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In 1999, a research team in Vienna demonstrated diffraction for molecules as large as fullerenes. The researchers calculated a de Broglie wavelength of the most probable C60 velocity as .
More recent experiments prove the quantum nature of molecules made of 810 atoms and with a mass of . As of 2019, this has been pushed to molecules of .
In these experiments the build-up of such interference patterns could be recorded in real time and with single molecule sensitivity.
Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e., to certain decoherence mechanisms.
#### Others
Matter wave was detected in van der Waals molecules, rho mesons, Bose-Einstein condensate.
## Traveling matter waves
Waves have more complicated concepts for velocity than solid objects.
The simplest approach is to focus on the description in terms of plane matter waves for a free particle, that is a wave function described by
$$
\psi (\mathbf{r}) = e^{ i \mathbf{k} \cdot \mathbf{r}-i \omega t },
$$
where _
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## Traveling matter waves
Waves have more complicated concepts for velocity than solid objects.
The simplest approach is to focus on the description in terms of plane matter waves for a free particle, that is a wave function described by
$$
\psi (\mathbf{r}) = e^{ i \mathbf{k} \cdot \mathbf{r}-i \omega t },
$$
where _ BLOCK1_ is a position in real space,
$$
\mathbf{k}
$$
is the wave vector in units of inverse meters, is the angular frequency with units of inverse time and
$$
t
$$
is time. (Here the physics definition for the wave vector is used, which is
$$
2 \pi
$$
times the wave vector used in crystallography, see wavevector.)
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BLOCK1_ is a position in real space,
$$
\mathbf{k}
$$
is the wave vector in units of inverse meters, is the angular frequency with units of inverse time and
$$
t
$$
is time. (Here the physics definition for the wave vector is used, which is
$$
2 \pi
$$
times the wave vector used in crystallography, see wavevector.) The de Broglie equations relate the wavelength to the modulus of the momentum
$$
|\mathbf{p}| = p
$$
, and frequency to the total energy of a free particle as written above:
$$
\begin{align}
& \lambda = \frac {2 \pi}{|\mathbf{k}|} = \frac{h}{p}\\
& f = \frac{\omega}{2 \pi}= \frac{E}{h}
\end{align}
$$
where is the Planck constant. The equations can also be written as
$$
\begin{align}
& \mathbf{p} = \hbar \mathbf{k}\\
& E = \hbar \omega ,\\
\end{align}
$$
Here, is the reduced Planck constant.
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The de Broglie equations relate the wavelength to the modulus of the momentum
$$
|\mathbf{p}| = p
$$
, and frequency to the total energy of a free particle as written above:
$$
\begin{align}
& \lambda = \frac {2 \pi}{|\mathbf{k}|} = \frac{h}{p}\\
& f = \frac{\omega}{2 \pi}= \frac{E}{h}
\end{align}
$$
where is the Planck constant. The equations can also be written as
$$
\begin{align}
& \mathbf{p} = \hbar \mathbf{k}\\
& E = \hbar \omega ,\\
\end{align}
$$
Here, is the reduced Planck constant. The second equation is also referred to as the Planck–Einstein relation.
### Group velocity
In the de Broglie hypothesis, the velocity of a particle equals the group velocity of the matter wave.
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The second equation is also referred to as the Planck–Einstein relation.
### Group velocity
In the de Broglie hypothesis, the velocity of a particle equals the group velocity of the matter wave.
In isotropic media or a vacuum the group velocity of a wave is defined by:
$$
\mathbf{v_g} = \frac{\partial \omega(\mathbf{k})}{\partial \mathbf{k}}
$$
The relationship between the angular frequency and wavevector is called the dispersion relationship. For the non-relativistic case this is:
$$
\omega(\mathbf{k}) \approx \frac{m_0 c^2}{\hbar} + \frac{\hbar k^2}{2m_{0} }\,.
$$
where
$$
m_0
$$
is the rest mass. Applying the derivative gives the (non-relativistic) matter wave group velocity:
$$
\mathbf{v_g} = \frac{\hbar \mathbf{k}}{m_0}
$$
For comparison, the group velocity of light, with a dispersion
$$
\omega(k)=ck
$$
, is the speed of light
$$
c
$$
.
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For the non-relativistic case this is:
$$
\omega(\mathbf{k}) \approx \frac{m_0 c^2}{\hbar} + \frac{\hbar k^2}{2m_{0} }\,.
$$
where
$$
m_0
$$
is the rest mass. Applying the derivative gives the (non-relativistic) matter wave group velocity:
$$
\mathbf{v_g} = \frac{\hbar \mathbf{k}}{m_0}
$$
For comparison, the group velocity of light, with a dispersion
$$
\omega(k)=ck
$$
, is the speed of light
$$
c
$$
.
As an alternative, using the relativistic dispersion relationship for matter waves
$$
\omega(\mathbf{k}) = \sqrt{k^2c^2 + \left(\frac{m_0c^2}{\hbar}\right)^2} \,.
$$
then
$$
\mathbf{v_g} = \frac{\mathbf{k}c^2}{\omega}
$$
This relativistic form relates to the phase velocity as discussed below.
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Applying the derivative gives the (non-relativistic) matter wave group velocity:
$$
\mathbf{v_g} = \frac{\hbar \mathbf{k}}{m_0}
$$
For comparison, the group velocity of light, with a dispersion
$$
\omega(k)=ck
$$
, is the speed of light
$$
c
$$
.
As an alternative, using the relativistic dispersion relationship for matter waves
$$
\omega(\mathbf{k}) = \sqrt{k^2c^2 + \left(\frac{m_0c^2}{\hbar}\right)^2} \,.
$$
then
$$
\mathbf{v_g} = \frac{\mathbf{k}c^2}{\omega}
$$
This relativistic form relates to the phase velocity as discussed below.
For non-isotropic media we use the Energy–momentum form instead:
$$
\begin{align}
\mathbf{v}_\mathrm{g} &= \frac{\partial \omega}{\partial \mathbf{k}} = \frac{\partial (E/\hbar)}{\partial (\mathbf{p} /\hbar)} = \frac{\partial E}{\partial \mathbf{p}} = \frac{\partial}{\partial \mathbf{p}} \left( \sqrt{p^2c^2+m_0^2c^4} \right)\\
_BLOCK0_\end{align}
$$
But (see below), since the phase velocity is
$$
\mathbf{v}_\mathrm{p} = E/\mathbf{p} = c^2/\mathbf{v}
$$
, then
$$
\begin{align}
\mathbf{v}_\mathrm{g} &= \frac{\mathbf{p}c^2}{E}\\
_BLOCK1_\end{align}
$$
where
$$
\mathbf{v}
$$
is the velocity of the center of mass of the particle, identical to the group velocity.
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As an alternative, using the relativistic dispersion relationship for matter waves
$$
\omega(\mathbf{k}) = \sqrt{k^2c^2 + \left(\frac{m_0c^2}{\hbar}\right)^2} \,.
$$
then
$$
\mathbf{v_g} = \frac{\mathbf{k}c^2}{\omega}
$$
This relativistic form relates to the phase velocity as discussed below.
For non-isotropic media we use the Energy–momentum form instead:
$$
\begin{align}
\mathbf{v}_\mathrm{g} &= \frac{\partial \omega}{\partial \mathbf{k}} = \frac{\partial (E/\hbar)}{\partial (\mathbf{p} /\hbar)} = \frac{\partial E}{\partial \mathbf{p}} = \frac{\partial}{\partial \mathbf{p}} \left( \sqrt{p^2c^2+m_0^2c^4} \right)\\
_BLOCK0_\end{align}
$$
But (see below), since the phase velocity is
$$
\mathbf{v}_\mathrm{p} = E/\mathbf{p} = c^2/\mathbf{v}
$$
, then
$$
\begin{align}
\mathbf{v}_\mathrm{g} &= \frac{\mathbf{p}c^2}{E}\\
_BLOCK1_\end{align}
$$
where
$$
\mathbf{v}
$$
is the velocity of the center of mass of the particle, identical to the group velocity.
### Phase velocity
The phase velocity in isotropic media is defined as:
$$
\mathbf{v_p} = \frac{\omega}{\mathbf{k}}
$$
Using the relativistic group velocity above:
$$
\mathbf{v_p} = \frac{c^2 }{\mathbf{v_g}}
$$
This shows that
$$
\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2
$$
as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952.
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For non-isotropic media we use the Energy–momentum form instead:
$$
\begin{align}
\mathbf{v}_\mathrm{g} &= \frac{\partial \omega}{\partial \mathbf{k}} = \frac{\partial (E/\hbar)}{\partial (\mathbf{p} /\hbar)} = \frac{\partial E}{\partial \mathbf{p}} = \frac{\partial}{\partial \mathbf{p}} \left( \sqrt{p^2c^2+m_0^2c^4} \right)\\
_BLOCK0_\end{align}
$$
But (see below), since the phase velocity is
$$
\mathbf{v}_\mathrm{p} = E/\mathbf{p} = c^2/\mathbf{v}
$$
, then
$$
\begin{align}
\mathbf{v}_\mathrm{g} &= \frac{\mathbf{p}c^2}{E}\\
_BLOCK1_\end{align}
$$
where
$$
\mathbf{v}
$$
is the velocity of the center of mass of the particle, identical to the group velocity.
### Phase velocity
The phase velocity in isotropic media is defined as:
$$
\mathbf{v_p} = \frac{\omega}{\mathbf{k}}
$$
Using the relativistic group velocity above:
$$
\mathbf{v_p} = \frac{c^2 }{\mathbf{v_g}}
$$
This shows that
$$
\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2
$$
as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952. Electromagnetic waves also obey
$$
\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2
$$
, as both
$$
|\mathbf{v_p}|=c
$$
and
$$
|\mathbf{v_g}|=c
$$
.
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### Phase velocity
The phase velocity in isotropic media is defined as:
$$
\mathbf{v_p} = \frac{\omega}{\mathbf{k}}
$$
Using the relativistic group velocity above:
$$
\mathbf{v_p} = \frac{c^2 }{\mathbf{v_g}}
$$
This shows that
$$
\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2
$$
as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952. Electromagnetic waves also obey
$$
\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2
$$
, as both
$$
|\mathbf{v_p}|=c
$$
and
$$
|\mathbf{v_g}|=c
$$
. Since for matter waves,
$$
|\mathbf{v_g}| < c
$$
, it follows that
$$
|\mathbf{v_p}| > c
$$
, but only the group velocity carries information. The superluminal phase velocity therefore does not violate special relativity, as it does not carry information.
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Since for matter waves,
$$
|\mathbf{v_g}| < c
$$
, it follows that
$$
|\mathbf{v_p}| > c
$$
, but only the group velocity carries information. The superluminal phase velocity therefore does not violate special relativity, as it does not carry information.
For non-isotropic media, then
$$
\mathbf{v}_\mathrm{p} = \frac{\omega}{\mathbf{k}} = \frac{E/\hbar}{\mathbf{p}/\hbar} = \frac{E}{\mathbf{p}}.
$$
Using the relativistic relations for energy and momentum yields
$$
\mathbf{v}_\mathrm{p} = \frac{E}{\mathbf{p}} = \frac{m c^2}{m \mathbf{v}} = \frac{\gamma m_0 c^2}{\gamma m_0 \mathbf{v}} = \frac{c^2}{\mathbf{v}}.
$$
The variable
$$
\mathbf{v}
$$
can either be interpreted as the speed of the particle or the group velocity of the corresponding matter wave—the two are the same.
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The superluminal phase velocity therefore does not violate special relativity, as it does not carry information.
For non-isotropic media, then
$$
\mathbf{v}_\mathrm{p} = \frac{\omega}{\mathbf{k}} = \frac{E/\hbar}{\mathbf{p}/\hbar} = \frac{E}{\mathbf{p}}.
$$
Using the relativistic relations for energy and momentum yields
$$
\mathbf{v}_\mathrm{p} = \frac{E}{\mathbf{p}} = \frac{m c^2}{m \mathbf{v}} = \frac{\gamma m_0 c^2}{\gamma m_0 \mathbf{v}} = \frac{c^2}{\mathbf{v}}.
$$
The variable
$$
\mathbf{v}
$$
can either be interpreted as the speed of the particle or the group velocity of the corresponding matter wave—the two are the same. Since the particle speed
$$
|\mathbf{v}| < c
$$
for any particle that has nonzero mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.,
$$
| \mathbf{v}_\mathrm{p} | > c ,
$$
which approaches c when the particle speed is relativistic.
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For non-isotropic media, then
$$
\mathbf{v}_\mathrm{p} = \frac{\omega}{\mathbf{k}} = \frac{E/\hbar}{\mathbf{p}/\hbar} = \frac{E}{\mathbf{p}}.
$$
Using the relativistic relations for energy and momentum yields
$$
\mathbf{v}_\mathrm{p} = \frac{E}{\mathbf{p}} = \frac{m c^2}{m \mathbf{v}} = \frac{\gamma m_0 c^2}{\gamma m_0 \mathbf{v}} = \frac{c^2}{\mathbf{v}}.
$$
The variable
$$
\mathbf{v}
$$
can either be interpreted as the speed of the particle or the group velocity of the corresponding matter wave—the two are the same. Since the particle speed
$$
|\mathbf{v}| < c
$$
for any particle that has nonzero mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.,
$$
| \mathbf{v}_\mathrm{p} | > c ,
$$
which approaches c when the particle speed is relativistic. The superluminal phase velocity does not violate special relativity, similar to the case above for non-isotropic media.
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Since the particle speed
$$
|\mathbf{v}| < c
$$
for any particle that has nonzero mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.,
$$
| \mathbf{v}_\mathrm{p} | > c ,
$$
which approaches c when the particle speed is relativistic. The superluminal phase velocity does not violate special relativity, similar to the case above for non-isotropic media. See the article on Dispersion (optics) for further details.
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The superluminal phase velocity does not violate special relativity, similar to the case above for non-isotropic media. See the article on Dispersion (optics) for further details.
### Special relativity
Using two formulas from special relativity, one for the relativistic mass energy and one for the relativistic momentum
$$
\begin{align}
E &= m c^2 = \gamma m_0 c^2 \\[1ex]
\mathbf{p} &= m\mathbf{v} = \gamma m_0 \mathbf{v}
\end{align}
$$
allows the equations for de Broglie wavelength and frequency to be written as
$$
\begin{align}
&\lambda =\,\, \frac {h}{\gamma m_0v}\, =\, \frac {h}{m_0v}\,\,\, \sqrt{1 - \frac{v^2}{c^2}} \\[2.38ex]
& f = \frac{\gamma\,m_0 c^2}{h} = \frac {m_0 c^2}{h\sqrt{1 - \frac{v^2}{c^2}}} ,
\end{align}
$$
where _
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See the article on Dispersion (optics) for further details.
### Special relativity
Using two formulas from special relativity, one for the relativistic mass energy and one for the relativistic momentum
$$
\begin{align}
E &= m c^2 = \gamma m_0 c^2 \\[1ex]
\mathbf{p} &= m\mathbf{v} = \gamma m_0 \mathbf{v}
\end{align}
$$
allows the equations for de Broglie wavelength and frequency to be written as
$$
\begin{align}
&\lambda =\,\, \frac {h}{\gamma m_0v}\, =\, \frac {h}{m_0v}\,\,\, \sqrt{1 - \frac{v^2}{c^2}} \\[2.38ex]
& f = \frac{\gamma\,m_0 c^2}{h} = \frac {m_0 c^2}{h\sqrt{1 - \frac{v^2}{c^2}}} ,
\end{align}
$$
where _ BLOCK2_ is the velocity,
$$
\gamma
$$
the Lorentz factor, and
$$
c
$$
the speed of light in vacuum. Williams, W.S.C. (2002).
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the speed of light in vacuum. Williams, W.S.C. (2002). Introducing Special Relativity, Taylor & Francis, London, , p. 192. This shows that as the velocity of a particle approaches zero (rest) the de Broglie wavelength approaches infinity.
### Four-vectors
Using four-vectors, the de Broglie relations form a single equation:
$$
\mathbf{P}= \hbar\mathbf{K} ,
$$
which is frame-independent.
Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by:
$$
\mathbf{K} = \left(\frac{\omega_0}{c^2}\right)\mathbf{U} ,
$$
where
- Four-momentum
$$
\mathbf{P} = \left(\frac{E}{c}, {\mathbf{p}} \right)
$$
- Four-wavevector _ BLOCK3_- Four-velocity
$$
\mathbf{U} = \gamma(c,{\mathbf{u}}) = \gamma(c,v_\mathrm{g} \hat{\mathbf{u}})
$$
## General matter waves
The preceding sections refer specifically to free particles for which the wavefunctions are plane waves.
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BLOCK3_- Four-velocity
$$
\mathbf{U} = \gamma(c,{\mathbf{u}}) = \gamma(c,v_\mathrm{g} \hat{\mathbf{u}})
$$
## General matter waves
The preceding sections refer specifically to free particles for which the wavefunctions are plane waves. There are significant numbers of other matter waves, which can be broadly split into three classes: single-particle matter waves, collective matter waves and standing waves.
### Single-particle matter waves
The more general description of matter waves corresponding to a single particle type (e.g. a single electron or neutron only) would have a form similar to
$$
\psi (\mathbf{r}) = u(\mathbf{r},\mathbf{k})\exp(i\mathbf{k}\cdot \mathbf{r} - iE(\mathbf{k})t/\hbar)
$$
where now there is an additional spatial term
$$
u(\mathbf{r},\mathbf{k})
$$
in the front, and the energy has been written more generally as a function of the wave vector. The various terms given before still apply, although the energy is no longer always proportional to the wave vector squared.
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### Single-particle matter waves
The more general description of matter waves corresponding to a single particle type (e.g. a single electron or neutron only) would have a form similar to
$$
\psi (\mathbf{r}) = u(\mathbf{r},\mathbf{k})\exp(i\mathbf{k}\cdot \mathbf{r} - iE(\mathbf{k})t/\hbar)
$$
where now there is an additional spatial term
$$
u(\mathbf{r},\mathbf{k})
$$
in the front, and the energy has been written more generally as a function of the wave vector. The various terms given before still apply, although the energy is no longer always proportional to the wave vector squared. A common approach is to define an effective mass which in general is a tensor
$$
m_{ij}^*
$$
given by
$$
{m_{ij}^*}^{-1} = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \partial k_j}
$$
so that in the simple case where all directions are the same the form is similar to that of a free wave above.
$$
E(\mathbf k) = \frac{\hbar^2 \mathbf k^2}{2 m^*}
$$
In
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The various terms given before still apply, although the energy is no longer always proportional to the wave vector squared. A common approach is to define an effective mass which in general is a tensor
$$
m_{ij}^*
$$
given by
$$
{m_{ij}^*}^{-1} = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \partial k_j}
$$
so that in the simple case where all directions are the same the form is similar to that of a free wave above.
$$
E(\mathbf k) = \frac{\hbar^2 \mathbf k^2}{2 m^*}
$$
In general the group velocity would be replaced by the probability current
$$
\mathbf{j}(\mathbf{r}) = \frac{\hbar}{2mi} \left( \psi^*(\mathbf{r}) \mathbf \nabla \psi(\mathbf{r}) - \psi(\mathbf{r}) \mathbf \nabla \psi^{*}(\mathbf{r}) \right)
$$
where
$$
\nabla
$$
is the del or gradient operator.
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A common approach is to define an effective mass which in general is a tensor
$$
m_{ij}^*
$$
given by
$$
{m_{ij}^*}^{-1} = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \partial k_j}
$$
so that in the simple case where all directions are the same the form is similar to that of a free wave above.
$$
E(\mathbf k) = \frac{\hbar^2 \mathbf k^2}{2 m^*}
$$
In general the group velocity would be replaced by the probability current
$$
\mathbf{j}(\mathbf{r}) = \frac{\hbar}{2mi} \left( \psi^*(\mathbf{r}) \mathbf \nabla \psi(\mathbf{r}) - \psi(\mathbf{r}) \mathbf \nabla \psi^{*}(\mathbf{r}) \right)
$$
where
$$
\nabla
$$
is the del or gradient operator. The momentum would then be described using the kinetic momentum operator,
$$
\mathbf{p} = -i\hbar\nabla
$$
The wavelength is still described as the inverse of the modulus of the wavevector, although measurement is more complex.
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general the group velocity would be replaced by the probability current
$$
\mathbf{j}(\mathbf{r}) = \frac{\hbar}{2mi} \left( \psi^*(\mathbf{r}) \mathbf \nabla \psi(\mathbf{r}) - \psi(\mathbf{r}) \mathbf \nabla \psi^{*}(\mathbf{r}) \right)
$$
where
$$
\nabla
$$
is the del or gradient operator. The momentum would then be described using the kinetic momentum operator,
$$
\mathbf{p} = -i\hbar\nabla
$$
The wavelength is still described as the inverse of the modulus of the wavevector, although measurement is more complex. There are many cases where this approach is used to describe single-particle matter waves:
- Bloch wave, which form the basis of much of band structure as described in Ashcroft and Mermin, and are also used to describe the diffraction of high-energy electrons by solids.
- Waves with angular momentum such as electron vortex beams.
- Evanescent waves, where the component of the wavevector in one direction is complex. These are common when matter waves are being reflected, particularly for grazing-incidence diffraction.
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- Evanescent waves, where the component of the wavevector in one direction is complex. These are common when matter waves are being reflected, particularly for grazing-incidence diffraction.
Collective matter waves
Other classes of matter waves involve more than one particle, so are called collective waves and are often quasiparticles. Many of these occur in solids – see Ashcroft and Mermin. Examples include:
- In solids, an electron quasiparticle is an electron where interactions with other electrons in the solid have been included. An electron quasiparticle has the same charge and spin as a "normal" (elementary particle) electron and, like a normal electron, it is a fermion. However, its effective mass can differ substantially from that of a normal electron. Its electric field is also modified, as a result of electric field screening.
- A hole is a quasiparticle which can be thought of as a vacancy of an electron in a state; it is most commonly used in the context of empty states in the valence band of a semiconductor. A hole has the opposite charge of an electron.
- A polaron is a quasiparticle where an electron interacts with the polarization of nearby atoms.
- An exciton is an electron and hole pair which are bound together.
- A Cooper pair is two electrons bound together so they behave as a single matter wave.
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A polaron is a quasiparticle where an electron interacts with the polarization of nearby atoms.
- An exciton is an electron and hole pair which are bound together.
- A Cooper pair is two electrons bound together so they behave as a single matter wave.
### Standing matter waves
The third class are matter waves which have a wavevector, a wavelength and vary with time, but have a zero group velocity or probability flux. The simplest of these, similar to the notation above would be
$$
\cos(\mathbf{k}\cdot\mathbf{r} - \omega t)
$$
These occur as part of the particle in a box, and other cases such as in a ring. This can, and arguably should be, extended to many other cases. For instance, in early work de Broglie used the concept that an electron matter wave must be continuous in a ring to connect to the Bohr–Sommerfeld condition in the early approaches to quantum mechanics. In that sense atomic orbitals around atoms, and also molecular orbitals are electron matter waves.
## Matter waves vs. electromagnetic waves (light)
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In that sense atomic orbitals around atoms, and also molecular orbitals are electron matter waves.
## Matter waves vs. electromagnetic waves (light)
Schrödinger applied Hamilton's optico-mechanical analogy to develop his wave mechanics for subatomic particles Consequently, wave solutions to the Schrödinger equation share many properties with results of light wave optics. In particular, Kirchhoff's diffraction formula works well for electron optics and for atomic optics. The approximation works well as long as the electric fields change more slowly than the de Broglie wavelength. Macroscopic apparatus fulfill this condition; slow electrons moving in solids do not.
Beyond the equations of motion, other aspects of matter wave optics differ from the corresponding light optics cases.
Sensitivity of matter waves to environmental condition.
Many examples of electromagnetic (light) diffraction occur in air under many environmental conditions. Obviously visible light interacts weakly with air molecules. By contrast, strongly interacting particles like slow electrons and molecules require vacuum: the matter wave properties rapidly fade when they are exposed to even low pressures of gas. With special apparatus, high velocity electrons can be used to study liquids and gases.
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By contrast, strongly interacting particles like slow electrons and molecules require vacuum: the matter wave properties rapidly fade when they are exposed to even low pressures of gas. With special apparatus, high velocity electrons can be used to study liquids and gases. Neutrons, an important exception, interact primarily by collisions with nuclei, and thus travel several hundred feet in air.
Dispersion. Light waves of all frequencies travel at the same speed of light while matter wave velocity varies strongly with frequency. The relationship between frequency (proportional to energy) and wavenumber or velocity (proportional to momentum) is called a dispersion relation. Light waves in a vacuum have linear dispersion relation between frequency:
$$
\omega = ck
$$
.
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The relationship between frequency (proportional to energy) and wavenumber or velocity (proportional to momentum) is called a dispersion relation. Light waves in a vacuum have linear dispersion relation between frequency:
$$
\omega = ck
$$
. For matter waves the relation is non-linear:
$$
\omega(k) \approx \frac{m_0 c^2}{\hbar} + \frac{\hbar k^2}{2m_{0} }\,.
$$
This non-relativistic matter wave dispersion relation says the frequency in vacuum varies with wavenumber (
$$
k=1/\lambda
$$
) in two parts: a constant part due to the de Broglie frequency of the rest mass (
$$
\hbar \omega_0 = m_{0}c^2
$$
) and a quadratic part due to kinetic energy. The quadratic term causes rapid spreading of wave packets of matter waves.
Coherence The visibility of diffraction features using an optical theory approach depends on the beam coherence, which at the quantum level is equivalent to a density matrix approach. As with light, transverse coherence (across the direction of propagation) can be increased by collimation.
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The visibility of diffraction features using an optical theory approach depends on the beam coherence, which at the quantum level is equivalent to a density matrix approach. As with light, transverse coherence (across the direction of propagation) can be increased by collimation. Electron optical systems use stabilized high voltage to give a narrow energy spread in combination with collimating (parallelizing) lenses and pointed filament sources to achieve good coherence. Because light at all frequencies travels the same velocity, longitudinal and temporal coherence are linked; in matter waves these are independent. For example, for atoms, velocity (energy) selection controls longitudinal coherence and pulsing or chopping controls temporal coherence.
Optically shaped matter waves
Optical manipulation of matter plays a critical role in matter wave optics: "Light waves can act as refractive, reflective, and absorptive structures for matter waves, just as glass interacts with light waves." Laser light momentum transfer can cool matter particles and alter the internal excitation state of atoms.
Multi-particle experiments
While single-particle free-space optical and matter wave equations are identical, multiparticle systems like coincidence experiments are not.
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Laser light momentum transfer can cool matter particles and alter the internal excitation state of atoms.
Multi-particle experiments
While single-particle free-space optical and matter wave equations are identical, multiparticle systems like coincidence experiments are not.
## Applications of matter waves
The following subsections provide links to pages describing applications of matter waves as probes of materials or of fundamental quantum properties. In most cases these involve some method of producing travelling matter waves which initially have the simple form
$$
\exp(i \mathbf{k}\cdot \mathbf{r} - i\omega t)
$$
, then using these to probe materials.
As shown in the table below, matter wave mass ranges over 6 orders of magnitude and energy over 9 orders but the wavelengths are all in the picometre range, comparable to atomic spacings. (Atomic diameters range from 62 to 520 pm, and the typical length of a carbon–carbon single bond is 154 pm.) Reaching longer wavelengths requires special techniques like laser cooling to reach lower energies; shorter wavelengths make diffraction effects more difficult to discern. Therefore, many applications focus on material structures, in parallel with applications of electromagnetic waves, especially X-rays.
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Reaching longer wavelengths requires special techniques like laser cooling to reach lower energies; shorter wavelengths make diffraction effects more difficult to discern. Therefore, many applications focus on material structures, in parallel with applications of electromagnetic waves, especially X-rays. Unlike light, matter wave particles may have mass, electric charge, magnetic moments, and internal structure, presenting new challenges and opportunities.
+ Various matter wave wavelengths matter mass kinetic energy wavelength referenceElectron 1/1823 Da Davisson–Germer experimentElectron 1/1823 Da Tonomura et al. He atom, H2 moleculeEstermann and SternNeutron Wollan and Shull Sodium atomMoskowitz et al.Helium Grisenti et al. Na2 Chapman et al. C60 fullereneArndt et al.C70 fullerene Brezger et al. polypeptide, Gramicidin AShayeghi et al.functionalized oligoporphyrinsFein et al.
Electrons
Electron diffraction patterns emerge when energetic electrons reflect or penetrate ordered solids; analysis of the patterns leads to models of the atomic arrangement in the solids.
They are used for imaging from the micron to atomic scale using electron microscopes, in transmission, using scanning, and for surfaces at low energies.
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Chapman et al. C60 fullereneArndt et al.C70 fullerene Brezger et al. polypeptide, Gramicidin AShayeghi et al.functionalized oligoporphyrinsFein et al.
Electrons
Electron diffraction patterns emerge when energetic electrons reflect or penetrate ordered solids; analysis of the patterns leads to models of the atomic arrangement in the solids.
They are used for imaging from the micron to atomic scale using electron microscopes, in transmission, using scanning, and for surfaces at low energies.
The measurements of the energy they lose in electron energy loss spectroscopy provides information about the chemistry and electronic structure of materials. Beams of electrons also lead to characteristic X-rays in energy dispersive spectroscopy which can produce information about chemical content at the nanoscale.
Quantum tunneling explains how electrons escape from metals in an electrostatic field at energies less than classical predictions allow: the matter wave penetrates of the work function barrier in the metal.
Scanning tunneling microscope leverages quantum tunneling to image the top atomic layer of solid surfaces.
Electron holography, the electron matter wave analog of optical holography, probes the electric and magnetic fields in thin films.
Neutrons
Neutron diffraction complements x-ray diffraction through the different scattering cross sections and sensitivity to magnetism.
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Electron holography, the electron matter wave analog of optical holography, probes the electric and magnetic fields in thin films.
Neutrons
Neutron diffraction complements x-ray diffraction through the different scattering cross sections and sensitivity to magnetism.
Small-angle neutron scattering provides way to obtain structure of disordered systems that is sensitivity to light elements, isotopes and magnetic moments.
Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films.
### Neutral atoms
Atom interferometers, similar to optical interferometers, measure the difference in phase between atomic matter waves along different paths.
Atom optics mimic many light optic devices, including mirrors, atom focusing zone plates.
Scanning helium microscopy uses He atom waves to image solid structures non-destructively.
Quantum reflection uses matter wave behavior to explain grazing angle atomic reflection, the basis of some atomic mirrors.
Quantum decoherence measurements rely on Rb atom wave interference.
Molecules
Quantum superposition revealed by interference of matter waves from large molecules probes the limits of wave–particle duality and quantum macroscopicity.
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Quantum decoherence measurements rely on Rb atom wave interference.
Molecules
Quantum superposition revealed by interference of matter waves from large molecules probes the limits of wave–particle duality and quantum macroscopicity.
Matter-wave interfererometers generate nanostructures on molecular beams that can be read with nanometer accuracy and therefore be used for highly sensitive force measurements, from which one can deduce a plethora of properties of individualized complex molecules.
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## In mathematics
, the term linear is used in two distinct senses for two different properties:
- linearity of a function (or mapping);
- linearity of a polynomial.
An example of a linear function is the function defined by
$$
f(x)=(ax,bx)
$$
that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables
$$
X,
$$
$$
Y
$$
and
$$
Z
$$
is
$$
aX+bY+cZ+d.
$$
Linearity of a mapping is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are nonlinear.
Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle.
Linearity of a polynomial means that its degree is less than two. The use of the term for polynomials stems from the fact that the graph of a polynomial in one variable is a straight line. In the term "linear equation", the word refers to the linearity of the polynomials involved.
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The use of the term for polynomials stems from the fact that the graph of a polynomial in one variable is a straight line. In the term "linear equation", the word refers to the linearity of the polynomials involved.
Because a function such as
$$
f(x)=ax+b
$$
is defined by a linear polynomial in its argument, it is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially confusing, but usually the intended meaning will be clear from the context.
The word linear comes from Latin linearis, "pertaining to or resembling a line".
In mathematics
### Linear maps
In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties:
- Additivity: .
- Homogeneity of degree 1: for all α.
These properties are known as the superposition principle. In this definition, x is not necessarily a real number, but can in general be an element of any vector space. A more special definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below).
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In this definition, x is not necessarily a real number, but can in general be an element of any vector space. A more special definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below).
Additivity alone implies homogeneity for rational α, since
$$
f(x+x)=f(x)+f(x)
$$
implies
$$
f(nx)=n f(x)
$$
for any natural number n by mathematical induction, and then
$$
n f(x) = f(nx)=f(m\tfrac{n}{m}x)= m f(\tfrac{n}{m}x)
$$
implies
$$
f(\tfrac{n}{m}x) = \tfrac{n}{m} f(x)
$$
. The density of the rational numbers in the reals implies that any additive continuous function is homogeneous for any real number α, and is therefore linear.
The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and other operators constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.
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Important examples of linear operators include the derivative considered as a differential operator, and other operators constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.
### Linear polynomials
In a different usage to the above definition, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a straight line.
Over the reals, a simple example of a linear equation is given by:
$$
y = m x + b\
$$
where m is often called the slope or gradient, and b the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.
Note that this usage of the term linear is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if the constant term – b in the example – equals 0. If , the function is called an affine function (see in greater generality affine transformation).
Linear algebra is the branch of mathematics concerned with systems of linear equations.
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If , the function is called an affine function (see in greater generality affine transformation).
Linear algebra is the branch of mathematics concerned with systems of linear equations.
### Boolean functions
In Boolean algebra, a linear function is a function
$$
f
$$
for which there exist
$$
a_0, a_1, \ldots, a_n \in \{0,1\}
$$
such that
$$
f(b_1, \ldots, b_n) = a_0 \oplus (a_1 \land b_1) \oplus \cdots \oplus (a_n \land b_n)
$$
, where
$$
b_1, \ldots, b_n \in \{0,1\}.
$$
Note that if
$$
a_0 = 1
$$
, the above function is considered affine in linear algebra (i.e. not linear).
A Boolean function is linear if one of the following holds for the function's truth table:
1. In every row in which the truth value of the function is T, there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, , and these functions correspond to linear maps over the Boolean vector space.
1.
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Specifically, , and these functions correspond to linear maps over the Boolean vector space.
1. In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the truth value of the function is F, there are an odd number of Ts assigned to arguments. In this case, .
Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference.
Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.
## Physics
In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation.
Linearity of a homogenous differential equation means that if two functions f and g are solutions of the equation, then any linear combination is, too.
In instrumentation, linearity means that a given change in an input variable gives the same change in the output of the measurement apparatus: this is highly desirable in scientific work. In general, instruments are close to linear over a certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain absolute threshold number of photons.
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In general, instruments are close to linear over a certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain absolute threshold number of photons.
Linear motion traces a straight line trajectory.
## Electronics
In electronics, the linear operating region of a device, for example a transistor, is where an output dependent variable (such as the transistor collector current) is directly proportional to an input dependent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a high fidelity audio amplifier, which must amplify a signal without changing its waveform. Others are linear filters, and linear amplifiers in general.
In most scientific and technological, as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value.
### Integral linearity
For an electronic device (or other physical device) that converts a quantity to another quantity
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In most scientific and technological, as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value.
### Integral linearity
For an electronic device (or other physical device) that converts a quantity to another quantity , Bertram S. Kolts writes:
There are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of full scale, or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance.
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Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.
In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.
## History
Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability. These areas of mathematics related directly to the development of Newtonian physics, and in fact, the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century.
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## History
Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability. These areas of mathematics related directly to the development of Newtonian physics, and in fact, the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a pedagogical legacy in the United States: until the early 20th century, subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments.
### Engineering
and computer science departments have traditionally made use of applied mathematics.
As time passed, Applied Mathematics grew alongside the advancement of science and technology. With the advent of modern times, the application of mathematics in fields such as science, economics, technology, and more became deeper and more timely. The development of computers and other technologies enabled a more detailed study and application of mathematical concepts in various fields.
Today, Applied Mathematics continues to be crucial for societal and technological advancement. It guides the development of new technologies, economic progress, and addresses challenges in various scientific fields and industries.
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Today, Applied Mathematics continues to be crucial for societal and technological advancement. It guides the development of new technologies, economic progress, and addresses challenges in various scientific fields and industries. The history of Applied Mathematics continually demonstrates the importance of mathematics in human progress.
## Divisions
Today, the term "applied mathematics" is used in a broader sense. It includes the classical areas noted above as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptography), though they are not generally considered to be part of the field of applied mathematics per se.
There is no consensus as to what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees.
Many mathematicians distinguish between "applied mathematics", which is concerned with mathematical methods, and the "applications of mathematics" within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it; however, mathematical biologists have posed problems that have stimulated the growth of pure mathematics.
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Many mathematicians distinguish between "applied mathematics", which is concerned with mathematical methods, and the "applications of mathematics" within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it; however, mathematical biologists have posed problems that have stimulated the growth of pure mathematics. Mathematicians such as Poincaré and Arnold deny the existence of "applied mathematics" and claim that there are only "applications of mathematics." Similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to solve industrial problems is also called "industrial mathematics".
The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics, computational science, and computational engineering, which use high-performance computing for the simulation of phenomena and the solution of problems in the sciences and engineering. These are often considered interdisciplinary.
### Applicable mathematics
Sometimes, the term applicable mathematics is used to distinguish between the traditional applied mathematics that developed alongside physics and the many areas of mathematics that are applicable to real-world problems today, although there is no consensus as to a precise definition.
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### Applicable mathematics
Sometimes, the term applicable mathematics is used to distinguish between the traditional applied mathematics that developed alongside physics and the many areas of mathematics that are applicable to real-world problems today, although there is no consensus as to a precise definition.
Mathematicians often distinguish between "applied mathematics" on the one hand, and the "applications of mathematics" or "applicable mathematics" both within and outside of science and engineering, on the other. Some mathematicians emphasize the term applicable mathematics to separate or delineate the traditional applied areas from new applications arising from fields that were previously seen as pure mathematics. For example, from this viewpoint, an ecologist or geographer using population models and applying known mathematics would not be doing applied, but rather applicable, mathematics. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptography), though they are not generally considered to be part of the field of applied mathematics per se. Such descriptions can lead to applicable mathematics being seen as a collection of mathematical methods such as real analysis, linear algebra, mathematical modelling, optimisation, combinatorics, probability and statistics, which are useful in areas outside traditional mathematics and not specific to mathematical physics.
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Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptography), though they are not generally considered to be part of the field of applied mathematics per se. Such descriptions can lead to applicable mathematics being seen as a collection of mathematical methods such as real analysis, linear algebra, mathematical modelling, optimisation, combinatorics, probability and statistics, which are useful in areas outside traditional mathematics and not specific to mathematical physics.
Other authors prefer describing applicable mathematics as a union of "new" mathematical applications with the traditional fields of applied mathematics. THOUGHTS ON APPLIED MATHEMATICS. With this outlook, the terms applied mathematics and applicable mathematics are thus interchangeable.
## Utility
Historically, mathematics was most important in the natural sciences and engineering. However, since World War II, fields outside the physical sciences have spawned the creation of new areas of mathematics, such as game theory and social choice theory, which grew out of economic considerations. Further, the utilization and development of mathematical methods expanded into other areas leading to the creation of new fields such as mathematical finance and data science.
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However, since World War II, fields outside the physical sciences have spawned the creation of new areas of mathematics, such as game theory and social choice theory, which grew out of economic considerations. Further, the utilization and development of mathematical methods expanded into other areas leading to the creation of new fields such as mathematical finance and data science.
The advent of the computer has enabled new applications: studying and using the new computer technology itself (computer science) to study problems arising in other areas of science (computational science) as well as the mathematics of computation (for example, theoretical computer science, computer algebra,Geddes, K. O., Czapor, S. R., & Labahn, G. (1992). Algorithms for computer algebra. Springer Science & Business Media. Mignotte, M. (2012). Mathematics for computer algebra. Springer Science & Business Media. numerical analysisConte, S. D., & De Boor, C. (2017). Elementary numerical analysis: an algorithmic approach. Society for Industrial and Applied Mathematics. Linz, P. (2019). Theoretical numerical analysis. Courier Dover Publications.).
### Statistics
is probably the most widespread mathematical science used in the social sciences.
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Courier Dover Publications.).
### Statistics
is probably the most widespread mathematical science used in the social sciences.
## Status in academic departments
Academic institutions are not consistent in the way they group and label courses, programs, and degrees in applied mathematics. At some schools, there is a single mathematics department, whereas others have separate departments for Applied Mathematics and (Pure) Mathematics. It is very common for Statistics departments to be separated at schools with graduate programs, but many undergraduate-only institutions include statistics under the mathematics department.
Many applied mathematics programs (as opposed to departments) consist primarily of cross-listed courses and jointly appointed faculty in departments representing applications. Some Ph.D. programs in applied mathematics require little or no coursework outside mathematics, while others require substantial coursework in a specific area of application. In some respects this difference reflects the distinction between "application of mathematics" and "applied mathematics".
Some universities in the U.K. host departments of Applied Mathematics and Theoretical Physics,Dept of Applied Mathematics & Theoretical Physics. Queen's University, Belfast. but it is now much less common to have separate departments of pure and applied mathematics.
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Queen's University, Belfast. but it is now much less common to have separate departments of pure and applied mathematics. A notable exception to this is the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, housing the Lucasian Professor of Mathematics whose past holders include Isaac Newton, Charles Babbage, James Lighthill, Paul Dirac, and Stephen Hawking.
Schools with separate applied mathematics departments range from Brown University, which has a large Division of Applied Mathematics that offers degrees through the doctorate, to Santa Clara University, which offers only the M.S. in applied mathematics. Research universities dividing their mathematics department into pure and applied sections include MIT. Students in this program also learn another skill (computer science, engineering, physics, pure math, etc.) to supplement their applied math skills.
## Associated mathematical sciences
Applied mathematics is associated with the following mathematical sciences:
Engineering
Mathematics is used in all branches of engineering and has subsequently developed as distinct specialties within the engineering profession.
For example, continuum mechanics is foundational to civil, mechanical and aerospace engineering, with courses in solid mechanics and fluid mechanics being important components of the engineering curriculum. Continuum mechanics is also an important branch of mathematics in its own right.
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For example, continuum mechanics is foundational to civil, mechanical and aerospace engineering, with courses in solid mechanics and fluid mechanics being important components of the engineering curriculum. Continuum mechanics is also an important branch of mathematics in its own right. It has served as the inspiration for a vast range of difficult research questions for mathematicians involved in the analysis of partial differential equations, differential geometry and the calculus of variations. Perhaps the most well-known mathematical problem posed by a continuum mechanical system is the question of Navier-Stokes existence and smoothness. Prominent career mathematicians rather than engineers who have contributed to the mathematics of continuum mechanics are Clifford Truesdell, Walter Noll, Andrey Kolmogorov and George Batchelor.
An essential discipline for many fields in engineering is that of control engineering. The associated mathematical theory of this specialism is control theory, a branch of applied mathematics that builds off the mathematics of dynamical systems. Control theory has played a significant enabling role in modern technology, serving a foundational role in electrical, mechanical and aerospace engineering. Like continuum mechanics, control theory has also become a field of mathematical research in its own right, with mathematicians such as Aleksandr Lyapunov, Norbert Wiener, Lev Pontryagin and fields medallist Pierre-Louis Lions contributing to its foundations.
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Control theory has played a significant enabling role in modern technology, serving a foundational role in electrical, mechanical and aerospace engineering. Like continuum mechanics, control theory has also become a field of mathematical research in its own right, with mathematicians such as Aleksandr Lyapunov, Norbert Wiener, Lev Pontryagin and fields medallist Pierre-Louis Lions contributing to its foundations.
### Scientific computing
Scientific computing includes applied mathematics (especially numerical analysis), computing science (especially high-performance computingGeshi, M. (2019). The Art of High Performance Computing for Computational Science, Springer.), and mathematical modelling in a scientific discipline.
### Computer science
Computer science relies on logic, algebra, discrete mathematics such as graph theory,Bondy, J. A., & Murty, U. S. R. (1976). Graph theory with applications (Vol. 290). London: Macmillan. and combinatorics.
### Operations research and management science
Operations research and management science are often taught in faculties of engineering, business, and public policy.
Statistics
Applied mathematics has substantial overlap with the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions.
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Statistics
Applied mathematics has substantial overlap with the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on probability and decision theory, and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorial design. Applied mathematicians and statisticians often work in a department of mathematical sciences (particularly at colleges and small universities).
### Actuarial science
Actuarial science applies probability, statistics, and economic theory to assess risk in insurance, finance and other industries and professions.
### Mathematical economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Na, N. (2016). Mathematical economics. Springer. The applied methods usually refer to nontrivial mathematical techniques or approaches. Mathematical economics is based on statistics, probability, mathematical programming (as well as other computational methods), operations research, game theory, and some methods from mathematical analysis. In this regard, it resembles (but is distinct from) financial mathematics, another part of applied mathematics.
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Mathematical economics is based on statistics, probability, mathematical programming (as well as other computational methods), operations research, game theory, and some methods from mathematical analysis. In this regard, it resembles (but is distinct from) financial mathematics, another part of applied mathematics.
According to the Mathematics Subject Classification (MSC), mathematical economics falls into the Applied mathematics/other classification of category 91:
Game theory, economics, social and behavioral sciences
with MSC2010 classifications for 'Game theory' at codes 91Axx and for 'Mathematical economics' at codes 91Bxx .
### Other disciplines
The line between applied mathematics and specific areas of application is often blurred. Many universities teach mathematical and statistical courses outside the respective departments, in departments and areas including business, engineering, physics, chemistry, psychology, biology, computer science, scientific computation, information theory, and mathematical physics.
## Applied Mathematics Societies
The Society for Industrial and Applied Mathematics is an international applied mathematics organization. As of 2024, the society has 14,000 individual members. The American Mathematics Society has its Applied Mathematics Group.
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An optical spectrometer (spectrophotometer, spectrograph or spectroscope) is an instrument used to measure properties of light over a specific portion of the electromagnetic spectrum, typically used in spectroscopic analysis to identify materials. The variable measured is most often the irradiance of the light but could also, for instance, be the polarization state. The independent variable is usually the wavelength of the light or a closely derived physical quantity, such as the corresponding wavenumber or the photon energy, in units of measurement such as centimeters, reciprocal centimeters, or electron volts, respectively.
A spectrometer is used in spectroscopy for producing spectral lines and measuring their wavelengths and intensities. Spectrometers may operate over a wide range of non-optical wavelengths, from gamma rays and X-rays into the far infrared. If the instrument is designed to measure the spectrum on an absolute scale rather than a relative one, then it is typically called a spectrophotometer. The majority of spectrophotometers are used in spectral regions near the visible spectrum.
A spectrometer that is calibrated for measurement of the incident optical power is called a spectroradiometer.
In general, any particular instrument will operate over a small portion of this total range because of the different techniques used to measure different portions of the spectrum.
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A spectrometer that is calibrated for measurement of the incident optical power is called a spectroradiometer.
In general, any particular instrument will operate over a small portion of this total range because of the different techniques used to measure different portions of the spectrum. Below optical frequencies (that is, at microwave and radio frequencies), the spectrum analyzer is a closely related electronic device.
Spectrometers are used in many fields. For example, they are used in astronomy to analyze the radiation from objects and deduce their chemical composition. The spectrometer uses a prism or a grating to spread the light into a spectrum. This allows astronomers to detect many of the chemical elements by their characteristic spectral lines. These lines are named for the elements which cause them, such as the hydrogen alpha, beta, and gamma lines. A glowing object will show bright spectral lines. Dark lines are made by absorption, for example by light passing through a gas cloud, and these absorption lines can also identify chemical compounds. Much of our knowledge of the chemical makeup of the universe comes from spectra.
## Spectroscopes
Spectroscopes are often used in astronomy and some branches of chemistry. Early spectroscopes were simply prisms with graduations marking wavelengths of light.
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## Spectroscopes
Spectroscopes are often used in astronomy and some branches of chemistry. Early spectroscopes were simply prisms with graduations marking wavelengths of light. Modern spectroscopes generally use a diffraction grating, a movable slit, and some kind of photodetector, all automated and controlled by a computer. Recent advances have seen increasing reliance of computational algorithms in a range of miniaturised spectrometers without diffraction gratings, for example, through the use of quantum dot-based filter arrays on to a CCD chip or a series of photodetectors realised on a single nanostructure.
Joseph von Fraunhofer developed the first modern spectroscope by combining a prism, diffraction slit and telescope in a manner that increased the spectral resolution and was reproducible in other laboratories. Fraunhofer also went on to invent the first diffraction spectroscope. Gustav Robert Kirchhoff and Robert Bunsen discovered the application of spectroscopes to chemical analysis and used this approach to discover caesium and rubidium. Kirchhoff and Bunsen's analysis also enabled a chemical explanation of stellar spectra, including Fraunhofer lines.
When a material is heated to incandescence it emits light that is characteristic of the atomic makeup of the material.
Particular light frequencies give rise to sharply defined bands on the scale which can be thought of as fingerprints.
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When a material is heated to incandescence it emits light that is characteristic of the atomic makeup of the material.
Particular light frequencies give rise to sharply defined bands on the scale which can be thought of as fingerprints. For example, the element sodium has a very characteristic double yellow band known as the Sodium D-lines at 588.9950 and 589.5924 nanometers, the color of which will be familiar to anyone who has seen a low pressure sodium vapor lamp.
In the original spectroscope design in the early 19th century, light entered a slit and a collimating lens transformed the light into a thin beam of parallel rays. The light then passed through a prism (in hand-held spectroscopes, usually an Amici prism) that refracted the beam into a spectrum because different wavelengths were refracted different amounts due to dispersion. This image was then viewed through a tube with a scale that was transposed upon the spectral image, enabling its direct measurement.
With the development of photographic film, the more accurate spectrograph was created. It was based on the same principle as the spectroscope, but it had a camera in place of the viewing tube. In recent years, the electronic circuits built around the photomultiplier tube have replaced the camera, allowing real-time spectrographic analysis with far greater accuracy.
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It was based on the same principle as the spectroscope, but it had a camera in place of the viewing tube. In recent years, the electronic circuits built around the photomultiplier tube have replaced the camera, allowing real-time spectrographic analysis with far greater accuracy. Arrays of photosensors are also used in place of film in spectrographic systems. Such spectral analysis, or spectroscopy, has become an important scientific tool for analyzing the composition of unknown material and for studying astronomical phenomena and testing astronomical theories.
In modern spectrographs in the UV, visible, and near-IR spectral ranges, the spectrum is generally given in the form of photon number per unit wavelength (nm or μm), wavenumber (μm−1, cm−1), frequency (THz), or energy (eV), with the units indicated by the abscissa. In the mid- to far-IR, spectra are typically expressed in units of Watts per unit wavelength (μm) or wavenumber (cm−1). In many cases, the spectrum is displayed with the units left implied (such as "digital counts" per spectral channel).
### In Gemology
Gemologists frequently use spectroscopes to determine the absorption spectra of gemstones, thereby allowing them to make inferences about what kind of gem they are examining.
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In many cases, the spectrum is displayed with the units left implied (such as "digital counts" per spectral channel).
### In Gemology
Gemologists frequently use spectroscopes to determine the absorption spectra of gemstones, thereby allowing them to make inferences about what kind of gem they are examining. A gemologist may compare the absorption spectrum they observe with a catalogue of spectra for various gems to help narrow down the exact identity of the gem.
## Spectrographs
A spectrograph is an instrument that separates light into its wavelengths and records the data. A spectrograph typically has a multi-channel detector system or camera that detects and records the spectrum of light.
The term was first used in 1876 by Dr. Henry Draper when he invented the earliest version of this device, and which he used to take several photographs of the spectrum of Vega. This earliest version of the spectrograph was cumbersome to use and difficult to manage.
There are several kinds of machines referred to as spectrographs, depending on the precise nature of the waves. The first spectrographs used photographic paper as the detector. The plant pigment phytochrome was discovered using a spectrograph that used living plants as the detector. More recent spectrographs use electronic detectors, such as CCDs which can be used for both visible and UV light.
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The plant pigment phytochrome was discovered using a spectrograph that used living plants as the detector. More recent spectrographs use electronic detectors, such as CCDs which can be used for both visible and UV light. The exact choice of detector depends on the wavelengths of light to be recorded.
A spectrograph is sometimes called polychromator, as an analogy to monochromator.
### Stellar and solar spectrograph
The star spectral classification and discovery of the main sequence, Hubble's law and the Hubble sequence were all made with spectrographs that used photographic paper. James Webb Space Telescope contains both a near-infrared spectrograph (NIRSpec) and a mid-infrared spectrograph (MIRI).
### Echelle spectrograph
An echelle-based spectrograph uses two diffraction gratings, rotated 90 degrees with respect to each other and placed close to one another. Therefore, an entrance point and not a slit is used and a CCD-chip records the spectrum. Both gratings have a wide spacing, and one is blazed so that only the first order is visible and the other is blazed with many higher orders visible, so a very fine spectrum is presented to the CCD.
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Therefore, an entrance point and not a slit is used and a CCD-chip records the spectrum. Both gratings have a wide spacing, and one is blazed so that only the first order is visible and the other is blazed with many higher orders visible, so a very fine spectrum is presented to the CCD.
### Slitless spectrograph
In conventional spectrographs, a slit is inserted into the beam to limit the image extent in the dispersion direction. A slitless spectrograph omits the slit; this results in images that convolve the image information with spectral information along the direction of dispersion. If the field is not sufficiently sparse, then spectra from different sources in the image field will overlap. The trade is that slitless spectrographs can produce spectral images much more quickly than scanning a conventional spectrograph. That is useful in applications such as solar physics where time evolution is important.
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In computer science, a linked data structure is a data structure which consists of a set of data records (nodes) linked together and organized by references (links or pointers). The link between data can also be called a connector.
In linked data structures, the links are usually treated as special data types that can only be dereferenced or compared for equality. Linked data structures are thus contrasted with arrays and other data structures that require performing arithmetic operations on pointers. This distinction holds even when the nodes are actually implemented as elements of a single array, and the references are actually array indices: as long as no arithmetic is done on those indices, the data structure is essentially a linked one.
Linking can be done in two ways using dynamic allocation and using array index linking.
Linked data structures include linked lists, search trees, expression trees, and many other widely used data structures. They are also key building blocks for many efficient algorithms, such as topological sort and set union-find.
## Common types of linked data structures
### Linked lists
A linked list is a collection of structures ordered not by their physical placement in memory but by logical links that are stored as part of the data in the structure itself. It is not necessary that it should be stored in the adjacent memory locations.
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### Linked lists
A linked list is a collection of structures ordered not by their physical placement in memory but by logical links that are stored as part of the data in the structure itself. It is not necessary that it should be stored in the adjacent memory locations. Every structure has a data field and an address field. The Address field contains the address of its successor.
Linked list can be singly, doubly or multiply linked and can either be linear or circular.
Basic properties
- Objects, called nodes, are linked in a linear sequence.
- A reference to the first node of the list is always kept. This is called the 'head' or 'front'.
A linked list with three nodes contain two fields each: an integer value and a link to the next node
#### Example in Java
This is an example of the node class used to store integers in a Java implementation of a linked list:
```java
public class IntNode {
public int value;
public IntNode link;
public IntNode(int v) { value = v; }
}
```
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A linked list with three nodes contain two fields each: an integer value and a link to the next node
#### Example in Java
This is an example of the node class used to store integers in a Java implementation of a linked list:
```java
public class IntNode {
public int value;
public IntNode link;
public IntNode(int v) { value = v; }
}
```
#### Example in C
This is an example of the structure used for implementation of linked list in C:
```c
struct node
{
int val;
struct node *next;
};
```
This is an example using typedefs:
```c
typedef struct node node;
struct node
{
int val;
node *next;
};
```
Note: A structure like this which contains a member that points to the same structure is called a self-referential structure.
#### Example in C++
This is an example of the node class structure used for implementation of linked list in C++:
```cpp
class Node
{
int val;
Node *next;
};
```
### Search trees
A search tree is a tree data structure in whose nodes data values can be stored from some ordered set, which is such that in an in-order traversal of the tree the nodes are visited in ascending order of the stored values.
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This is an example of the node class structure used for implementation of linked list in C++:
```cpp
class Node
{
int val;
Node *next;
};
```
### Search trees
A search tree is a tree data structure in whose nodes data values can be stored from some ordered set, which is such that in an in-order traversal of the tree the nodes are visited in ascending order of the stored values.
Basic properties
- Objects, called nodes, are stored in an ordered set.
- In-order traversal provides an ascending readout of the data in the tree.
## Advantages and disadvantages
### Linked list versus arrays
Compared to arrays, linked data structures allow more flexibility in organizing the data and in allocating space for it. In arrays, the size of the array must be specified precisely at the beginning, which can be a potential waste of memory, or an arbitrary limitation which would later hinder functionality in some way. A linked data structure is built dynamically and never needs to be bigger than the program requires. It also requires no guessing at creation time, in terms of how much space must be allocated. This is a feature that is key in avoiding wastes of memory.
In an array, the array elements have to be in a contiguous (connected and sequential) portion of memory.
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This is a feature that is key in avoiding wastes of memory.
In an array, the array elements have to be in a contiguous (connected and sequential) portion of memory. But in a linked data structure, the reference to each node gives users the information needed to find the next one. The nodes of a linked data structure can also be moved individually to different locations within physical memory without affecting the logical connections between them, unlike arrays. With due care, a certain process or thread can add or delete nodes in one part of a data structure even while other processes or threads are working on other parts.
On the other hand, access to any particular node in a linked data structure requires following a chain of references that are stored in each node. If the structure has n nodes, and each node contains at most b links, there will be some nodes that cannot be reached in less than logb n steps, slowing down the process of accessing these nodes - this sometimes represents a considerable slowdown, especially in the case of structures containing large numbers of nodes. For many structures, some nodes may require worst case up to n−1 steps. In contrast, many array data structures allow access to any element with a constant number of operations, independent of the number of entries.
Broadly the implementation of these linked data structure is through dynamic data structures. It gives us the chance to use particular space again.
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Broadly the implementation of these linked data structure is through dynamic data structures. It gives us the chance to use particular space again. Memory can be utilized more efficiently by using these data structures. Memory is allocated as per the need and when memory is not further needed, deallocation is done.
### General disadvantages
Linked data structures may also incur in substantial memory allocation overhead (if nodes are allocated individually) and frustrate memory paging and processor caching algorithms (since they generally have poor locality of reference). In some cases, linked data structures may also use more memory (for the link fields) than competing array structures. This is because linked data structures are not contiguous. Instances of data can be found all over in memory, unlike arrays.
In arrays, nth element can be accessed immediately, while in a linked data structure we have to follow multiple pointers so element access time varies according to where in the structure the element is.
In some theoretical models of computation that enforce the constraints of linked structures, such as the pointer machine, many problems require more steps than in the unconstrained random-access machine model.
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In computer science, a type class is a type system construct that supports ad hoc polymorphism. This is achieved by adding constraints to type variables in parametrically polymorphic types. Such a constraint typically involves a type class `T` and a type variable `a`, and means that `a` can only be instantiated to a type whose members support the overloaded operations associated with `T`.
Type classes were first implemented in the Haskell programming language after first being proposed by Philip Wadler and Stephen Blott as an extension to "eqtypes" in Standard ML, and were originally conceived as a way of implementing overloaded arithmetic and equality operators in a principled fashion.
In contrast with the "eqtypes" of Standard ML, overloading the equality operator through the use of type classes in Haskell does not need extensive modification of the compiler frontend or the underlying type system.
## Overview
Type classes are defined by specifying a set of function or constant names, together with their respective types, that must exist for every type that belongs to the class.
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In contrast with the "eqtypes" of Standard ML, overloading the equality operator through the use of type classes in Haskell does not need extensive modification of the compiler frontend or the underlying type system.
## Overview
Type classes are defined by specifying a set of function or constant names, together with their respective types, that must exist for every type that belongs to the class. In Haskell, types can be parameterized; a type class `Eq` intended to contain types that admit equality would be declared in the following way:
```haskell
class Eq a where
(==) :: a -> a -> Bool
(/=) :: a -> a -> Bool
```
where `a` is one instance of the type class `Eq`, and `a` defines the function signatures for 2 functions (the equality and inequality functions), which each take 2 arguments of type `a` and return a Boolean.
The type variable `a` has kind
$$
$$
(
$$
$$
is also known as `Type` in the latest Glasgow Haskell Compiler (GHC) release), meaning that the kind of `Eq` is
```haskell
Eq :: Type -> Constraint
```
The declaration may be read as stating a "type `a` belongs to type class `Eq` if there are functions named `(==)`, and `(/=)`, of the appropriate types, defined on it".
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In Haskell, types can be parameterized; a type class `Eq` intended to contain types that admit equality would be declared in the following way:
```haskell
class Eq a where
(==) :: a -> a -> Bool
(/=) :: a -> a -> Bool
```
where `a` is one instance of the type class `Eq`, and `a` defines the function signatures for 2 functions (the equality and inequality functions), which each take 2 arguments of type `a` and return a Boolean.
The type variable `a` has kind
$$
$$
(
$$
$$
is also known as `Type` in the latest Glasgow Haskell Compiler (GHC) release), meaning that the kind of `Eq` is
```haskell
Eq :: Type -> Constraint
```
The declaration may be read as stating a "type `a` belongs to type class `Eq` if there are functions named `(==)`, and `(/=)`, of the appropriate types, defined on it". A programmer could then define a function `elem` (which determines if an element is in a list) in the following way:
```haskell
elem :: Eq a => a -> [a] -> Bool
elem y [] = False
elem y (x:xs) = (x == y) || elem y xs
```
The function `elem` has the type `a -> [a] -> Bool` with the context `Eq a`, which constrains the types which `a` can range over to those `a` which belong to the `Eq` type class.
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The type variable `a` has kind
$$
$$
(
$$
$$
is also known as `Type` in the latest Glasgow Haskell Compiler (GHC) release), meaning that the kind of `Eq` is
```haskell
Eq :: Type -> Constraint
```
The declaration may be read as stating a "type `a` belongs to type class `Eq` if there are functions named `(==)`, and `(/=)`, of the appropriate types, defined on it". A programmer could then define a function `elem` (which determines if an element is in a list) in the following way:
```haskell
elem :: Eq a => a -> [a] -> Bool
elem y [] = False
elem y (x:xs) = (x == y) || elem y xs
```
The function `elem` has the type `a -> [a] -> Bool` with the context `Eq a`, which constrains the types which `a` can range over to those `a` which belong to the `Eq` type class. (Haskell `=>` can be called a 'class constraint'.)
Any type `t` can be made a member of a given type class `C` by using an instance declaration that defines implementations of all of `C`'s methods for the particular type `t`.
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(Haskell `=>` can be called a 'class constraint'.)
Any type `t` can be made a member of a given type class `C` by using an instance declaration that defines implementations of all of `C`'s methods for the particular type `t`. For example, if a new data type `t` is defined, this new type can be made an instance of `Eq` by providing an equality function over values of type `t` in any way that is useful. Once this is done, the function `elem` can be used on `[t]`, that is, lists of elements of type `t`.
Type classes are different from classes in object-oriented programming languages. In particular, `Eq` is not a type: there is no such thing as a value of type `Eq`.
Type classes are closely related to parametric polymorphism. For example, the type of `elem` as specified above would be the parametrically polymorphic type `a -> [a] -> Bool` were it not for the type class constraint "`Eq a =>`".
## Higher-kinded polymorphism
A type class need not take a type variable of kind `Type` but can take one of any kind. These type classes with higher kinds are sometimes called constructor classes (the constructors referred to are type constructors such as `Maybe`, rather than data constructors such as `Just`).
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## Higher-kinded polymorphism
A type class need not take a type variable of kind `Type` but can take one of any kind. These type classes with higher kinds are sometimes called constructor classes (the constructors referred to are type constructors such as `Maybe`, rather than data constructors such as `Just`). An example is the `Monad` class:
```haskell
class Monad m where
return :: a -> m a
(>>=) :: m a -> (a -> m b) -> m b
```
That m is applied to a type variable indicates that it has kind `Type -> Type`, i.e., it takes a type and returns a type, the kind of `Monad` is thus:
```haskell
Monad :: (Type -> Type) -> Constraint
```
## Multi-parameter type classes
Type classes permit multiple type parameters, and so type classes can be seen as relations on types. For example, in the GHC standard library, the class `IArray` expresses a general immutable array interface. In this class, the type class constraint `IArray a e` means that `a` is an array type that contains elements of type `e`. (This restriction on polymorphism is used to implement unboxed array types, for example.)
Like multimethods, multi-parameter type classes support calling different implementations of a method depending on the types of multiple arguments, and indeed return types.
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(This restriction on polymorphism is used to implement unboxed array types, for example.)
Like multimethods, multi-parameter type classes support calling different implementations of a method depending on the types of multiple arguments, and indeed return types. Multi-parameter type classes do not require searching for the method to call on every call at runtime; rather the method to call is first compiled and stored in the dictionary of the type class instance, just as with single-parameter type classes.
Haskell code that uses multi-parameter type classes is not portable, as this feature is not part of the Haskell 98 standard. The popular Haskell implementations, GHC and Hugs, support multi-parameter type classes.
## Functional dependencies
In Haskell, type classes have been refined to allow the programmer to declare functional dependencies between type parameters—a concept inspired from relational database theory. Haskell' page FunctionalDependencies. That is, the programmer can assert that a given assignment of some subset of the type parameters uniquely determines the remaining type parameters. For example, a general monad `m` which carries a state parameter of type `s` satisfies the type class constraint `Monad. State s m`. In this constraint, there is a functional dependency `m -> s`. This means that for a given monad `m` of type class `Monad.
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In this constraint, there is a functional dependency `m -> s`. This means that for a given monad `m` of type class `Monad. State`, the state type accessible from `m` is uniquely determined. This aids the compiler in type inference, as well as aiding the programmer in type-directed programming.
Simon Peyton Jones has objected to the introduction of functional dependencies in Haskell on grounds of complexity.
## Type classes and implicit parameters
Type classes and implicit parameters are very similar in nature, although not quite the same. A polymorphic function with a type class constraint such as:
```hs
sum :: Num a => [a] -> a
```
can be intuitively treated as a function that implicitly accepts an instance of `Num`:
```hs
sum_ :: Num_ a -> [a] -> a
```
The instance `Num_ a` is essentially a record that contains the instance definition of `Num a`. (This is in fact how type classes are implemented under the hood by the Glasgow Haskell Compiler.)
However, there is a crucial difference: implicit parameters are more flexible; different instances of `Num Int` can be passed. In contrast, type classes enforce the so-called coherence property, which requires that there should only be one unique choice of instance for any given type.
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However, there is a crucial difference: implicit parameters are more flexible; different instances of `Num Int` can be passed. In contrast, type classes enforce the so-called coherence property, which requires that there should only be one unique choice of instance for any given type. The coherence property makes type classes somewhat antimodular, which is why orphan instances (instances that are defined in a module that neither contains the class nor the type of interest) are strongly discouraged. However, coherence adds another level of safety to a language, providing a guarantee that two disjoint parts of the same code will share the same instance.
As an example, an ordered set (of type `Set a`) requires a total ordering on the elements (of type `a`) to function. This can be evidenced by a constraint `Ord a`, which defines a comparison operator on the elements. However, there can be numerous ways to impose a total order. Since set algorithms are generally intolerant of changes in the ordering once a set has been constructed, passing an incompatible instance of `Ord a` to functions that operate on the set may lead to incorrect results (or crashes). Thus, enforcing coherence of `Ord a` in this particular scenario is crucial.
Instances (or "dictionaries") in Scala type classes are just ordinary values in the language, rather than a completely separate kind of entity.
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Thus, enforcing coherence of `Ord a` in this particular scenario is crucial.
Instances (or "dictionaries") in Scala type classes are just ordinary values in the language, rather than a completely separate kind of entity. While these instances are by default supplied by finding appropriate instances in scope to be used as the implicit parameters for explicitly-declared implicit formal parameters, that they are ordinary values means that they can be supplied explicitly, to resolve ambiguity. As a result, Scala type classes do not satisfy the coherence property and are effectively a syntactic sugar for implicit parameters.
This is an example taken from the Cats documentation:
```scala
// A type class to provide textual representation
trait Show[A] {
def show(f: A): String
}
// A polymorphic function that works only when there is an implicit
// instance of Show[A] available
def log[A](a: A)(implicit s: Show[A]) = println(s.show(a))
// An instance for String
implicit val stringShow = new Show[String] {
def show(s: String) = s
}
// The parameter stringShow was inserted by the compiler.
scala> log("a string")
a string
```
Rocq (previously known as Coq), version 8.2 onward, also supports type classes by inferring the appropriate instances.
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As a result, Scala type classes do not satisfy the coherence property and are effectively a syntactic sugar for implicit parameters.
This is an example taken from the Cats documentation:
```scala
// A type class to provide textual representation
trait Show[A] {
def show(f: A): String
}
// A polymorphic function that works only when there is an implicit
// instance of Show[A] available
def log[A](a: A)(implicit s: Show[A]) = println(s.show(a))
// An instance for String
implicit val stringShow = new Show[String] {
def show(s: String) = s
}
// The parameter stringShow was inserted by the compiler.
scala> log("a string")
a string
```
Rocq (previously known as Coq), version 8.2 onward, also supports type classes by inferring the appropriate instances. Recent versions of Agda 2 also provide a similar feature, called "instance arguments".
## Other approaches to operator overloading
In Standard ML, the mechanism of "equality types" corresponds roughly to Haskell's built-in type class `Eq`, but all equality operators are derived automatically by the compiler. The programmer's control of the process is limited to designating which type components in a structure are equality types and which type variables in a polymorphic type range over equality types.
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## Other approaches to operator overloading
In Standard ML, the mechanism of "equality types" corresponds roughly to Haskell's built-in type class `Eq`, but all equality operators are derived automatically by the compiler. The programmer's control of the process is limited to designating which type components in a structure are equality types and which type variables in a polymorphic type range over equality types.
SML's and OCaml's modules and functors can play a role similar to that of Haskell's type classes, the principal difference being the role of type inference, which makes type classes suitable for ad hoc polymorphism.
The object oriented subset of OCaml is yet another approach which is somewhat comparable to the one of type classes.
## Related notions
An analogous notion for overloaded data (implemented in GHC) is that of type family.
In C++, notably C++20, has support for type classes using Concepts (C++). As an illustration, the above mentioned Haskell example of typeclass Eq would be written as
```Cpp
template <typename T>
concept Equal =
requires (T a, T b) {
{ a == b } -> std::convertible_to<bool>;
{ a != b } -> std::convertible_to<bool>;
};
```
In Clean typeclasses are similar to Haskell, but have a slightly different syntax.
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As an illustration, the above mentioned Haskell example of typeclass Eq would be written as
```Cpp
template <typename T>
concept Equal =
requires (T a, T b) {
{ a == b } -> std::convertible_to<bool>;
{ a != b } -> std::convertible_to<bool>;
};
```
In Clean typeclasses are similar to Haskell, but have a slightly different syntax.
Rust supports traits, which are a limited form of type classes with coherence.
Mercury has typeclasses, although they are not exactly the same as in Haskell.
In Scala, type classes are a programming idiom that can be implemented with existing language features such as implicit parameters, not a separate language feature per se. Because of the way they are implemented in Scala, it is possible to explicitly specify which type class instance to use for a type at a particular place in the code, in case of ambiguity. However, this is not necessarily a benefit as ambiguous type class instances can be error-prone.
The proof assistant Rocq has also supported type classes in recent versions. Unlike in ordinary programming languages, in Rocq, any laws of a type class (such as the monad laws) that are stated within the type class definition, must be mathematically proved of each type class instance before using them.
## References
-
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Unlike in ordinary programming languages, in Rocq, any laws of a type class (such as the monad laws) that are stated within the type class definition, must be mathematically proved of each type class instance before using them.
## References
-
## External links
-
- Advanced Functional Programming course at Utrecht University, 74 lecture slides on Advanced Type Classes. 2005-06-07.
- Implementing, and Understanding Type Classes. 2014-11-13.
Category:Functional programming
Category: Type theory
Category:Data types
Category:Programming language comparisons
Category: Articles with example C++ code
Category: Articles with example Haskell code
Category: Articles with example Scala code
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A versioning file system is any computer file system which allows a computer file to exist in several versions at the same time. Thus it is a form of revision control. Most common versioning file systems keep a number of old copies of the file. Some limit the number of changes per minute or per hour to avoid storing large numbers of trivial changes.
### Others
instead take periodic snapshots whose contents can be accessed using methods similar as those for normal file access.
## Similar technologies
### Backup
A versioning file system is similar to a periodic backup, with several key differences.
- Backups are normally triggered on a timed basis, while versioning occurs when the file changes.
- Backups are usually system-wide or partition-wide, while versioning occurs independently on a file-by-file basis.
- Backups are normally written to separate media, while versioning file systems write to the same hard drive (and normally the same folder, directory, or local partition).
### In comparison to revision control systems
Versioning file systems provide some of the features of revision control systems. However, unlike most revision control systems, they are transparent to users, not requiring a separate "commit" step to record a new revision.
### Journaling file system
Versioning file systems should not be confused with journaling file systems.
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However, unlike most revision control systems, they are transparent to users, not requiring a separate "commit" step to record a new revision.
### Journaling file system
Versioning file systems should not be confused with journaling file systems. Whereas journaling file systems work by keeping a log of the changes made to a file before committing those changes to that file system (and overwriting the prior version), a versioning file system keeps previous copies of a file when saving new changes. The two features serve different purposes and are not mutually exclusive.
### Object storage
Some object storage implementations offers object versioning, such as Amazon S3.
## Implementations
### ITS
An early implementation of versioning, possibly the first, was in MIT's ITS. In ITS, a filename consisted of two six-character parts; if the second part was numeric (consisted only of digits), it was treated as a version number. When specifying a file to open for read or write, one could supply a second part of ">"; when reading, this meant to open the highest-numbered version of the file; when writing, it meant to increment the highest existing version number and create the new version for writing.
Another early implementation of versioning was in TENEX, which became TOPS-20.
### Files-11 (RSX-11 and OpenVMS)
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Another early implementation of versioning was in TENEX, which became TOPS-20.
### Files-11 (RSX-11 and OpenVMS)
A powerful example of a file versioning system is built into the RSX-11 and OpenVMS operating system from Digital Equipment Corporation. In essence, whenever an application opens a file for writing, the file system automatically creates a new instance of the file, with a version number appended to the name. Version numbers start at 1 and count upward as new instances of a file are created. When an application opens a file for reading, it can either specify the exact file name including version number, or just the file name without the version number, in which case the most recent instance of the file is opened.
The "purge" DCL/CCL command can be used at any time to manage the number of versions in a specific directory. By default, all but the highest numbered versions of all files in the current directory will be deleted; this behavior can be overridden with the /keep=n switch and/or by specifying directory path(s) and/or filename patterns. VMS systems are often scripted to purge user directories on a regular schedule; this is sometimes misconstrued by end-users as a property of the versioning system.
### Linux
- NILFS – A log-structured file system supporting versioning of the entire file system and continuous snapshotting.
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VMS systems are often scripted to purge user directories on a regular schedule; this is sometimes misconstrued by end-users as a property of the versioning system.
### Linux
- NILFS – A log-structured file system supporting versioning of the entire file system and continuous snapshotting. In this list, this is the only one that is stable and included in the mainline kernel.
- Tux3 – Most recent change was in 2014.
- Next3 – Most recent update was in 2012.
- ext3cow – Most recent release was in 2005.
On February 8, 2004, Kiran-Kumar Muniswamy-Reddy, Charles P. Wright, Andrew Himmer, and Erez Zadok (all from Stony Brook University) proposed a stackable file system Versionfs, providing a versioning layer on top of any other Linux file systems.
### LMFS
The Lisp Machine File System supports versioning. This was provided by implementations from MIT, LMI, Symbolics and Texas Instruments. Such an operating system was Symbolics Genera.
### macOS
Starting with Lion (10.7), macOS has a feature called Versions which allows Time Machine-like saving and browsing of past versions of documents for applications written to use Versions. This functionality, however, takes place at the application layer, not the filesystem layer; Lion and later releases do not incorporate a true versioning file system.
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### macOS
Starting with Lion (10.7), macOS has a feature called Versions which allows Time Machine-like saving and browsing of past versions of documents for applications written to use Versions. This functionality, however, takes place at the application layer, not the filesystem layer; Lion and later releases do not incorporate a true versioning file system.
### SCO OpenServer
HTFS, adopted as the primary filesystem for SCO OpenServer in 1995, supports file versioning. Versioning is enabled on a per-directory basis by setting the directory's setuid bit, which is inherited when subdirectories are created. If versioning is enabled, a new file version is created when a file or directory is removed, or when an existing file is opened with truncation. Non-current versions remain in the filesystem namespace, under the name of the original file but with a suffix attached consisting of a semicolon and version sequence number. All but the current version are hidden from directory reads (unless the SHOWVERSIONS environment variable is set), but versions are otherwise accessible for all normal operations. The environment variable and general accessibility allow versions to be managed with the usual filesystem utilities, though there is also an "undelete" command that can be used to purge and restore files, enable and disable versioning on directories, etc.
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All but the current version are hidden from directory reads (unless the SHOWVERSIONS environment variable is set), but versions are otherwise accessible for all normal operations. The environment variable and general accessibility allow versions to be managed with the usual filesystem utilities, though there is also an "undelete" command that can be used to purge and restore files, enable and disable versioning on directories, etc.
Others
- Subversion has a feature called "autoversioning" where a WebDAV source with a subversion backend can be mounted as a file system on systems that support this kind of mount (Linux, Windows and others do) and saves to that file system generate new revisions on the revision control system.
- The commercial Clearcase configuration management and revision control software has also supported "MVFS" (multi version file system) on HP-UX, AIX and Windows since the early 1990s.
### Related software
The following are not versioning filesystems, but allow similar functionality.
- APFS and ZFS support instantaneous snapshots and clones.
- Btrfs supports snapshots.
- HAMMER in DragonFlyBSD has the ability to store revisions in the filesystem.
- NILFS, which supports snapshotting.
- Plan 9's Fossil file system can provide a similar feature, taking periodic snapshots (often hourly) and making them available in .
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- NILFS, which supports snapshotting.
- Plan 9's Fossil file system can provide a similar feature, taking periodic snapshots (often hourly) and making them available in . Fossil can forever archive a snapshot into Venti (usually one snapshot each day) and make them available in . If multiple changes are made to a file during the interval between snapshots, only the most recent will be recorded in the next snapshot.
- Write Anywhere File Layout - NetApp's storage solutions implement a file system called WAFL, which uses snapshot technology to keep different versions of all files in a volume around.
- pdumpfs, authored by Satoru Takabayashi, is a simple daily backup system similar to Plan 9's /n/dump, implemented in Ruby. It functions as a snapshotting tool, which makes it possible to copy a whole directory to another location by using hardlinks. Used regularly, this can produce an effect similar to versioning.
- Microsoft Windows
- Shadow Copy - is a feature introduced by Microsoft with Windows Server 2003. Shadow Copy allows for taking manual or automatic backup copies or snapshots of a file or folder on a specific volume at a specific point in time.
- RollBack Rx - Allows snapshots of disk partitions to be taken. Each snapshot contains only the differences between previous snapshots, and take only seconds to create.
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- RollBack Rx - Allows snapshots of disk partitions to be taken. Each snapshot contains only the differences between previous snapshots, and take only seconds to create. Can be reliably used to keep a Windows OS stable and/or protected from malware.
- GoBack (discontinued) - The GoBack software for Windows from Symantec enables reversion of files, directories or disks to previous states. It can record a maximum of 8GB in changes, and temporarily stops recording each change in the event of high I/O activity.
- Versomatic - Versomatic software by Acertant automatically tracks file changes and preemptively archives a copy of a file before it is modified.
- Cascade File System exposes a Subversion or Perforce repository via a file system driver. The user must still explicitly decide when to commit changes.
- git implementation documents call git a "content addressable filesystem with a VCS user interface written on top of it." There's also a 3rd-party FUSE implementation exists that may extend git as a mountable, read-write versioning filesystem.
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus. They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable.
Infinitesimal calculus was formulated separately in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. The concepts and techniques found in calculus have diverse applications in science, engineering, and other branches of mathematics.
## Etymology
In mathematics education, calculus is an abbreviation of both infinitesimal calculus and integral calculus, which denotes courses of elementary mathematical analysis.
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The concepts and techniques found in calculus have diverse applications in science, engineering, and other branches of mathematics.
## Etymology
In mathematics education, calculus is an abbreviation of both infinitesimal calculus and integral calculus, which denotes courses of elementary mathematical analysis.
In Latin, the word calculus means “small pebble”, (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to be the Latin word for calculation. In this sense, it was used in English at least as early as 1672, several years before the publications of Leibniz and Newton, who wrote their mathematical texts in Latin.
In addition to differential calculus and integral calculus, the term is also used for naming specific methods of computation or theories that imply some sort of computation. Examples of this usage include propositional calculus, Ricci calculus, calculus of variations, lambda calculus, sequent calculus, and process calculus. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus, and the ethical calculus.
## History
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Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus, and the ethical calculus.
## History
### Modern
calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it first appeared in ancient
#### Egypt
and later
#### Greece
, then in
#### China
and the
#### Middle East
, and still later again in medieval Europe and
#### India
.
### Ancient precursors
Egypt
Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (), but the formulae are simple instructions, with no indication as to how they were obtained.
Greece
Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus () developed the method of exhaustion to prove the formulas for cone and pyramid volumes.
During the Hellenistic period, this method was further developed by Archimedes (BC), who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems now treated by integral calculus.
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Greece
Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus () developed the method of exhaustion to prove the formulas for cone and pyramid volumes.
During the Hellenistic period, this method was further developed by Archimedes (BC), who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems now treated by integral calculus. In The Method of Mechanical Theorems he describes, for example, calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines.
China
The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method Extract of page 27 that would later be called Cavalieri's principle to find the volume of a sphere.
### Medieval
Middle East
In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen (AD) derived a formula for the sum of fourth powers.
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In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method Extract of page 27 that would later be called Cavalieri's principle to find the volume of a sphere.
### Medieval
Middle East
In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen (AD) derived a formula for the sum of fourth powers. He determined the equations to calculate the area enclosed by the curve represented by
$$
y=x^k
$$
(which translates to the integral _ BLOCK1_ in contemporary notation), for any given non-negative integer value of
$$
k
$$
.He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.
India
Bhāskara II () was acquainted with some ideas of differential calculus and suggested that the "differential coefficient" vanishes at an extremum value of the function. In his astronomical work, he gave a procedure that looked like a precursor to infinitesimal methods. Namely, if
$$
x \approx y
$$
then
$$
\sin(y) - \sin(x) \approx (y - x)\cos(y).
$$
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In his astronomical work, he gave a procedure that looked like a precursor to infinitesimal methods. Namely, if
$$
x \approx y
$$
then
$$
\sin(y) - \sin(x) \approx (y - x)\cos(y).
$$
This can be interpreted as the discovery that cosine is the derivative of sine. In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics stated components of calculus. They studied series equivalent to the Maclaurin expansions of , , and more than two hundred years before their introduction in Europe. According to Victor J. Katz they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".
Modern
Johannes Kepler's work Stereometria Doliorum (1615) formed the basis of integral calculus. Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.
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Modern
Johannes Kepler's work Stereometria Doliorum (1615) formed the basis of integral calculus. Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.
Significant work was a treatise, the origin being Kepler's methods, written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise is believed to have been lost in the 13th century and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670.
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Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670.
The product rule and chain rule, the notions of higher derivatives and Taylor series, and of analytic functions were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics. In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.
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In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.
These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation.
Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics. Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series.
When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit.
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The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series.
When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his Method of Fluxions), but Leibniz published his "Nova Methodus pro Maximis et Minimis" first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions", a term that endured in English schools into the 19th century. The first complete treatise on calculus to be written in English and use the
### Leibniz notation
was not published until 1815.
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