell_{1,1} x_1 & & & & & & & = & b_1 \\
\ell_{2,1} x_1 & + & \ell_{2,2} x_2 & & & & & = & b_2 \\
_BLOCK0_ \ell_{m,1} x_1 & + & \ell_{m,2} x_2 & + & \dotsb & + & \ell_{m,m} x_m & = & b_m \\
\end{matrix}
$$
Observe that the first equation (
$$
\ell_{1,1} x_1 = b_1
$$
) only involves
$$
x_1
$$
, and thus one can solve for
$$
x_1
$$
directly. The second equation only involves
$$
x_1
$$
and
$$
x_2
$$
, and thus can be solved once one substitutes in the already solved value for
$$
x_1
$$
.
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### Forward substitution
The matrix equation Lx = b can be written as a system of linear equations
$$
\begin{matrix}
\ell_{1,1} x_1 & & & & & & & = & b_1 \\
\ell_{2,1} x_1 & + & \ell_{2,2} x_2 & & & & & = & b_2 \\
_BLOCK0_ \ell_{m,1} x_1 & + & \ell_{m,2} x_2 & + & \dotsb & + & \ell_{m,m} x_m & = & b_m \\
\end{matrix}
$$
Observe that the first equation (
$$
\ell_{1,1} x_1 = b_1
$$
) only involves
$$
x_1
$$
, and thus one can solve for
$$
x_1
$$
directly. The second equation only involves
$$
x_1
$$
and
$$
x_2
$$
, and thus can be solved once one substitutes in the already solved value for
$$
x_1
$$
. Continuing in this way, the
$$
k
$$
-th equation only involves
$$
x_1,\dots,x_k
$$
, and one can solve for
$$
x_k
$$
using the previously solved values for
$$
x_1,\dots,x_{k-1}
$$
.
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The second equation only involves
$$
x_1
$$
and
$$
x_2
$$
, and thus can be solved once one substitutes in the already solved value for
$$
x_1
$$
. Continuing in this way, the
$$
k
$$
-th equation only involves
$$
x_1,\dots,x_k
$$
, and one can solve for
$$
x_k
$$
using the previously solved values for
$$
x_1,\dots,x_{k-1}
$$
. The resulting formulas are:
$$
\begin{align}
x_1 &= \frac{b_1}{\ell_{1,1}}, \\
x_2 &= \frac{b_2 - \ell_{2,1} x_1}{\ell_{2,2}}, \\
_BLOCK1_ x_m &= \frac{b_m - \sum_{i=1}^{m-1} \ell_{m,i}x_i}{\ell_{m,m}}.
\end{align}
$$
A matrix equation with an upper triangular matrix U can be solved in an analogous way, only working backwards.
### Applications
Forward substitution is used in financial bootstrapping to construct a yield curve.
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The resulting formulas are:
$$
\begin{align}
x_1 &= \frac{b_1}{\ell_{1,1}}, \\
x_2 &= \frac{b_2 - \ell_{2,1} x_1}{\ell_{2,2}}, \\
_BLOCK1_ x_m &= \frac{b_m - \sum_{i=1}^{m-1} \ell_{m,i}x_i}{\ell_{m,m}}.
\end{align}
$$
A matrix equation with an upper triangular matrix U can be solved in an analogous way, only working backwards.
### Applications
Forward substitution is used in financial bootstrapping to construct a yield curve.
## Properties
The transpose of an upper triangular matrix is a lower triangular matrix and vice versa.
A matrix which is both symmetric and triangular is diagonal.
In a similar vein, a matrix which is both normal (meaning A*A = AA*, where A* is the conjugate transpose) and triangular is also diagonal. This can be seen by looking at the diagonal entries of A*A and AA*.
The determinant and permanent of a triangular matrix equal the product of the diagonal entries, as can be checked by direct computation.
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This can be seen by looking at the diagonal entries of A*A and AA*.
The determinant and permanent of a triangular matrix equal the product of the diagonal entries, as can be checked by direct computation.
In fact more is true: the eigenvalues of a triangular matrix are exactly its diagonal entries. Moreover, each eigenvalue occurs exactly k times on the diagonal, where k is its algebraic multiplicity, that is, its multiplicity as a root of the characteristic polynomial
$$
p_A(x)=\det(xI-A)
$$
of A. In other words, the characteristic polynomial of a triangular n×n matrix A is exactly
$$
p_A(x) = (x-a_{11})(x-a_{22})\cdots(x-a_{nn})
$$
,
that is, the unique degree n polynomial whose roots are the diagonal entries of A (with multiplicities).
To see this, observe that _ BLOCK2_ is also triangular and hence its determinant
$$
\det(xI-A)
$$
is the product of its diagonal entries
$$
(x-a_{11})(x-a_{22})\cdots(x-a_{nn})
$$
.
## Special forms
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BLOCK2_ is also triangular and hence its determinant
$$
\det(xI-A)
$$
is the product of its diagonal entries
$$
(x-a_{11})(x-a_{22})\cdots(x-a_{nn})
$$
.
## Special forms
### Unitriangular matrix
If the entries on the main diagonal of a (lower or upper) triangular matrix are all 1, the matrix is called (lower or upper) unitriangular.
Other names used for these matrices are unit (lower or upper) triangular, or very rarely normed (lower or upper) triangular. However, a unit triangular matrix is not the same as the unit matrix, and a normed triangular matrix has nothing to do with the notion of matrix norm.
All finite unitriangular matrices are unipotent.
### Strictly triangular matrix
If all of the entries on the main diagonal of a (lower or upper) triangular matrix are also 0, the matrix is called strictly (lower or upper) triangular.
All finite strictly triangular matrices are nilpotent of index at most n as a consequence of the Cayley-Hamilton theorem.
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### Strictly triangular matrix
If all of the entries on the main diagonal of a (lower or upper) triangular matrix are also 0, the matrix is called strictly (lower or upper) triangular.
All finite strictly triangular matrices are nilpotent of index at most n as a consequence of the Cayley-Hamilton theorem.
### Atomic triangular matrix
An atomic (lower or upper) triangular matrix is a special form of unitriangular matrix, where all of the off-diagonal elements are zero, except for the entries in a single column. Such a matrix is also called a Frobenius matrix, a Gauss matrix, or a Gauss transformation matrix.
### Block triangular matrix
A block triangular matrix is a block matrix (partitioned matrix) that is a triangular matrix.
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Such a matrix is also called a Frobenius matrix, a Gauss matrix, or a Gauss transformation matrix.
### Block triangular matrix
A block triangular matrix is a block matrix (partitioned matrix) that is a triangular matrix.
#### Upper block triangular
A matrix
$$
A
$$
is upper block triangular if
$$
A = \begin{bmatrix}
A_{11} & A_{12} & \cdots & A_{1k} \\
0 & A_{22} & \cdots & A_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & A_{kk}
\end{bmatrix}
$$
,
where
$$
A_{ij} \in \mathbb{F}^{n_i \times n_j}
$$
for all
$$
i, j = 1, \ldots, k
$$
.
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### Block triangular matrix
A block triangular matrix is a block matrix (partitioned matrix) that is a triangular matrix.
#### Upper block triangular
A matrix
$$
A
$$
is upper block triangular if
$$
A = \begin{bmatrix}
A_{11} & A_{12} & \cdots & A_{1k} \\
0 & A_{22} & \cdots & A_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & A_{kk}
\end{bmatrix}
$$
,
where
$$
A_{ij} \in \mathbb{F}^{n_i \times n_j}
$$
for all
$$
i, j = 1, \ldots, k
$$
.
#### Lower block triangular
A matrix
$$
A
$$
is lower block triangular if
$$
A = \begin{bmatrix}
A_{11} & 0 & \cdots & 0 \\
A_{21} & A_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
A_{k1} & A_{k2} & \cdots & A_{kk}
\end{bmatrix}
$$
,
where
$$
A_{ij} \in \mathbb{F}^{n_i \times n_j}
$$
for all
$$
i, j = 1, \ldots, k
$$
.
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#### Upper block triangular
A matrix
$$
A
$$
is upper block triangular if
$$
A = \begin{bmatrix}
A_{11} & A_{12} & \cdots & A_{1k} \\
0 & A_{22} & \cdots & A_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & A_{kk}
\end{bmatrix}
$$
,
where
$$
A_{ij} \in \mathbb{F}^{n_i \times n_j}
$$
for all
$$
i, j = 1, \ldots, k
$$
.
#### Lower block triangular
A matrix
$$
A
$$
is lower block triangular if
$$
A = \begin{bmatrix}
A_{11} & 0 & \cdots & 0 \\
A_{21} & A_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
A_{k1} & A_{k2} & \cdots & A_{kk}
\end{bmatrix}
$$
,
where
$$
A_{ij} \in \mathbb{F}^{n_i \times n_j}
$$
for all
$$
i, j = 1, \ldots, k
$$
.
## Triangularisability
A matrix that is similar to a triangular matrix is referred to as triangularizable.
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#### Lower block triangular
A matrix
$$
A
$$
is lower block triangular if
$$
A = \begin{bmatrix}
A_{11} & 0 & \cdots & 0 \\
A_{21} & A_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
A_{k1} & A_{k2} & \cdots & A_{kk}
\end{bmatrix}
$$
,
where
$$
A_{ij} \in \mathbb{F}^{n_i \times n_j}
$$
for all
$$
i, j = 1, \ldots, k
$$
.
## Triangularisability
A matrix that is similar to a triangular matrix is referred to as triangularizable. Abstractly, this is equivalent to stabilizing a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis
$$
(e_1,\ldots,e_n)
$$
and the resulting flag
$$
0 < \left\langle e_1\right\rangle < \left\langle e_1,e_2\right\rangle < \cdots < \left\langle e_1,\ldots,e_n \right\rangle = K^n.
$$
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## Triangularisability
A matrix that is similar to a triangular matrix is referred to as triangularizable. Abstractly, this is equivalent to stabilizing a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis
$$
(e_1,\ldots,e_n)
$$
and the resulting flag
$$
0 < \left\langle e_1\right\rangle < \left\langle e_1,e_2\right\rangle < \cdots < \left\langle e_1,\ldots,e_n \right\rangle = K^n.
$$
All flags are conjugate (as the general linear group acts transitively on bases), so any matrix that stabilises a flag is similar to one that stabilizes the standard flag.
Any complex square matrix is triangularizable. In fact, a matrix A over a field containing all of the eigenvalues of A (for example, any matrix over an algebraically closed field) is similar to a triangular matrix. This can be proven by using induction on the fact that A has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that A stabilizes a flag, and is thus triangularizable with respect to a basis for that flag.
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In fact, a matrix A over a field containing all of the eigenvalues of A (for example, any matrix over an algebraically closed field) is similar to a triangular matrix. This can be proven by using induction on the fact that A has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that A stabilizes a flag, and is thus triangularizable with respect to a basis for that flag.
A more precise statement is given by the Jordan normal form theorem, which states that in this situation, A is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.
In the case of complex matrices, it is possible to say more about triangularization, namely, that any square matrix A has a Schur decomposition. This means that A is unitarily equivalent (i.e. similar, using a unitary matrix as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag.
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A has a Schur decomposition. This means that A is unitarily equivalent (i.e. similar, using a unitary matrix as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag.
### Simultaneous triangularisability
A set of matrices
$$
A_1, \ldots, A_k
$$
are said to be if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix P. Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the
$$
A_i,
$$
denoted
$$
K[A_1,\ldots,A_k].
$$
Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a Borel subalgebra.
The basic result is that (over an algebraically closed field), the commuting matrices
$$
A,B
$$
or more generally
$$
A_1,\ldots,A_k
$$
are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before.
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The basic result is that (over an algebraically closed field), the commuting matrices
$$
A,B
$$
or more generally
$$
A_1,\ldots,A_k
$$
are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. This was proven by Frobenius, starting in 1878 for a commuting pair, as discussed at commuting matrices. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices.
The fact that commuting matrices have a common eigenvector can be interpreted as a result of Hilbert's Nullstellensatz: commuting matrices form a commutative algebra
$$
K[A_1,\ldots,A_k]
$$
over
$$
K[x_1,\ldots,x_k]
$$
which can be interpreted as a variety in k-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz. In algebraic terms, these operators correspond to an algebra representation of the polynomial algebra in k variables.
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The fact that commuting matrices have a common eigenvector can be interpreted as a result of Hilbert's Nullstellensatz: commuting matrices form a commutative algebra
$$
K[A_1,\ldots,A_k]
$$
over
$$
K[x_1,\ldots,x_k]
$$
which can be interpreted as a variety in k-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz. In algebraic terms, these operators correspond to an algebra representation of the polynomial algebra in k variables.
This is generalized by Lie's theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable, the case of commuting matrices being the abelian Lie algebra case, abelian being a fortiori solvable.
More generally and precisely, a set of matrices
$$
A_1,\ldots,A_k
$$
is simultaneously triangularisable if and only if the matrix
$$
p(A_1,\ldots,A_k)[A_i,A_j]
$$
is nilpotent for all polynomials p in k non-commuting variables, where _
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This is generalized by Lie's theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable, the case of commuting matrices being the abelian Lie algebra case, abelian being a fortiori solvable.
More generally and precisely, a set of matrices
$$
A_1,\ldots,A_k
$$
is simultaneously triangularisable if and only if the matrix
$$
p(A_1,\ldots,A_k)[A_i,A_j]
$$
is nilpotent for all polynomials p in k non-commuting variables, where _ BLOCK9_ is the commutator; for commuting
$$
A_i
$$
the commutator vanishes so this holds. This was proven by Drazin, Dungey, and Gruenberg in 1951; a brief proof is given by Prasolov in 1994. One direction is clear: if the matrices are simultaneously triangularisable, then
$$
[A_i, A_j]
$$
is strictly upper triangularizable (hence nilpotent), which is preserved by multiplication by any
$$
A_k
$$
or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.
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This was proven by Drazin, Dungey, and Gruenberg in 1951; a brief proof is given by Prasolov in 1994. One direction is clear: if the matrices are simultaneously triangularisable, then
$$
[A_i, A_j]
$$
is strictly upper triangularizable (hence nilpotent), which is preserved by multiplication by any
$$
A_k
$$
or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.
## Algebras of triangular matrices
Upper triangularity is preserved by many operations:
- The sum of two upper triangular matrices is upper triangular.
- The product of two upper triangular matrices is upper triangular.
- The inverse of an upper triangular matrix, if it exists, is upper triangular.
- The product of an upper triangular matrix and a scalar is upper triangular.
Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the Lie algebra of square matrices of a fixed size, where the Lie bracket [a, b] given by the commutator .
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Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the Lie algebra of square matrices of a fixed size, where the Lie bracket [a, b] given by the commutator . The Lie algebra of all upper triangular matrices is a solvable Lie algebra. It is often referred to as a Borel subalgebra of the Lie algebra of all square matrices.
All these results hold if upper triangular is replaced by lower triangular throughout; in particular the lower triangular matrices also form a Lie algebra. However, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a lower triangular matrix can be any matrix; the product of a lower triangular with an upper triangular matrix is not necessarily triangular either.
The set of unitriangular matrices forms a Lie group.
The set of strictly upper (or lower) triangular matrices forms a nilpotent Lie algebra, denoted
$$
\mathfrak{n}.
$$
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The set of unitriangular matrices forms a Lie group.
The set of strictly upper (or lower) triangular matrices forms a nilpotent Lie algebra, denoted
$$
\mathfrak{n}.
$$
This algebra is the derived Lie algebra of
$$
\mathfrak{b}
$$
, the Lie algebra of all upper triangular matrices; in symbols,
$$
\mathfrak{n} = [\mathfrak{b},\mathfrak{b}].
$$
In addition,
$$
\mathfrak{n}
$$
is the Lie algebra of the Lie group of unitriangular matrices.
In fact, by Engel's theorem, any finite-dimensional nilpotent Lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices, that is to say, a finite-dimensional nilpotent Lie algebra is simultaneously strictly upper triangularizable.
Algebras of upper triangular matrices have a natural generalization in functional analysis which yields nest algebras on Hilbert spaces.
### Borel subgroups and Borel subalgebras
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Algebras of upper triangular matrices have a natural generalization in functional analysis which yields nest algebras on Hilbert spaces.
### Borel subgroups and Borel subalgebras
The set of invertible triangular matrices of a given kind (lower or upper) forms a group, indeed a Lie group, which is a subgroup of the general linear group of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero).
Over the real numbers, this group is disconnected, having
$$
2^n
$$
components accordingly as each diagonal entry is positive or negative. The identity component is invertible triangular matrices with positive entries on the diagonal, and the group of all invertible triangular matrices is a semidirect product of this group and the group of diagonal matrices with
$$
\pm 1
$$
on the diagonal, corresponding to the components.
The Lie algebra of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a solvable Lie algebra. These are, respectively, the standard Borel subgroup B of the Lie group GLn and the standard Borel subalgebra
$$
\mathfrak{b}
$$
of the Lie algebra gln.
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The Lie algebra of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a solvable Lie algebra. These are, respectively, the standard Borel subgroup B of the Lie group GLn and the standard Borel subalgebra
$$
\mathfrak{b}
$$
of the Lie algebra gln.
The upper triangular matrices are precisely those that stabilize the standard flag. The invertible ones among them form a subgroup of the general linear group, whose conjugate subgroups are those defined as the stabilizer of some (other) complete flag. These subgroups are Borel subgroups. The group of invertible lower triangular matrices is such a subgroup, since it is the stabilizer of the standard flag associated to the standard basis in reverse order.
The stabilizer of a partial flag obtained by forgetting some parts of the standard flag can be described as a set of block upper triangular matrices (but its elements are not all triangular matrices). The conjugates of such a group are the subgroups defined as the stabilizer of some partial flag. These subgroups are called parabolic subgroups.
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The conjugates of such a group are the subgroups defined as the stabilizer of some partial flag. These subgroups are called parabolic subgroups.
Examples
The group of 2×2 upper unitriangular matrices is isomorphic to the additive group of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic Möbius transformations; the 3×3 upper unitriangular matrices form the Heisenberg group.
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Data exhaust or exhaust data is the trail of data left by the activities of an Internet or other computer system users during their online activity, behavior, and transactions. This is part of a broader category of unconventional data that includes geospatial, network, and time-series data and may be useful for predictive analytics. Every visited website, clicked link, and even hovering with a mouse is collected, leaving behind a trail of data. An enormous amount of often raw data are created, which can be in the form of cookies, temporary files, logfiles, storable choices, and more. This information can help to improve the online experience, for example through customized content. It can be used to improve tracking trends and studying data exhaust also improves the user interface and the layout design. On the other hand, they can also compromise privacy, as they offer a valuable insight into the user's habits. For example, as the world's most popular website, Google, uses this data exhaust to refine the predictive value of their products.
The data that is collected by companies is often information that does not seem immediately useful. Although the information is not used by the company right away, it can be stored for future use or sold to someone else who can use the information. The data can help with quality control, performance, and revenue. Unlike primary content, these data are not purposefully created by the user, who is often unaware of their very existence.
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The data can help with quality control, performance, and revenue. Unlike primary content, these data are not purposefully created by the user, who is often unaware of their very existence. A bank for example would consider as primary data information concerning the sums and parties of a transaction, whilst secondary data might include the percentage of transactions carried out at a cash machine instead of a real bank.
## Medical exhaust data
Most medical devices emit some form of exhaust data, such as many pacemakers, dialysis machines, and cameras used during surgery. The majority of this data is never captured, and is primarily abandoned after the surgery is completed, or the device makes its next routine check. Some issues have arisen regarding the use of the data captured by devices like pacemakers. This can lead to larger issues surrounding the use of this exhaust data. Using electronic health records (EMR) for research poses a large number of challenges, the most prevalent being the amount of data there is. This surplus of data is too much for people to sort through and analyze, thus creating a need for algorithms.
## Solutions
Although data exhaust is not a new concept, the ubiquity of Internet-enabled gadgetry has exacerbated the scope and impacts of our passive digital trail. The collection and distribution of data thus generated is not illegal, but there are steps that must be taken to ensure that the use of this data is ethical.
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https://en.wikipedia.org/wiki/Data_exhaust
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Although data exhaust is not a new concept, the ubiquity of Internet-enabled gadgetry has exacerbated the scope and impacts of our passive digital trail. The collection and distribution of data thus generated is not illegal, but there are steps that must be taken to ensure that the use of this data is ethical. In order to ensure privacy of users, when the information is sold it can be anonymized. Also, users can be given the opportunity to opt-out of the selling of their information if they choose. Lastly, to build trust, websites can update their privacy policies so that they include all the data they will be collecting about the user.
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Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
## Main branches
Below are some of the main areas studied in algebraic topology:
### Homotopy groups
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
### Homology
In algebraic topology and abstract algebra, homology (in part from Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.
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Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
### Homology
In algebraic topology and abstract algebra, homology (in part from Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.
### Cohomology
In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign "quantities" to the chains of homology theory.
### Manifolds
A manifold is a topological space that near each point resembles Euclidean space.
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In less abstract language, cochains in the fundamental sense should assign "quantities" to the chains of homology theory.
### Manifolds
A manifold is a topological space that near each point resembles Euclidean space. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be embedded in three dimensions, but can be embedded in four dimensions. Typically, results in algebraic topology focus on global, non-differentiable aspects of manifolds; for example Poincaré duality.
### Knot theory
Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in three-dimensional Euclidean space,
$$
\mathbb{R}^3
$$
. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of
$$
\mathbb{R}^3
$$
upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
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In precise mathematical language, a knot is an embedding of a circle in three-dimensional Euclidean space,
$$
\mathbb{R}^3
$$
. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of
$$
\mathbb{R}^3
$$
upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
### Complexes
A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.
A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex).
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A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex).
## Method of algebraic invariants
An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW complex). In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces.
In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statements easier to prove. Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups.
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This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statements easier to prove. Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with. The fundamental group of a (finite) simplicial complex does have a finite presentation.
Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are completely classified and are particularly easy to work with.
## Setting in category theory
In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond—a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.
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