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Bose–Einstein statistics

In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting identical particles may occupy a set of available discrete energy states at thermodynamic equilibrium. The aggregation of particles in the same state, which is a characteristic of particles obeying Bose–Einstein statistics, accounts for the cohesive streaming of laser light and the frictionless creeping of superfluid helium. The theory of this behaviour was developed (1924–25) by Satyendra Nath Bose, who recognized that a collection of identical and indistinguishable particles can be distributed in this way. The idea was later adopted and extended by Albert Einstein in collaboration with Bose.

Bose–Einstein statistics apply only to particles that do not follow the Pauli exclusion principle restrictions. Particles that follow Bose-Einstein statistics are called bosons, which have integer values of spin. In contrast, particles that follow Fermi-Dirac statistics are called fermions and have half-integer spins.

Bose–Einstein distribution
At low temperatures, bosons behave differently from fermions (which obey the Fermi–Dirac statistics) in a way that an unlimited number of them can "condense" into the same energy state. This apparently unusual property also gives rise to the special state of matter – the Bose–Einstein condensate. Fermi–Dirac and Bose–Einstein statistics apply when quantum effects are important and the particles are "indistinguishable". Quantum effects appear if the concentration of particles satisfies $$\frac{N}{V} \ge n_\text{q}, $$ where $N$ is the number of particles, $V$ is the volume, and $n_{q}$ is the quantum concentration, for which the interparticle distance is equal to the thermal de Broglie wavelength, so that the wavefunctions of the particles are barely overlapping.

Fermi–Dirac statistics applies to fermions (particles that obey the Pauli exclusion principle), and Bose–Einstein statistics applies to bosons. As the quantum concentration depends on temperature, most systems at high temperatures obey the classical (Maxwell–Boltzmann) limit, unless they also have a very high density, as for a white dwarf. Both Fermi–Dirac and Bose–Einstein become Maxwell–Boltzmann statistics at high temperature or at low concentration.

Bose–Einstein statistics was introduced for photons in 1924 by Bose and generalized to atoms by Einstein in 1924–25.

The expected number of particles in an energy state $i$ for Bose–Einstein statistics is: $$ with $ε_{i} > μ$ and where $n_{i}$ is the occupation number (the number of particles) in state $i$, $$g_i$$ is the degeneracy of energy level $i$, $ε_{i}$ is the energy of the $i$-th state, μ is the chemical potential (zero for a photon gas), $k_{B}$ is the Boltzmann constant, and $T$ is the absolute temperature.

The variance of this distribution $$V(n)$$ is calculated directly from the expression above for the average number. $$ V(n) = kT\frac{\partial}{\partial \mu}\bar{n}_i = \langle n\rangle(1+\langle n\rangle) = \bar{n} + \bar{n}^2 $$

For comparison, the average number of fermions with energy $$\varepsilon_i$$ given by Fermi–Dirac particle-energy distribution has a similar form: $$ \bar{n}_i(\varepsilon_i) = \frac{g_i}{e^{(\varepsilon_i - \mu)/k_\text{B}T} + 1}.$$

As mentioned above, both the Bose–Einstein distribution and the Fermi–Dirac distribution approaches the Maxwell–Boltzmann distribution in the limit of high temperature and low particle density, without the need for any ad hoc assumptions: \bar{n}_i = \frac{g_i}{e^{(\varepsilon_i-\mu)/k_\text{B}T} \pm 1} \ll 1$$. This again reduces to Maxwell–Boltzmann statistics.
 * In the limit of low particle density, $$\bar{n}_i = \frac{g_i}{e^{(\varepsilon_i-\mu)/k_\text{B}T}\pm 1} \ll 1$$, therefore $$e^{(\varepsilon_i-\mu)/k_\text{B}T} \pm 1 \gg 1$$ or equivalently $$e^{(\varepsilon_i-\mu)/k_\text{B}T} \gg 1$$. In that case, $$\bar{n}_i \approx \frac{g_i}{e^{(\varepsilon_i-\mu)/k_\text{B}T}}=\frac{1}{Z}e^{-(\varepsilon_i - \mu)/k_\text{B} T}$$, which is the result from Maxwell–Boltzmann statistics.
 * In the limit of high temperature, the particles are distributed over a large range of energy values, therefore the occupancy on each state (especially the high energy ones with $$ \varepsilon_i - \mu \gg k_\text{B}T $$) is again very small, $$

In addition to reducing to the Maxwell–Boltzmann distribution in the limit of high $$T$$ and low density, Bose–Einstein statistics also reduces to Rayleigh–Jeans law distribution for low energy states with $$ \varepsilon_i - \mu \ll k_\text{B}T $$, namely $$ \begin{align} \bar{n}_i & = \frac{g_i}{e^{(\varepsilon_i-\mu)/k_{\text{B}}T}-1} \\ &\approx \frac{g_i}{(\varepsilon_i-\mu)/k_\text{B}T} = \frac{g_i k_\text{B}T}{\varepsilon_i - \mu}. \end{align} $$

History
Władysław Natanson in 1911 concluded that Planck's law requires indistinguishability of "units of energy", although he did not frame this in terms of Einstein's light quanta.

While presenting a lecture at the University of Dhaka (in what was then British India and is now Bangladesh) on the theory of radiation and the ultraviolet catastrophe, Satyendra Nath Bose intended to show his students that the contemporary theory was inadequate, because it predicted results not in accordance with experimental results. During this lecture, Bose committed an error in applying the theory, which unexpectedly gave a prediction that agreed with the experiment. The error was a simple mistake—similar to arguing that flipping two fair coins will produce two heads one-third of the time—that would appear obviously wrong to anyone with a basic understanding of statistics (remarkably, this error resembled the famous blunder by d'Alembert known from his Croix ou Pile article ). However, the results it predicted agreed with experiment, and Bose realized it might not be a mistake after all. For the first time, he took the position that the Maxwell–Boltzmann distribution would not be true for all microscopic particles at all scales. Thus, he studied the probability of finding particles in various states in phase space, where each state is a little patch having phase volume of h3, and the position and momentum of the particles are not kept particularly separate but are considered as one variable.

Bose adapted this lecture into a short article called "Planck's law and the hypothesis of light quanta" and submitted it to the Philosophical Magazine. However, the referee's report was negative, and the paper was rejected. Undaunted, he sent the manuscript to Albert Einstein requesting publication in the Zeitschrift für Physik. Einstein immediately agreed, personally translated the article from English into German (Bose had earlier translated Einstein's article on the general theory of relativity from German to English), and saw to it that it was published. Bose's theory achieved respect when Einstein sent his own paper in support of Bose's to Zeitschrift für Physik, asking that they be published together. The paper came out in 1924.

The reason Bose produced accurate results was that since photons are indistinguishable from each other, one cannot treat any two photons having equal quantum numbers (e.g., polarization and momentum vector) as being two distinct identifiable photons. Bose originally had a factor of 2 for the possible spin states, but Einstein changed it to polarization. By analogy, if in an alternate universe coins were to behave like photons and other bosons, the probability of producing two heads would indeed be one-third, and so is the probability of getting a head and a tail which equals one-half for the conventional (classical, distinguishable) coins. Bose's "error" leads to what is now called Bose–Einstein statistics.

Bose and Einstein extended the idea to atoms and this led to the prediction of the existence of phenomena which became known as Bose–Einstein condensate, a dense collection of bosons (which are particles with integer spin, named after Bose), which was demonstrated to exist by experiment in 1995.

Derivation from the microcanonical ensemble
In the microcanonical ensemble, one considers a system with fixed energy, volume, and number of particles. We take a system composed of $N = \sum_i n_i$ identical bosons, $$n_i$$ of which have energy $$\varepsilon_i$$ and are distributed over $$g_i$$ levels or states with the same energy $$\varepsilon_i$$, i.e. $$g_i$$ is the degeneracy associated with energy $$\varepsilon_i$$ of total energy $E = \sum_i n_i \varepsilon_i$. Calculation of the number of arrangements of $$n_i$$ particles distributed among $$g_i$$ states is a problem of combinatorics. Since particles are indistinguishable in the quantum mechanical context here, the number of ways for arranging $$n_i$$ particles in $$g_i$$ boxes (for the $$i$$th energy level) would be (see image): $$ w_{i,\text{BE}} = \frac{(n_i+g_i-1)!}{n_i! (g_i-1)!} = C^{n_i+g_i-1}_{n_i}, $$ where $$ C^m_k $$ is the k-combination of a set with m elements. The total number of arrangements in an ensemble of bosons is simply the product of the binomial coefficients $$C^{n_i+g_i-1}_{n_i}$$ above over all the energy levels, i.e. $$ W_\text{BE} =\prod_i w_{i,\text{BE}}=\prod_i\frac{(n_i +g_i -1)!}{(g_i-1)! n_i!},$$

The maximum number of arrangements determining the corresponding occupation number $$n_i$$ is obtained by maximizing the entropy, or equivalently, setting $$\mathrm{d}(\ln W_\text{BE}) = 0$$ and taking the subsidiary conditions $N=\sum n_i, E=\sum_i n_i\varepsilon_i$ into account (as Lagrange multipliers). The result for $$n_i\gg 1$$, $$g_i\gg 1$$, $$n_i/g_i=O(1)$$ is the Bose–Einstein distribution.

Derivation from the grand canonical ensemble
The Bose–Einstein distribution, which applies only to a quantum system of non-interacting bosons, is naturally derived from the grand canonical ensemble without any approximations. In this ensemble, the system is able to exchange energy and exchange particles with a reservoir (temperature T and chemical potential μ fixed by the reservoir).

Due to the non-interacting quality, each available single-particle level (with energy level ϵ) forms a separate thermodynamic system in contact with the reservoir. That is, the number of particles within the overall system that occupy a given single particle state form a sub-ensemble that is also grand canonical ensemble; hence, it may be analysed through the construction of a grand partition function.

Every single-particle state is of a fixed energy, $$\varepsilon$$. As the sub-ensemble associated with a single-particle state varies by the number of particles only, it is clear that the total energy of the sub-ensemble is also directly proportional to the number of particles in the single-particle state; where $$N$$ is the number of particles, the total energy of the sub-ensemble will then be $$N\varepsilon$$. Beginning with the standard expression for a grand partition function and replacing $$E$$ with $$N \varepsilon$$, the grand partition function takes the form $$\mathcal Z = \sum_N \exp((N\mu - N\varepsilon)/k_\text{B} T) = \sum_N \exp(N(\mu - \varepsilon)/k_\text{B} T)$$

This formula applies to fermionic systems as well as bosonic systems. Fermi–Dirac statistics arises when considering the effect of the Pauli exclusion principle: whilst the number of fermions occupying the same single-particle state can only be either 1 or 0, the number of bosons occupying a single particle state may be any integer. Thus, the grand partition function for bosons can be considered a geometric series and may be evaluated as such: $$ \begin{align}\mathcal Z & = \sum_{N=0}^\infty \exp(N(\mu - \varepsilon)/k_\text{B} T) = \sum_{N=0}^\infty [\exp((\mu - \varepsilon)/k_\text{B}T)]^N \\ & = \frac{1}{1 - \exp((\mu - \varepsilon)/k_\text{B} T)}.\end{align}$$

Note that the geometric series is convergent only if $$e^{(\mu - \varepsilon)/k_\text{B}T}<1$$, including the case where $$\varepsilon = 0$$. This implies that the chemical potential for the Bose gas must be negative, i.e., $$\mu<0$$, whereas the Fermi gas is allowed to take both positive and negative values for the chemical potential.

The average particle number for that single-particle substate is given by $$ \langle N\rangle = k_\text{B} T \frac{1}{\mathcal Z} \left(\frac{\partial \mathcal Z}{\partial \mu}\right)_{V,T} = \frac{1}{\exp((\varepsilon-\mu)/k_\text{B} T)-1} $$ This result applies for each single-particle level and thus forms the Bose–Einstein distribution for the entire state of the system.

The variance in particle number, $\sigma_N^2 = \langle N^2 \rangle - \langle N \rangle^2$, is: $$ \sigma_N^2 = k_\text{B} T \left(\frac{d\langle N\rangle}{d\mu}\right)_{V,T} = \frac{\exp((\varepsilon-\mu)/k_\text{B} T)}{(\exp((\varepsilon-\mu)/k_\text{B} T)-1)^2} = \langle N\rangle(1 + \langle N\rangle). $$

As a result, for highly occupied states the standard deviation of the particle number of an energy level is very large, slightly larger than the particle number itself: $$\sigma_N \approx \langle N\rangle$$. This large uncertainty is due to the fact that the probability distribution for the number of bosons in a given energy level is a geometric distribution; somewhat counterintuitively, the most probable value for N is always 0. (In contrast, classical particles have instead a Poisson distribution in particle number for a given state, with a much smaller uncertainty of $ \sigma_{N,{\rm classical}} = \sqrt{\langle N\rangle}$, and with the most-probable N value being near $$\langle N \rangle$$.)

Derivation in the canonical approach
It is also possible to derive approximate Bose–Einstein statistics in the canonical ensemble. These derivations are lengthy and only yield the above results in the asymptotic limit of a large number of particles. The reason is that the total number of bosons is fixed in the canonical ensemble. The Bose–Einstein distribution in this case can be derived as in most texts by maximization, but the mathematically best derivation is by the Darwin–Fowler method of mean values as emphasized by Dingle. See also Müller-Kirsten. The fluctuations of the ground state in the condensed region are however markedly different in the canonical and grand-canonical ensembles.

Suppose we have a number of energy levels, labeled by index $$ i$$, each level having energy $$ \varepsilon_i$$ and containing a total of $$ n_i$$ particles. Suppose each level contains $$ g_i$$ distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of $$ g_i$$ associated with level $$ i$$ is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel.

Let $$ w(n,g)$$ be the number of ways of distributing $$ n$$ particles among the $$ g$$ sublevels of an energy level. There is only one way of distributing $$ n$$ particles with one sublevel, therefore $$ w(n,1)=1$$. It is easy to see that there are $$ (n+1)$$ ways of distributing $$ n$$ particles in two sublevels which we will write as: $$w(n,2)=\frac{(n+1)!}{n!1!}.$$

With a little thought (see Notes below) it can be seen that the number of ways of distributing $$ n$$ particles in three sublevels is $$w(n,3) = w(n,2) + w(n-1,2) + \cdots + w(1,2) + w(0,2)$$ so that $$w(n,3)=\sum_{k=0}^n w(n-k,2) = \sum_{k=0}^n\frac{(n-k+1)!}{(n-k)!1!}=\frac{(n+2)!}{n!2!}$$ where we have used the following theorem involving binomial coefficients: $$\sum_{k=0}^n\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}.$$

Continuing this process, we can see that $$ w(n,g)$$ is just a binomial coefficient (See Notes below) $$w(n,g)=\frac{(n+g-1)!}{n!(g-1)!}.$$

For example, the population numbers for two particles in three sublevels are 200, 110, 101, 020, 011, or 002 for a total of six which equals 4!/(2!2!). The number of ways that a set of occupation numbers $$ n_i$$ can be realized is the product of the ways that each individual energy level can be populated: $$ W = \prod_i w(n_i,g_i) = \prod_i \frac{(n_i + g_i-1)!}{n_i!(g_i-1)!} \approx \prod_i \frac{(n_i+g_i)!}{n_i!(g_i)!} $$ where the approximation assumes that $$n_i \gg 1$$.

Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of $$ n_i$$ for which W is maximised, subject to the constraint that there be a fixed total number of particles, and a fixed total energy. The maxima of $$ W$$ and $$ \ln(W)$$ occur at the same value of $$ n_i$$ and, since it is easier to accomplish mathematically, we will maximise the latter function instead. We constrain our solution using Lagrange multipliers forming the function: $$f(n_i)=\ln(W)+\alpha(N-\sum n_i)+ \beta(E-\sum n_i \varepsilon_i)$$

Using the $$n_i \gg 1$$ approximation and using Stirling's approximation for the factorials $$\left(x!\approx x^x\,e^{-x}\,\sqrt{2\pi x}\right)$$ gives $$f(n_i)=\sum_i (n_i + g_i) \ln(n_i + g_i) - n_i \ln(n_i) +\alpha\left(N-\sum n_i\right)+\beta\left(E-\sum n_i \varepsilon_i\right)+K ,$$ where K is the sum of a number of terms which are not functions of the $$n_i$$. Taking the derivative with respect to $$ n_i$$, and setting the result to zero and solving for $$ n_i$$, yields the Bose–Einstein population numbers: $$n_i = \frac{g_i}{e^{\alpha+\beta \varepsilon_i}-1}.$$

By a process similar to that outlined in the Maxwell–Boltzmann statistics article, it can be seen that: $$d\ln W = \alpha\,dN + \beta\,dE$$ which, using Boltzmann's famous relationship $$S=k_\text{B}\,\ln W$$ becomes a statement of the second law of thermodynamics at constant volume, and it follows that $$\beta = \frac{1}{k_\text{B}T}$$ and $$\alpha = - \frac{\mu}{k_\text{B}T}$$ where S is the entropy, $$\mu$$ is the chemical potential, kB is the Boltzmann constant and T is the temperature, so that finally: $$n_i = \frac{g_i}{e^{(\varepsilon_i-\mu)/k_\text{B}T}-1}.$$

Note that the above formula is sometimes written: $$n_i = \frac{g_i}{e^{\varepsilon_i/k_\text{B}T}/z-1},$$ where $$ z=\exp(\mu/k_\text{B}T)$$ is the absolute activity, as noted by McQuarrie.

Also note that when the particle numbers are not conserved, removing the conservation of particle numbers constraint is equivalent to setting $$\alpha$$ and therefore the chemical potential $$\mu$$ to zero. This will be the case for photons and massive particles in mutual equilibrium and the resulting distribution will be the Planck distribution.

A much simpler way to think of Bose–Einstein distribution function is to consider that n particles are denoted by identical balls and g shells are marked by g-1 line partitions. It is clear that the permutations of these n balls and g − 1 partitions will give different ways of arranging bosons in different energy levels. Say, for 3 (= n) particles and 3 (= g) shells, therefore (g − 1) = 2, the arrangement might be |●●|●, or ||●●●, or |●|●● , etc. Hence the number of distinct permutations of n + (g − 1) objects which have n identical items and (g − 1) identical items will be: $$ \frac{(g-1 + n)!}{(g-1)! n!} $$

See the image for a visual representation of one such distribution of n particles in g boxes that can be represented as g − 1 partitions.

OR

The purpose of these notes is to clarify some aspects of the derivation of the Bose–Einstein distribution for beginners. The enumeration of cases (or ways) in the Bose–Einstein distribution can be recast as follows. Consider a game of dice throwing in which there are $$ n$$ dice, with each die taking values in the set$$ \{ 1, \dots, g \}$$, for $$g \ge 1$$. The constraints of the game are that the value of a die $$ i$$, denoted by $$ m_i$$, has to be greater than or equal to the value of die $$ (i-1)$$, denoted by $$m_{i-1}$$, in the previous throw, i.e., $$m_i \ge m_{i-1}$$. Thus a valid sequence of die throws can be described by an n-tuple $$ ( m_1, m_2 , \dots , m_n)$$, such that $$m_i \ge m_{i-1}$$. Let $$ S(n,g)$$ denote the set of these valid n-tuples:

Then the quantity $$ w(n,g)$$ (defined above as the number of ways to distribute $$ n$$ particles among the $$ g$$ sublevels of an energy level) is the cardinality of $$ S(n,g)$$, i.e., the number of elements (or valid n-tuples) in $$ S(n,g)$$. Thus the problem of finding an expression for $$ w(n,g)$$ becomes the problem of counting the elements in $$ S(n,g)$$.

Example n = 4, g = 3: $$  S(4,3) = \left\{ \underbrace{(1111), (1112), (1113)}_{(a)}, \underbrace{(1122), (1123), (1133)}_{(b)}, \underbrace{(1222), (1223), (1233), (1333)}_{(c)}, \underbrace{(2222), (2223), (2233), (2333), (3333)}_{(d)} \right\} $$ $$ w(4,3) = 15$$ (there are $$15$$ elements in $$ S(4,3)$$)

Subset $$ (a)$$ is obtained by fixing all indices $$ m_i$$ to $$ 1$$, except for the last index, $$ m_n$$, which is incremented from $$1$$ to $$ g=3$$. Subset $$(b)$$ is obtained by fixing $$ m_1 = m_2 = 1$$, and incrementing $$ m_3$$ from $$ 2$$ to $$ g=3$$. Due to the constraint $$ m_i \ge m_{i-1}$$ on the indices in $$ S(n,g)$$, the index $$ m_4$$ must automatically take values in $$ \left\{ 2, 3 \right\}$$. The construction of subsets $$ (c)$$ and $$ (d)$$ follows in the same manner.

Each element of $$ S(4,3)$$ can be thought of as a multiset of cardinality $$ n=4$$; the elements of such multiset are taken from the set $$ \left\{ 1, 2, 3 \right\}$$ of cardinality $$ g=3$$, and the number of such multisets is the multiset coefficient $$  \left\langle \begin{matrix} 3 \\ 4 \end{matrix} \right\rangle = {3 + 4 - 1 \choose 3-1} = {3 + 4 - 1 \choose 4} = \frac {6!} {4! 2!}  = 15 $$

More generally, each element of $$ S(n,g)$$ is a multiset of cardinality $$ n$$ (number of dice) with elements taken from the set $$ \left\{ 1, \dots, g \right\}$$ of cardinality $$ g$$ (number of possible values of each die), and the number of such multisets, i.e., $$ w(n,g)$$is the multiset coefficient

which is exactly the same as the formula for $$w(n,g)$$, as derived above with the aid of a theorem involving binomial coefficients, namely

To understand the decomposition

or for example, $$ n=4$$ and $$ g=3$$ $$ w(4,3) = w(4,2) + w(3,2) + w(2,2) + w(1,2) + w(0,2), $$

let us rearrange the elements of $$ S(4,3)$$ as follows $$ S(4,3) = \left\{ \underbrace{ (1111), 	 (1112), 	 (1122), 	 (1222), 	 (2222)     }_{(\alpha)}, \underbrace{ (111{\color{Red}{\underset{=}{3}}}), (112{\color{Red}{\underset{=}{3}}}), (122{\color{Red}{\underset{=}{3}}}), (222{\color{Red}{\underset{=}{3}}}) }_{(\beta)}, \underbrace{ (11{\color{Red}{\underset{==}{33}}}), (12{\color{Red}{\underset{==}{33}}}), (22{\color{Red}{\underset{==}{33}}}) }_{(\gamma)}, \underbrace{ (1{\color{Red}{\underset{===}{333}}}), (2{\color{Red}{\underset{===}{333}}}) }_{(\delta)} \underbrace{ ({\color{Red}{\underset{====}{3333}}}) }_{(\omega)} \right\}. $$

Clearly, the subset $$ (\alpha)$$ of $$ S(4,3)$$ is the same as the set $$ S(4,2) = \left\{ (1111), (1112), (1122), (1222), (2222) \right\}.$$

By deleting the index $$ m_4=3$$ (shown in red with double underline ) in the subset $$(\beta)$$ of $$S(4,3)$$, one obtains the set $$ S(3,2) = \left\{ (111), (112), (122), (222) \right\}.$$

In other words, there is a one-to-one correspondence between the subset $$(\beta)$$ of $$S(4,3)$$ and the set $$ S(3,2)$$. We write $$ (\beta)  \longleftrightarrow   S(3,2) .$$

Similarly, it is easy to see that $$ (\gamma)  \longleftrightarrow   S(2,2) =  \left\{ (11), (12), (22) \right\}$$ $$ (\delta)  \longleftrightarrow   S(1,2) =  \left\{ (1), (2) \right\}$$ $$ (\omega)  \longleftrightarrow   S(0,2) = \{\} = \varnothing.$$

Thus we can write $$ S(4,3) = \bigcup_{k=0}^{4}  S(4-k,2)$$ or more generally,

and since the sets $$ S(i,g-1), \text{ for } i = 0, \dots, n$$ are non-intersecting, we thus have

with the convention that

Continuing the process, we arrive at the following formula $$ w(n,g) = \sum_{k_1=0}^n \sum_{k_2=0}^{n-k_1} w(n - k_1 - k_2, g-2) = \sum_{k_1=0}^n \sum_{k_2=0}^{n-k_1} \cdots \sum_{k_g=0}^{n-\sum_{j=1}^{g-1} k_j} w(n - \sum_{i=1}^{g} k_i, 0). $$ Using the convention (7)2 above, we obtain the formula

keeping in mind that for $$ q$$ and $$p$$ being constants, we have

It can then be verified that (8) and (2) give the same result for $$w(4,3)$$, $$w(3,3)$$, $$w(3,2)$$, etc.

Interdisciplinary applications
Viewed as a pure probability distribution, the Bose–Einstein distribution has found application in other fields:
 * In recent years, Bose–Einstein statistics has also been used as a method for term weighting in information retrieval. The method is one of a collection of DFR ("Divergence From Randomness") models, the basic notion being that Bose–Einstein statistics may be a useful indicator in cases where a particular term and a particular document have a significant relationship that would not have occurred purely by chance. Source code for implementing this model is available from the Terrier project at the University of Glasgow.
 * The evolution of many complex systems, including the World Wide Web, business, and citation networks, is encoded in the dynamic web describing the interactions between the system's constituents. Despite their irreversible and nonequilibrium nature these networks follow Bose statistics and can undergo Bose–Einstein condensation. Addressing the dynamical properties of these nonequilibrium systems within the framework of equilibrium quantum gases predicts that the "first-mover-advantage", "fit-get-rich" (FGR) and "winner-takes-all" phenomena observed in competitive systems are thermodynamically distinct phases of the underlying evolving networks.

title: Bose–Einstein statistics


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Richard Bentley

Richard Bentley FRS (27 January 1662 – 14 July 1742) was an English classical scholar, critic, and theologian. Considered the "founder of historical philology", Bentley is widely credited with establishing the English school of Hellenism. In 1892, A. E. Housman called Bentley "the greatest scholar that England or perhaps that Europe ever bred".

Bentley's Dissertation upon the Epistles of Phalaris, published in 1699, proved that the letters in question, supposedly written in the 6th century BCE by the Sicilian tyrant Phalaris, were actually a forgery produced by a Greek sophist in the 2nd century CE. Bentley's investigation of the subject is still regarded as a landmark of textual criticism. He also showed that the sound represented in transcriptions of some Greek dialects by the letter digamma appeared also in Homeric poetry, even though it was not represented there in writing by any letter.

Bentley became Master of Trinity College, Cambridge in 1700. His autocratic manner and contemptuous treatment of the college fellows led to extensive controversy and litigation, but he remained in that post until his death, more than four decades later. In 1717 Bentley was appointed as the Regius Professor of Divinity at the University of Cambridge. As professor at Cambridge, Bentley introduced the first competitive written examinations in a Western university.

A fellow of the Royal Society, Bentley was interested in natural theology and the new physical sciences, subjects on which he corresponded with Isaac Newton. Bentley was in charge of the second edition of Newton's Principia Mathematica, although he delegated most of the scientific work involved to his pupil Roger Cotes.

Early life and education

Richard Bentley was born at his maternal grandparents' home at Oulton near Rothwell, Leeds, West Yorkshire, in northern England. A blue plaque near his birthplace commemorates the fact. His father was Thomas Bentley, a yeoman farmer of Oulton. His grandfather, Captain James Bentley, is said to have suffered for the Royalist cause following the English Civil War, leaving the family in reduced circumstances. Bentley's mother, the daughter of a stonemason, had some education, and was able to give her son his first lessons in Latin.

After attending grammar school in Wakefield, Bentley entered St John's College, Cambridge in 1676. He afterwards obtained a scholarship and received the degrees of BA in 1680 and MA in 1683.

Academic work
Bentley never became a college fellow, which would have been the more natural course to an academic career. Instead, he was appointed headmaster of Spalding Grammar School before he was 21 years old. Edward Stillingfleet, the dean of St Paul's Cathedral in London, hired Bentley as tutor to his son. This allowed Bentley to meet eminent scholars, have access to the best private library in England, and become familiar with Dean Stillingfleet. During his six years as tutor, Bentley also made a comprehensive study of Greek and Latin writers, storing up knowledge which he would use later in his scholarship.

In 1689, Stillingfleet became bishop of Worcester and Bentley's pupil went up to Wadham College, Oxford, accompanied by his tutor. At Oxford, Bentley soon met John Mill, Humphrey Hody, and Edward Bernard. He studied the manuscripts of the Bodleian, Corpus Christi, and other college libraries. He collected material for literary studies. Among these are a corpus of the fragments of the Greek poets and an edition of the Greek lexicographers.

The Oxford (Sheldonian) press was about to bring out an edition (the editio princeps) from the unique manuscript of the Chronographia in the Bodleian Library. It was a universal history (down to AD 560) in Greek by John Malalas or "John the Rhetor" of Antioch (date uncertain, between 600 and 1000). The editor, John Mill, principal of St Edmund Hall, asked Bentley to review it and make any pertinent remarks on the text.

Bentley wrote the Epistola ad Johannem Millium, which is about a hundred pages long and was included at the end of the Oxford Malalas (1691). That short treatise placed Bentley ahead of all living English scholars. The ease with which he restored corrupted passages, the certainty of his emendation and command over the relevant material, are in a style totally different from the careful and laborious learning of Hody, Mill or Edmund Chilmead. To the small circle of classical students (lacking the great critical dictionaries of modern times), it was obvious that he was a critic beyond the ordinary.

In 1690, Bentley had taken deacon's orders. In 1692 he was nominated first Boyle lecturer, a nomination repeated in 1694. He was offered the appointment a third time in 1695 but declined it, as he was involved in too many other activities. In the first series of lectures ("A Confutation of Atheism"), he endeavours to present Newtonian physics in a popular form, and to frame them (especially in opposition to Hobbes) into a proof of the existence of an intelligent Creator. He had some correspondence with Newton, then living in Trinity College, Cambridge, on the subject. The second series, preached in 1694, has not been published and is believed to be lost.

After being ordained, Bentley was promoted to a prebendal stall in Worcester Cathedral. In 1693 the curator of the royal library became vacant, and his friends tried to obtain the position for Bentley, but did not have enough influence. The new librarian, a Mr Thynne, resigned in favour of Bentley, on condition that he receive an annuity of £130 for life out of the £200 salary. In 1695 Bentley received a royal chaplaincy and the living of Hartlebury.

That same year, he was elected a fellow of the Royal Society, and in 1696 earned the degree of Doctor of Divinity. The scholar Johann Georg Graevius of Utrecht made a dedication to him, prefixed to a dissertation on the seventeenth-century scholar Albert Rubens, De Vita Fl. Mallii Theodori (1694).

Dissertation on the Epistles of Phalaris
Bentley had official apartments in St. James's Palace and his first care was the royal library in Ashburnham House. He worked to restore the collection from a dilapidated condition. He persuaded the Earl of Marlborough to ask for additional rooms in the palace for the books. This was granted, but Marlborough kept them for his personal use. Bentley enforced the law, ensuring that publishers delivered nearly one thousand volumes that had been purchased but not delivered.

The University of Cambridge commissioned Bentley to obtain Greek and Latin fonts for their classical books; he had these made in Holland. He assisted John Evelyn in his Numismata. Bentley did not settle down to the steady execution of any of the major projects he had started. In 1694, he designed an edition of Philostratus, but abandoned it to Gottfried Olearius (1672–1715), "to the joy," says F. A. Wolf, "of Olearius and of no one else." He supplied Graevius with collations of Cicero, and Joshua Barnes with a warning as to the spuriousness of the Epistles of Euripides. Barnes printed the epistles anyway and declared that no one could doubt their authenticity but a man who was perfrictae frontis aut judicii imminuti (boldfaced and lacking in judgment). For Graevius's Callimachus (1697), Bentley added a collection of the fragments with notes.

He wrote the Dissertation on the Epistles of Phalaris (1699), his major academic work, almost accidentally. In 1697, William Wotton, about to bring out a second edition of his Ancient and Modern Learning, asked Bentley to write out a paper exposing the spuriousness of the Epistles of Phalaris, long a subject of academic controversy. The Christ Church editor of Phalaris, Charles Boyle, resented Bentley's paper. He had already quarrelled with Bentley in trying to get the manuscript in the royal library collated for his edition (1695). Boyle wrote a response which was accepted by the reading public, although it was much later criticised as showing only superficial learning. The demand for Boyle's book required a second printing. When Bentley responded, it was with his dissertation. The truth of its conclusions was not immediately recognised, but it has a high reputation.

Master of Trinity College
In 1700, the commissioners of ecclesiastical patronage recommended Bentley to the Crown for the mastership of Trinity College, Cambridge. He arrived an outsider and proceeded to reform the college administration. He started a programme of renovations to the buildings, and used his position to promote learning. He is also credited by the British mathematician Rouse Ball with starting the first written examinations in the West in 1702, all those prior to this being oral in nature. At the same time, he antagonised the fellows, and the capital programme caused reductions in their incomes, which they resented.

After ten years of stubborn but ineffectual resistance, the fellows appealed to the Visitor, the bishop of Ely (John Moore). Their petition was full of general complaints. Bentley's reply (The Present State of Trinity College, etc., 1710) is in his most crushing style. The fellows amended their petition and added a charge of Bentley's having committed 54 breaches of the statutes. Bentley appealed directly to the Crown, and backed his application with a dedication of his Horace to the lord treasurer (Harley).

The Crown lawyers decided against him; the case was heard (1714) and a sentence of expulsion from the mastership was drawn up. Before it was executed, the bishop of Ely died and the process lapsed. The feud continued in various forms at lower levels. In 1718 Cambridge rescinded Bentley's degrees, as punishment for failing to appear in the vice-chancellor's court in a civil suit. It was not until 1724 that he had them restored under the law.

In 1733 the fellows of Trinity again brought Bentley to trial before the bishop of Ely (then Thomas Greene), and he was sentenced to deprivation. The college statutes required the sentence to be executed by the vice-master Richard Walker, who was a friend of Bentley and refused to act. Although the feud continued until 1738 or 1740 (about thirty years in all), Bentley remained in his post.

Later studies
During his mastership, except for the first two years, Bentley continuously pursued his studies, although he did not publish much. In 1709 he contributed a critical appendix to John Davies's edition of Cicero's Tusculan Disputations. In the following year, he published his emendations on the Plutus and Nubes of Aristophanes, and on the fragments of Menander and Philemon. He published the last work under the pen name of "Phileleutherus Lipsiensis." He used it again two years later in his Remarks on a late Discourse of Freethinking, a reply to Anthony Collins the deist. The university thanked him for this work and its support of the Anglican Church and clergy.

Although he had long studied Horace, Bentley wrote his edition quickly in the end, publishing it in 1711 to gain public support at a critical period of the Trinity quarrel. In the preface, he declared his intention of confining his attention to criticism and correction of the text. Some of his 700 or 800 emendations have been accepted, but the majority were rejected by the early 20th century as unnecessary, although scholars acknowledged they showed his wide learning.

In 1716, in a letter to William Wake, Archbishop of Canterbury, Bentley announced his plan to prepare a critical edition of the New Testament. During the next four years, assisted by J. J. Wetstein, an eminent biblical critic, he collected materials for the work. In 1720 he published Proposals for a New Edition of the Greek Testament, with examples of how he intended to proceed. By comparing the text of the Vulgate with that of the oldest Greek manuscripts, Bentley proposed to restore the Greek text as received by the church at the time of the Council of Nicaea. Bentley's lead manuscript was Codex Alexandrinus, which he described as "the oldest and best in the world." The manuscript was so precious to him that he rescued it from perishing in the Ashburnham House fire of 1731, during which many other Cotton Library manuscripts were destroyed. Bentley used also manuscripts: 51, 54, 60, 113, 440, 507, and 508. John Walker worked over many manuscripts for the project, particularly in Paris with the help of the Maurists. Numerous subscribers were obtained to support publication of the work, but he never completed it.

His Terence (1726) is more important than his Horace; next to the Phalaris, this most determined his reputation. In 1726 he also published the Fables of Phaedrus and the Sententiae of Publilius Syrus.

His Paradise Lost (1732), suggested by Queen Caroline, has been criticised as the weakest of his work. He suggested that the poet John Milton had employed both an amanuensis and an editor, who were responsible for clerical errors and interpolations, but it is unclear whether Bentley believed his own position. A. E. Housman, who called him "the greatest scholar that England or perhaps that Europe ever bred" nevertheless criticised his poetic sensibility severely: "we are not all so easily found out as Bentley, because we have not Bentley's intrepid candour. There is a sort of savage nobility about his firm reliance on his own bad taste".

Bentley never published his planned edition of Homer, but some of his manuscript and marginal notes are held by Trinity College. Their chief importance is in his attempt to restore the metre by the insertion of the lost digamma.

Relationships and personal life
According to the anonymous author of his biography in the Encyclopaedia Britannica Eleventh Edition, Bentley was self-assertive and presumptuous, which alienated some people. But, James Henry Monk, Bentley's biographer, charged him (in his first edition, 1830) with an indecorum of which he was not guilty. Bentley seemed to inspire mixed feelings of admiration and repugnance.

Marriage and family
In 1701, Bentley married Joanna (died 1740), daughter of Sir John Bernard, 2nd Baronet of Brampton, Huntingdonshire. They had three children together: Richard (1708–1782), an eccentric, playwright and artist, and two daughters. Their daughter Johanna married Denison Cumberland in 1728, a grandson of Richard Cumberland the bishop of Peterborough, and himself later a bishop of the Church of Ireland. Their son Richard Cumberland developed as a prolific dramatist while earning his living as a civil servant.

Later life
In old age, Bentley continued to read and enjoyed the society of his friends and of several rising scholars, including Jeremiah Markland, John Taylor, and his nephews Richard and Thomas Bentley, with whom he discussed classical subjects. He died of pleurisy on 14 July 1742, at the age of 80.

Legacy and honours
Bentley left about £5,000 in his estate (which would have the buying power of nearly £500,000 in 2010). He bequeathed a few Greek manuscripts, brought from Mount Athos, to the Trinity College library and the remainder of his books and papers to his nephew Richard Bentley, a fellow of Trinity. At his own death in 1786, the younger Bentley left the papers to the Trinity College library. The British Museum eventually purchased the books, many of which had valuable manuscript notes, and holds them in its collection.

Bentley is honoured to this day at Spalding Grammar School, where he was once headmaster. One of the 6 houses that students are sorted into is named Bentley after him, with the students wearing the colour blue on their ties and on sporting items. Furthermore, the school releases an annual magazine named the Bentlian, also named after him.

According to the anonymous author of his biography in the Encyclopaedia Britannica Eleventh Edition, Bentley was the first Englishman to be ranked with the great heroes of classical learning. Before him there were only John Selden, and, in a more restricted field, Thomas Gataker and John Pearson. "Bentley inaugurated a new era of the art of criticism. He opened a new path. With him criticism attained its majority. Where scholars had hitherto offered suggestions and conjectures, Bentley, with unlimited control over the whole material of learning, gave decisions"

The modern German school of philology recognised his genius. Bunsen wrote that Bentley "was the founder of historical philology." Jakob Bernays says of his corrections of the Tristia, "corruptions which had hitherto defied every attempt even of the mightiest, were removed by a touch of the fingers of this British Samson".

Bentley was credited with creating the English school of Hellenists, by which the 18th century was distinguished, including scholars such as Richard Dawes, Jeremiah Markland, John Taylor, Jonathan Toup, Thomas Tyrwhitt, Richard Porson, Peter Paul Dobree, Thomas Kidd and James Henry Monk. Although the Dutch school of the period had its own tradition, it was also influenced by Bentley. His letters to Tiberius Hemsterhuis on his edition of Julius Pollux made the latter one of Bentley's most devoted admirers.

Bentley inspired a following generation of scholars. Self-taught, he created his own discipline; but no contemporary English guild of learning could measure his power or check his eccentricities. He defeated his academic adversaries in the Phalaris controversy. The attacks by Alexander Pope (he was assigned a niche in The Dunciad), John Arbuthnot and others demonstrated their inability to appreciate his work, as they considered textual criticism as pedantry. His classical controversies also called forth Jonathan Swift's Battle of the Books.

In a university where the instruction of youth or the religious controversy of the day was the chief occupation, Bentley was unique. His learning and original views seem to have been developed before 1700. After this period, he acquired little and made only spasmodic efforts to publish. However A. E. Housman believed that the edition of Manilius (1739) was Bentley's greatest work.

Major

 * Works of Richard Bentley, collected by Alexander Dyce, 1836. v. 1–2. Dissertations upon the epistles of Phalaris, Themistocles, Socrates, Euripides, and upon the fables of Aesop; also, Epistola ad Joannem Millium – v. 3. Sermons preached at Boyle's lecture; remarks upon a discourse of free-thinking; proposals for an edition of the Greek testament.

Minor

 * Astronomica of Manilius (1739)
 * a letter on the Sigean inscription on a marble slab found in the Troad, now in the British Museum
 * notes on the Theriaca of Nicander and on Lucan, published after his death by his grandson, Richard Cumberland
 * emendations of Plautus (in his copies of the editions by Pareus, Camerarius and Gronovius, edited by Schroder, 1880, and Sonnenschein, 1883)
 * Bentleii Critica Sacra (1862), edited by A. A. Ellis, contains the epistle to the Galatians (and excerpts), printed from an interleaved folio copy of the Greek and Latin Vulgate in Trinity College.
 * a collection of his Opuscula Philologica published at Leipzig in 1781.
 * A Letter to the Reverend Master of Trinity College in Cambridge, Editor of a New Greek and Latin Testament (1721)
 * A confutation of atheism from the origin and frame of the world, a sermon, Volume 20 (1692)
 * A Defence of natural and revealed religion: being an abridgment of the lecture founded by the Honble Robert Boyle (1737)
 * Eight Sermons Preach'd at the Honourable Robert Boyle's Lecture (1724)
 * Remarks upon a late Discourse of free-thinking: in a letter to N. N. (1725)
 * The Folly and Unreasonableness of Atheism: Demonstrated from the Advantage and Pleasure of a Religious Life, The faculties of Human Souls, The Structure of Animate Bodies, & The Origin and Frame of the World: In Eight Sermons (1693)
 * The Present State of Trinity College in Cambridg[e], in a Letter from Dr. Bentley, London: A. Baldwin 1710.
 * Reflections on the Scandalous Aspersions cast on the Clergy, [London:] J. Morphew 1717.

Letters

 * Bentlei et doctorum-virorum ad eum Epistolae (1807)
 * The Correspondence of Richard Bentley, edited by C. Wordsworth (1842)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Richard Bentley


text:
Appius Claudius

Appius Claudius may refer to:

 * Appius Claudius Caecus
 * Appius Claudius Caudex
 * Appius Claudius Crassus Inregillensis Sabinus
 * Appius Claudius Pulcher (disambiguation)
 * Appius Claudius Sabinus Inregillensis

title: Appius Claudius


text:
Wikipedia talk:Canonization

It´s very important exemplarize the use of wiktionary from wikipedia, because some articles are not really encyclopedic, but dyctionarial.

Because of this, I include clearly how to write a link to wiktionary.

I don't know if this is the place to ask, but what about other protocols? IRC channel would like to make an [irc://]-style link for Mozilla users and others with the appropriate hooks. -Geoffrey

 * I love the idea !!! I suggest include it in the wiki links

Capitalizing the first letter of article titles
I just made an article titled I'noGo tied but the link is from a lower-case "i" at the beginning, and I'm pretty sure it's supposed to be, at least according to my source (even at the beginning of a sentence, so I assume it's a pronunciation thing in Inuit). The system automatically switched it. Is there anyway around that? Not really a huge deal, I guess; it's still in lowercase in the article itself.--User: Tokerboy

You can link to it with leading lowercase, but you should talk to LDC about whether there's anyway to make it display the title with leading lowercase. --KQ

There isn't one, but you could talk with him about making one. --Brion

Old discussion relating to an old version of the software
A better strategy might be
 * 1) remove all accents, umlauts and other diacriticals
 * 2) convert all non-alphanumerics to underscores
 * 3) delete all leading and trailing underscores, and replace any consecutive underscores with a single underscore
 * 4) uppercase the first letter, and any letter that follows an underscore
 * 5) lowercase all characters that were not explicitly uppercased in #4

What happened finally with the wiki canonization discussed here?? The new scheme was implemented on the non-english wikis, but the new version (magnus' one) used the ols scheme. what will happen? (specially since the links and article names in the non-english wikipedias are made using the all uppercase scheme... AN

Basically, we never switched to the newest version of UseModWiki, because we never solved the problem of the namespace crunch, where "Foo Bar" and "foo bar" were distinct articles in Wikipedia but one of which would be lost in converting to the newest version of UseModWiki.

After working on the Nupedia Chalkboard, which does use the latest version of UseModWiki, as for myself, I've more or less come to the conclusion that it's rather better to have the old format, which does distinguish between upper and lower case in titles. I'm not directly responsible for the fact that Magnus' software does distinguish between upper and lower case in titles, but neither did I speak up when I noticed that it still does. Frankly, I was a bit relieved that it did, because the titles do look nicer according to the older standard. I think we'd have even more of a problem (a wetware problem) with people making links All Upper Case, since they think that'd be necessary for the link to work. --LMS

er....what?
I have no idea from reading this article what Canonization means, what its use is, and why it exists as an idea. Please help me to understand, and please update the page so others can learn too. thanks. Kingturtle 02:59, 13 Dec 2003 (UTC)

 * It appears to be trying to refer to the process of transforming a given title to its canonical (standard, official) form, though I'm not convinced "canonization" is the correct word for this. --Brion 03:18, 13 Dec 2003 (UTC)

"Canonicalization"?
I'm pretty sure this page should be at Canonicalization. Yes, "canonicalize" is a neologism, but it's quite common in computer programming circles. There's really no definition of "canonize" which means "to render canonical" -- it seems like a well-intentioned but fault back formation, here. "Canonize" mostly means "to recognize as a saint", and although there are some Wikipedia saints, that's not what this page seem to be about. --ESP 20:04, 6 Jan 2004 (UTC)

title: Wikipedia talk:Canonization


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Jeremiah Markland

Jeremiah Markland (18 October (or 29) 1693 – 7 July 1776) was an English classical scholar.

Life
He was born in Childwall in Lancashire (now Liverpool) on 29 (or 18) October 1693. He was educated at Christ's Hospital and Peterhouse, Cambridge.

He left Cambridge in 1728 to act as private tutor to the son of W. Strode of Ponsbourne, Hertfordshire, returning to the university in 1733. At a later date he lived at Twyford, and in 1744 went to Uckfield, Sussex, in order to superintend the education of the son of his former pupil, Mr. Strode. In 1752 he fixed his abode at Milton Court, near Dorking, Surrey, and remained there, living in great privacy, to the end of his days.

He died at Milton Court, near Dorking.

Works
His most important works are
 * Epistola critica: ad ... Franciscum Hare ... in qua Horatii loca aliquot et aliorum veterum emendantur (1723)
 * the Sylvae of Statius (1728)
 * notes to the editions of Lysias by Taylor, of Maximus of Tyre by Davies, of Euripides's Hippolytus by Musgrave
 * editions of Euripides's Supplices, Iphigenia in Tauride and in Aulide (ed. T. Gaisford 1811)
 * Remarks on the Epistles of Cicero to Brutus, and of Brutus to Cicero (1745).

title: Jeremiah Markland


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Jonathan Toup

Jonathan Oannes Toup (19 December 1713 – 19 January 1785) was an English philologist, classical scholar and critic.

Early life and education
Toup was born at St Ives, Cornwall in December 1713 and baptised on 5 January 1714. After the death of his father in 1721, Toup's mother Prudence (née Busvargus) remarried, and Toup was adopted by her brother, William, the last male of the Busvargus family. Toup had one full sister, Mary, and two half-sisters, Prudence and Ann. After the death of Toup's stepfather in 1763, Toup's mother and all three of her daughters lived with him.

Toup was educated at St. Ives grammar school, and afterwards by the Rev. John Gurney, who kept a private school at St Merryn in Cornwall. From 15 March 1733 to 13 November 1739, Toup was battellar of Exeter College, Oxford, where John Upton was his tutor. He graduated B.A. on 14 October 1736, and later received his M.A. from Cambridge in 1756. He never married.

Church positions
Toup was ordained a deacon on 6 March 1736, and three days later was licensed to the curacy of Philleigh in his native county. After two years, on 29 May 1738, he was licensed as curate of Buryan, also in Cornwall, having proceeded to priest's orders on the previous day. Through the influence of his uncle Busvargus, he was presented on 28 July 1750 to the rectory of St. Martin's-by-Looe, which position he held until his death.

When his name was once mentioned to the bishop of Exeter the response was "Toup,--who is Toup?" the reply being "A poor curate in your diocese, my lord, but the greatest Greek scholar in Europe".

Achievements
Toup's reputation was established by his Emendationes in Suidam, the first part of which was published in 1760, the second part in 1764, and the third part in 1766. The Emendationes were followed by an Epistola Critica to Bishop Warburton in 1767, in which Toup made some derisive comments about Bishop Lowth, while flattering Warburton for his learning. A volume of Curae novissimae sive appendicula notarum et emendationum in Suidam was published in 1775.

Although the critical power and skill of these works earned Toup an immense reputation at home and abroad, he was also criticised for his "immoderate language" and "boorish conduct." One scholar called Toup "a piece of a coxcomb" having "superior airs"; another called him a "homo truculentus et maledicus" ("aggressive and slanderous man"). Although Toup was reviled by some, others allowed that he was very charitable to the poor of his parish, and that his excessive self-confidence could be attributed to the fact that he lived apart, without sufficient personal intercourse with other scholars. He was said to have possessed an "uncompromising independence of mind and a hatred of servility," and censure of others was with him more frequent than praise.

After a preparation of thirty-five years, Toup's edition of Longinus, in Greek and Latin, was published in 1778. This edition included notes and emendations by David Ruhnken, whose assistance was mentioned on the title page. However, Ruhnken later regretted providing assistance to Toup, feeling that Toup had taken credit for his work and had not even sent him a presentation copy of the work when it was completed.

Later years
On 14 May 1774, when Toup was more than sixty years old, he was appointed by Bishop Keppel to a prebendal stall at Exeter and was admitted on 29 July 1776 to the vicarage of St Merryn, the parish in which he had been partly educated. These preferments he held, with his rectory, to his death.

For some years before his death, Toup's mental capacity was diminished, and he was cared for by his half-sister Ann and her three daughters. He died at St Martin's rectory on 19 January 1785 and was buried under the communion table of the church. His property was left to his nephew John Toup Nicolas.

A small marble tablet was erected to his memory on the south wall of the church by his niece Phillis Blake. The tablet states that the excellence of Toup's scholarship was "known to the learned throughout Europe." The inscription on a round brass plate beneath the tablet records that the cost was defrayed by the delegates of the Oxford University Press.

title: Jonathan Toup


text:
Battle of Jemappes

The Battle of Jemappes (6 November 1792) took place near the town of Jemappes in Hainaut, Austrian Netherlands (now Belgium), near Mons during the War of the First Coalition, part of the French Revolutionary Wars. One of the first major offensive battles of the war, it was a victory for the armies of the infant French Republic, and saw the French Armée du Nord, which included many inexperienced volunteers, defeat a substantially smaller regular Austrian army.

General Charles François Dumouriez, in command of an army of French Revolutionary volunteers, faced the Imperial army of Field Marshal Duke Albert of Saxe-Teschen and his second-in-command François de Croix, Count of Clerfayt. The French, who outnumbered their opponents by about three-to-one, launched a series of enthusiastic but uncoordinated attacks against the Austrian position on a ridge. At length, the French seized a portion of the ridge and the Austrians were unable to drive them away. Saxe-Teschen conceded defeat by ordering a withdrawal.

Jemappes was won by costly but effective charges against the Austrians' prepared position. Dumouriez overran the Austrian Netherlands within a month, but lost it at the Battle of Neerwinden in March. The French would not reconquer the Austrian Netherlands until the summer of 1794.

Background
In the summer of 1792 Charles Dumouriez, the French foreign minister and commander of the Armée du Nord, had believed that the best way to prevent an Austrian and Prussian invasion of France was to invade the Austrian Netherlands, but the Allies had launched their invasion before Dumouriez was ready to move, and he had been forced to move south. The Allied invasion had been at Valmy on 20 September where the French army stood up to an artillery bombardment, and proved that it would not flee at the first sign of opposition The Allied commander, the Duke of Brunswick, was not willing to risk a full-scale assault on the French line, and withdrew after it.

This left Dumouriez free to move north, to first raise the siege to Lille in late September and into early October, and then to launch his long-planned invasion of the Austrian Netherlands. His original plan for a three-pronged invasion had to be changed, as the promised resources to achieve it proved unavailable, and instead, at the end of October, he concentrated most of his men in front of Valenciennes and marched towards Mons, and the way to Brussels.

Austrians
The Austrian army was commanded by Duke Albert of Saxe-Teschen, the governor of the Austrian Netherlands. Although he had more than 20,000 troops available, they were scattered in a long defensive line, and so at Jemappes he fought with only 11,600 infantry, 2,170 cavalry and 56 guns. With this power, he tried to defend the 5 mi long Cuesmes ridge which ran from Mons in the Austrian left to Jemappes on the right side.

The Austrian right was commanded by Franz Freiherr von Lilien, the center by Franz Sebastian de Croix, Count of Clerfayt and the left by Johann Peter Freiherr von Beaulieu. Lilien had seven companies and four infantry battalions and three squadrons of cavalry on their left while Clerfayt had three infantry battalions and four squadrons around the village of Cuesmes and Beaulieu had three battalions of infantry on the hills south of Bertaimont with five companies of infantry and a squadron of cavalry guarding his left. Two other companies were further to the left around Mont Palisel and an infantry battalion was at Mons.

The Austrian army positioned themselves on the marshes around the Trouille groves and rivers, with two dams to their rear. The only other way for a retreat was via Mons.

French
Dumouriez had twice as many men as the Austrians. His own Armée du Nord contained 32,000 infantry, 3,800 cavalry and 100 guns and was supported in Jemappes by a further 4,000 men and 15 guns under General François Harville. Dumouriez's infantry battalions contained thirteen volunteers from 1792. Harville's men were also volunteers, but most of the older commanders were either experienced soldiers or aristocrats. The most obvious example was the commander of the French center, the Duke of Chartres, who had assumed the name of General Egalite, and would later become King Louis-Philippe of France. The right wing was commanded by General Pierre de Ruel, marquis de Beurnonville and the left by General Louis Marie de la Caussade Ferrand who also carried the name Jean Henri Becays Ferrand. Harville was to reinforce the right.

Dumouriez planned to use his army's numbers to overtake the Austrian position. The plan was for Harville and Beurnonville to attack first, and surround the weak Austrian left. Ferrand would then capture Quaregnon before Jemappes. Beurnonville would then attack the Austrian center while Harville moved to Mont Palisel to cut off the Austrian retreat.

Battle

Saxe-Teschen entrenched his 11,628 infantry, 2,168 cavalry and 56 guns along the Cuesmes Ridge just a few kilometers west of Mons. The Austrian artillery included fourteen 12-lb cannon, thirty-six 6-lb and 3-lb cannon and six 7-lb howitzers. The north end of the position, defended by Feldmarschall-Leutnant Franz Freiherr von Lilien, was anchored on the village of Jemappes. Feldzeugmeister Count Clerfayt commanded the center and Feldmarschall-Leutnant Johann Peter Beaulieu led the left wing. The Austrian right wing faced to the west, while the center and the left wings faced toward the southwest. The village of Cuesmes lay behind the Austrian left. One flaw in the position was that an Austrian retreat could only be made across a single bridge over the Hain River.

Dumouriez had 32,000 infantry, 3,800 cavalry and 100 artillery pieces. He expected to be joined by an additional 4,000 troops on the right under General Louis Auguste Juvénal des Ursins d'Harville. (Digby Smith gave a total of 40,000 infantry and 3,000 cavalry.) Dumouriez planned to turn both Austrian flanks. Accordingly, he divided his army into two wings, giving General Jean Henri Becays Ferrand command of the left wing and General Pierre de Ruel, marquis de Beurnonville control of the right wing. The French army was made up of a motley collection of royal army, volunteer, and national guard units.

The French made a series of "ill-coordinated but enthusiastic" attacks which began at dawn and continued throughout the morning. With momentum stalling Dumouriez ordered a renewed assault at noon. The Duke of Chartres sent a massive French column at the center of the ridge. This gained a foothold which the Austrians could not dislodge. Some French soldiers also enveloped the enemy right, threatening the Austrian rear. In response, Saxe-Teschen withdrew his right and center into Mons. Beaulieu ably covered the retreat with his left wing.

Aftermath
The French reported approximately 650 dead and 1,300 wounded. One of the dead being Armand Gagné, the foster son of Marie Antoinette. The Austrians reported 305 dead, 513 wounded, plus 423 men and five guns captured. Many of the Austrian casualties were caused by the plentiful French artillery. The Bender Infantry Regiment Nr. 41 suffered especially heavy casualties, losing 14 officers and 400 rank and file. Mons surrendered to the French the day after the battle and Brussels fell on 14 November. The French populace "went wild with joy" at this first offensive victory of the war.

At first glance Jemappes was not an impressive French victory. The Austrians had suffered 818 casualties, and lost another 423 men taken prisoner, while France had a higher casualty rate and had failed to prevent the escape of a much smaller army to defend a position of danger. However, in the context of the situation in 1792, with the French army in chaos due to exile of many of its experienced officers, it was a great success. The victory at Jemappes, achieved by inexperienced volunteers over the Austrian regulars, greatly increased the confidence of the revolutionary government in Paris, and encouraged their tendency to aggressive warfare.

In the short-term Jemappes gave the French control of the Austrian Netherlands. Mons opened its doors to Dumouriez, and he remained there until 12 November. He then moved to Brussels, fighting a rearguard action in Anderlecht on 13 November, before capturing the city on 14 November. This first French occupation of Belgium would be short-lived, but in the few months that the revolutionaries managed to alienate the population, imposing their ideas of freedom on a conservative population. In 1793 Dumouriez was forced to flee into exile, but his victory at Jemappes was an important step in the direction of the military triumphs of the French Republic. In addition, it ensured that the majority of the battles fought in 1793 would occur outside the borders of France.

title: Battle of Jemappes


text:
Athlon 64

The Athlon 64 is a ninth-generation, AMD64-architecture microprocessor produced by Advanced Micro Devices (AMD), released on September 23, 2003. It is the third processor to bear the name Athlon, and the immediate successor to the Athlon XP. The Athlon 64 was the second processor to implement the AMD64 architecture (after the Opteron) and the first 64-bit processor targeted at the average consumer. Variants of the Athlon 64 have been produced for Socket 754, Socket 939, Socket 940, and Socket AM2. It was AMD's primary consumer CPU, and primarily competed with Intel's Pentium 4, especially the Prescott and Cedar Mill core revisions.

The Athlon 64 is AMD's first K8, eighth-generation processor core for desktop and mobile computers. Despite being natively 64-bit, the AMD64 architecture is backward-compatible with 32-bit x86 instructions. The Athlon 64 line was succeeded by the dual-core Athlon 64 X2 and Athlon X2 lines.

Background
The Athlon 64 was originally codenamed ClawHammer by AMD, and was referred to as such internally and in press releases. The first Athlon 64 FX was based on the first Opteron core, SledgeHammer. Both cores, produced on a 130 nanometer process, were first introduced on September 23, 2003. The models first available were the FX-51, fitting Socket 940, and the 3200+, fitting Socket 754. Like the Opteron, on which it was based, the Athlon FX-51 required buffered random-access memory (RAM), increasing the final cost of an upgrade. The week of the Athlon 64's launch, Intel released the Pentium 4 Extreme Edition, a CPU designed to compete with the Athlon 64 FX. The Extreme Edition was widely considered a marketing ploy to draw publicity away from AMD, and was quickly nicknamed among some circles the "Emergency Edition". Despite a very strong demand for the chip, AMD experienced early manufacturing difficulties that made it difficult to deliver Athlon 64s in quantity. In the early months of the Athlon 64 lifespan, AMD could only produce 100,000 chips per month. However, it was very competitive in terms of performance to the Pentium 4, with magazine PC World calling it the "fastest yet". The Athlon FX-51 also outperforming the Pentium 4 3.2C in Quake III Arena and Unreal Tournament 2003 Benchmark, According to Maximum PC. "Newcastle" was released soon after ClawHammer, with half the Level 2 cache.

Single-core Athlon 64
All the 64-bit processors sold by AMD so far have their genesis in the K8 or Hammer project. On June 1, 2004, AMD released new versions of both the ClawHammer and Newcastle core revisions for the newly introduced Socket 939, an altered Socket 940 without the need for buffered memory. Socket 939 offered two main improvements over Socket 754: the memory controller was altered with dual-channel architecture, doubling peak memory bandwidth, and the HyperTransport bus was increased in speed from 800 MHz to 1000 MHz. Socket 939 also was introduced in the FX series in the form of the FX-55. At the same time, AMD also began to ship the "Winchester" core, based on a 90 nanometer process.

Core revisions "Venice" and "San Diego" succeeded all prior revisions on April 15, 2005. Venice, the lower-end part, was produced for both Sockets 754 and 939, and included 512 kB of L2 cache. San Diego, the higher-end chip, was produced only for Socket 939 and doubled Venice's L2 cache to 1 MB. Both were produced on the 90 nm fabrication process. Both also included support for the SSE3 instruction set, a new feature that had been included in the rival Pentium 4 since the release of the Prescott core in February 2004. In addition, AMD overhauled the memory controller for this revision, resulting in performance improvements as well as support for newer DDR RAM.

Dual-core Athlon 64
On April 21, 2005, less than a week after the release of Venice and San Diego, AMD announced its next addition to the Athlon 64 line, the Athlon 64 X2. Released on May 31, 2005, it also initially had two different core revisions available to the public, Manchester and Toledo, the only appreciable difference between them being the amount of L2 cache. Both were released only for Socket 939. The Athlon 64 X2 was received very well by reviewers and the general public, with a general consensus emerging that AMD's implementation of multi-core was superior to that of the competing Pentium D. Some felt initially that the X2 would cause market confusion with regard to price points since the new processor was targeted at the same "enthusiast," US$350 and above market already occupied by AMD's existing socket 939 Athlon 64s. AMD's official breakdown of the chips placed the Athlon X2 aimed at a segment they called the "prosumer", along with digital media fans. The Athlon 64 was targeted at the mainstream consumer, and the Athlon FX at gamers. The Sempron budget processor was targeted at value-conscious consumers. Following the launch of the Athlon 64 X2, AMD surpassed Intel in US retail sales for a period of time, although Intel retained overall market leadership because of its exclusive relationships with direct sellers such as Dell.

DDR2
The Athlon 64 had been maligned by some critics for some time because of its lack of support for DDR2 SDRAM, an at the time emerging technology that had been adopted much earlier by Intel. AMD's official position was that the CAS latency on DDR2 had not progressed to a point where it would be advantageous for the consumer to adopt it. AMD finally remedied this gap with the "Orleans" core revision, the first Athlon 64 to fit Socket AM2, released on May 23, 2006. "Windsor", an Athlon 64 X2 revision for Socket AM2, was released concurrently. Both Orleans and Windsor have either 512 kB or 1 MB of L2 cache per core. The Athlon 64 FX-62 was also released concurrently on the Socket AM2 platform. Socket AM2 also uses less power than prior platforms, and supports AMD-V.

The memory controller used in all DDR2 SDRAM capable processors (Socket AM2), has extended column address range of 11 columns instead of conventional 10 columns, and the support of 16 kB page size, with at most 2048 individual entries supported. An OCZ unbuffered DDR2 kit, optimized for 64-bit operating systems, was released to exploit the functionality provided by the memory controller in socket AM2 processors, allowing the memory controller to stay longer on the same page, thus benefitting graphics intensive applications.

Moving to the subnotebook space
The Athlon architecture was further extended with the release of Athlon Neo processors on January 9, 2009. Based on the same architecture as the other Athlon 64 variants, the new processor features a small package footprint targeting Ultra-portable notebook market.

Features
There are four variants: Athlon 64, Athlon 64 FX, Mobile Athlon 64 (later renamed "Turion 64") and the dual-core Athlon 64 X2. Common among the Athlon 64 line are a variety of instruction sets including MMX, 3DNow!, SSE, SSE2, and SSE3. All Athlon 64s also support the NX bit, a security feature named "Enhanced Virus Protection" by AMD. And as implementations of the AMD64 architecture, all Athlon 64 variants are able to run 16 bit, 32 bit x86, and AMD64 code, through two different modes the processor can run in: "Legacy mode" and "long mode". Legacy mode runs 16-bit and 32-bit programs natively, and long mode runs 64-bit programs natively, but also allows for 32-bit programs running inside a 64-bit operating system. All Athlon 64 processors feature 128 Kilobytes of level 1 cache, and at least 512 kB of level 2 cache.

On-die memory controller
The Athlon 64 features an on-die memory controller, a feature formerly seen on only the Transmeta Crusoe. This means the controller runs at the same clock rate as the CPU, and that the electrical signals have a shorter physical distance to travel compared to the old northbridge interfaces. The result is a significant reduction in latency (response time) for access requests to main memory. The lower latency was often cited as one of the advantages of the Athlon 64's architecture over those of its competitors at the time.

Memory and HT Northbridge buses
As the memory controller is integrated onto the CPU die, there is no FSB for the system memory to base its speed upon. Instead, system memory speed is obtained by using the following formula (using the ceiling function):

 * $$\frac{\mathrm{CPU~speed}}{\left\lceil\frac{\mathrm{CPU~multiplier}}{\mathrm{DRAM~divider}}\right\rceil}=\mathrm{DRAM~speed}$$

In simpler terms, the memory is always running at a set fraction of the CPU speed, with the divisor being a whole number. An 'FSB' figure is still used to determine the CPU speed, but the RAM speed is no longer directly related to this 'FSB' figure (known otherwise as the LDT).

A second bus, the northbridge, connected the CPU to the chipset and device attachment bus (PCIe, AGP, PCI). This was implemented using a new high-performance standard, HyperTransport. AMD attempted, with some success, to make this an industry standard. It was also useful in building multi-processor systems without additional glue chips.

Translation lookaside buffers
Translation lookaside buffers (TLBs) have also been enlarged (40 4k/2M/4M entries in L1 cache, 512 4k entries), with reduced latencies and improved branch prediction, with four times the number of bimodal counters in the global history counter. This and other architectural enhancements, especially as regards SSE implementation, improve instructions per cycle (|IPC) performance over the prior Athlon XP generation. To make this easier for consumers to understand, AMD has chosen to market the Athlon 64 using a PR (Performance Rating) system, where the numbers roughly map to Pentium 4 performance equivalents, rather than actual clock speed.

Cool'n'Quiet
Athlon 64 also features CPU speed throttling technology branded Cool'n'Quiet, a feature similar to Intel's SpeedStep that can throttle the processor's clock speed back to facilitate lower power use and heat output. When the user is running undemanding applications and the load on the processor is light, the processor clock speed and voltage are reduced. This in turn reduces its peak power use (max thermal design power (TDP) set at 89 W by AMD) to as low as 32 W (stepping level C0, clock speed reduced to 800 MHz) or 22W (stepping CG, clock speed reduced to 1 GHz). The Athlon 64 also has an integrated heat spreader (IHS) which prevents the CPU die from being damaged accidentally when mounting and unmounting heat sinks. With prior AMD CPUs, a CPU shim could be used by people worried about damaging the die.

NX bit
The No Execute bit (NX bit) supported by Windows XP Service Pack 2 and future versions of Windows, Linux 2.6.8 and higher and FreeBSD 5.3 and higher is also included, for improved protection from malicious buffer overflow security threats. Hardware-set permission levels make it much more difficult for malicious code to take control of the system. It is intended to make 64-bit computing a more secure environment.

Semiconductor Technology
The Athlon 64 CPUs have been produced with 130 and 90 nm silicon on insulator (SOI) process technologies. All of the latest chips (Winchester, Venice, and San Diego models) are on 90 nm. The Venice and San Diego models also incorporate dual stress liner technology (an amalgam of strained silicon and 'squeezed silicon', the latter of which is not actually a technology) co-developed with IBM.

Athlon 64 FX
The Athlon 64 FX is positioned as a hardware enthusiast product, marketed by AMD especially toward gamers. Unlike the standard Athlon 64, all of the Athlon 64 FX processors have their multipliers completely unlocked. Starting with the FX-60, the FX line became dual-core. The FX always has the highest clock speed of all Athlons at its release. From FX-70 onwards, the line of processors will also support dual-processor setup with NUMA, named AMD Quad FX platform.

Athlon 64 X2
The Athlon 64 X2 is the first dual-core desktop CPU manufactured by AMD. In 2007, AMD released two final Athlon 64 X2 versions: the AMD Athlon 64 X2 6400+ and 5000+ Black Editions. Both processors feature an unlocked multiplier, which allows for a large range of overclocked settings. The 6400+ is based on a 90 nm Windsor core (3.2 GHz, 2x1 MB L2, 125 W TDP) while the 5000+ is based on a 65 nm Brisbane core (2.6 GHz, 2 x 512 kB L2, 65 W TDP). These Black Edition processors are available at retail, but AMD does not include heatsinks in the retail package.

Turion 64 (formerly Mobile Athlon 64)
Formerly introduced as Mobile Athlon 64, Turion 64 is now the brand name AMD applies to its 64-bit low-power use (mobile) processors. The Turion 64 and Turion 64 X2 processors compete with Intel's mobile processors, initially the Pentium M and later the Intel Core and Intel Core 2 processors.

Earlier Turion 64 processors are compatible with AMD's Socket 754. The newer "Richmond" models are designed for AMD's Socket S1. They are equipped with 512 or 1024 kB of L2 cache, a 64-bit single channel on-die memory controller, and an 800 MHz HyperTransport bus. Battery saving features, like PowerNow!, are central to the marketing and usefulness of these CPUs.

Model naming methodology
The model naming scheme does not make it obvious how to compare one Turion with another, or even an Athlon 64. The model name is two letters, a dash, and a two digit number (for example, ML-34). The two letters together designate a processor class, while the number represents a PR rating. The first letter is M for single core processors and T for dual core Turion 64 X2 processors. The later in the alphabet that the second letter appears, the more the model has been designed for mobility (low power use). Take for example, an MT-30 and an ML-34. Since the T in the MT-30 is later in the alphabet than the L in ML-34, the MT-30 uses less power than the ML-34. But since 34 is greater than 30, the ML-34 is faster than the MT-30.

Athlon Neo
With 27 mm × 27 mm in size and 2.5 mm in thickness, the Athlon Neo processors utilize a new package called "ASB1", essentially a BGA package, for smaller footprint to allow smaller designs for notebooks and lowering the cost. The clock of the processors is significantly lower than desktop and other mobile counterparts to reach a low TDP, at 15W maximum for a single core x86-64 CPU at 1.6 GHz. The Athlon Neo processors are equipped with 512 kB of L2 cache and HyperTransport 1.0 running at 800 MHz frequency.

Sockets

 * Socket 754: The Athlon 64 value/budget line, 64-bit memory interface (Single-Channel)
 * Socket 939: Athlon 64 performance line, Athlon 64 X2s, and newer Athlon 64 FXs, Opteron, 128-bit memory interface (Dual-channel)
 * Socket 940: Opteron and old Athlon 64 FX, 128-bit memory interface - requires registered DDR memory
 * Socket AM2: Athlon 64/Athlon 64 FX/Athlon 64 X2/Sempron, 940 Pins (Not compatible with Socket 940); the first AMD socket to use DDR2 SDRAM.
 * Socket F: Opteron, 1207 Pins
 * Socket F (1207 FX): Athlon 64 FX on AMD Quad FX platform, also compatible for dual-processor Opteron 2200 series

At the introduction of Athlon 64 in September 2003, only Socket 754 and Socket 940 (Opteron) were ready and available. The onboard memory controller was not capable of running unbuffered (non-registered) memory in dual-channel mode at the time of release; as a stopgap measure, they introduced the Athlon 64 on Socket 754, and brought out a non-multiprocessor version of the Opteron called the Athlon 64 FX, as a multiplier unlocked enthusiast part for Socket 940, comparable to Intel's Pentium 4 Extreme Edition for the high end market.

In June 2004, AMD released Socket 939 as the mainstream Athlon 64 with dual-channel memory interface, leaving Socket 940 solely for the server market (Opterons), and relegating Socket 754 as a value/budget line, for Semprons and slower versions of the Athlon 64. Eventually Socket 754 replaced Socket A for Semprons.

In May 2006, AMD released Socket AM2, which provided support for the DDR2 memory interface. Also, this marked the release of AMD-V.

In August 2006, AMD released Socket F for Opteron server CPU which uses the LGA chip form factor.

In November 2006, AMD released a specialized version of Socket F, called 1207 FX, for dual-socket, dual-core Athlon FX processors on the Quad FX platform. While Socket F Opterons already allowed for four processor cores, Quad FX allowed unbuffered RAM and expanded CPU/chipset configuration in the BIOS. Consequentially, Socket F and F 1207 FX are incompatible and require different processors, chipsets, and motherboards.

Sledgehammer (130 nm SOI)

 * Stepping level: C0, CG
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 1024 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, AMD64
 * Socket 940, 800 MHz HyperTransport (HT800)
 * Registered DDR-SDRAM required
 * CPU core voltage (VCore): 1.50 or 1.55 Volts
 * Power use (TDP): 89 Watt max
 * First release: September 23, 2003
 * Clock rate: 2200 MHz (FX-51, C0), 2400 MHz (FX-53, C0 and CG)

Clawhammer (130 nm SOI)

 * Stepping level:  CG
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 1024 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, AMD64
 * Socket 939, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.50 Volts
 * Power use (TDP): 89 Watt (FX-55:104 Watt)
 * First release: June 1, 2004
 * Clock rate: 2400 MHz (FX-53), 2600 MHz (FX-55)

San Diego (90 nm SOI)

 * Stepping level: E4, E6
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 1024 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit
 * Socket 939, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.35 or 1.40 Volts
 * Power use (TDP): 104 Watt max
 * First release: April 15, 2005
 * Clock rate: 2600 MHz (FX-55), 2800 MHz (FX-57)

Toledo (90 nm SOI)
Dual-core CPU
 * Stepping level: E6
 * L1 cache: 64 + 64 kB (data + instructions), per core
 * L2 cache: 1024 kB full speed, per core
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit
 * Socket 939, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.30 or 1.35 Volts
 * Power use (TDP): 110 Watt max
 * First release: January 10, 2006
 * Clock rate: 2600 MHz (FX-60)

Windsor (90 nm SOI)
Dual-core CPU
 * Stepping level: F2, F3
 * L1 cache: 64 + 64 kB (data + instructions), per core
 * L2 cache: 512 - 1024 kB full speed, per core
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit, AMD-V
 * Socket AM2, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.30 V or 1.40 Volts
 * Power use (TDP): 125 Watt max
 * First release: May 23, 2006
 * Clock rate: 2000 - 3200 MHz (6400+), 2800 MHz (FX-62)

Windsor (90 nm SOI) - Quad FX platform
Dual-core, dual CPUs (four cores total)
 * Stepping level: F3
 * L1 cache: 64 + 64 kB (data + instructions), per core
 * L2 cache: 1024 kB full speed, per core
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit, AMD-V
 * Socket F (1207 FX), 2000 MHz HyperTransport (HT2000)
 * CPU core voltage (VCore): 1.35 or 1.40 Volts
 * Power use (TDP): 125 Watt max per CPU
 * First release: November 30, 2006
 * Clock rate: 2600 MHz (FX-70), 2800 MHz (FX-72), 3000 MHz (FX-74)

Clawhammer (130 nm SOI)

 * Stepping level: C0, CG
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 1024 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, AMD64, Cool'n'Quiet, NX bit (only CG)
 * Socket 754, 800 MHz HyperTransport (HT800)
 * Socket 939, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.50 Volts
 * Power use (TDP): 89 Watt max
 * First release: September 23, 2003
 * Clock rate: 2000–2600 MHz

Newcastle (130 nm SOI)
Also possible: ClawHammer-512 (Clawhammer with partially disabled L2 cache)
 * Stepping level: CG
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 512 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, AMD64, Cool'n'Quiet, NX bit
 * Socket 754, 800 MHz HyperTransport (HT800)
 * Socket 939, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.50 Volts
 * Power use (TDP): 89 Watt max
 * First release: 2004
 * Clock rate: 1800–2400 MHz

Winchester (90 nm SOI)

 * Stepping level: D0
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 512 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, AMD64, Cool'n'Quiet, NX bit
 * Socket 939, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.40 Volts
 * Power use (TDP): 67 Watt max
 * First release: 2004
 * Clock rate: 1800–2200 MHz

Venice (90 nm SOI)

 * Stepping level: E3, E6
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 512 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit
 * Socket 754, 800 MHz HyperTransport (HT800)
 * Socket 939, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.35 or 1.40 Volts
 * Power use (TDP): 89 Watt max
 * First release: April 4, 2005
 * Clock rate: 1800–2400 MHz

San Diego (90 nm SOI)

 * Stepping level: E4, E6
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 1024 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit
 * Socket 939, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.35 or 1.40 Volts
 * Power use (TDP): 89 Watt max
 * First release: April 15, 2005
 * Clock rate: 2200–2600 MHz

Manchester (90 nm SOI)

 * Stepping level: F1
 * L1 cache: 2 x 64 + 2 x 64 kB (data + instructions)
 * L2 cache: 2 x 512 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit
 * Socket 939, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.35 Volts
 * Power use (TDP): 89 Watt max
 * First release: April 15, 2005
 * Clock rate: 2200–2600 MHz

Orleans (90 nm SOI)

 * Stepping level: F2, F3
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 512 kB, 1 MB
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit, AMD-V
 * Socket AM2, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.25 or 1.40 Volts
 * Power use (TDP): 62 Watt max
 * First release: May 23, 2006
 * Clock rate: 1800–2600 MHz

Lima (65 nm SOI)

 * Stepping level: G1
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 512 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit, AMD-V
 * Socket AM2, 1000 MHz HyperTransport (HT1000)
 * CPU core voltage (VCore): 1.25, 1.35, or 1.40 Volts
 * Power use (TDP): 45 Watt max
 * First release: February 20, 2007
 * Clock rate: 2000–2800 MHz

Huron (65 nm SOI)

 * Stepping level: G2
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 512 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit, AMD-V
 * ASB1 package (BGA), 800 MHz HyperTransport (HT800)
 * CPU core voltage (VCore): 1.1 Volts
 * Power use (TDP): 15 Watt max
 * First release: January 8, 2009
 * Clock rate: 1600 MHz

Athlon X2 Dual Core Processor L310

 * Generation: K8
 * 65 nm SOI
 * Stepping level: G
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: 512 kB, full speed
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, NX bit, AMD-V
 * ASB1 package (BGA), 800 MHz HyperTransport (HT800)
 * Power use (TDP): 13 Watt max
 * PowerNow!: No
 * P-States: 1
 * Clock rate: 1200 MHz

Athlon X2 Dual Core Processor L335

 * Generation: K8
 * 65 nm SOI
 * Stepping level: G
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: (2*256 kB), full speed
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet(?), NX bit, AMD-V
 * ASB1 package (BGA), 800 MHz HyperTransport (HT800)
 * Power use (TDP): 18 Watt max
 * PowerNow!: Yes
 * Clock rate: 1600 MHz

Turion Neo X2 Dual Core Processor L625

 * Generation: K8
 * 65 nm SOI
 * Stepping level: G2
 * L1 cache: 64 + 64 kB (data + instructions)
 * L2 cache: (2*512 kB), full speed
 * MMX, Extended 3DNow!, SSE, SSE2, SSE3, AMD64, Cool'n'Quiet, NX bit, AMD-V
 * ASB1 package (BGA), 800 MHz HyperTransport (HT800)
 * Power use (TDP): 18 Watt max
 * PowerNow!: Yes
 * Clock rate: 1600 MHz

Successors
The Athlon 64 was succeeded by the K10 architecture in 2007, including but not limited to the Phenom and Phenom II processors. These successors feature higher core counts per CPU, and implement Hypertransport 3.0 and Socket AM2+/AM3.

As of February 2012, Athlon 64 X2 processors were still available for sale.

title: Athlon 64


text:
Disjoint union

In mathematics, the disjoint union (or discriminated union) $$A \sqcup B$$ of the sets $A$ and $B$ is the set formed from the elements of $A$ and $B$ labelled (indexed) with the name of the set from which they come. So, an element belonging to both $A$ and $B$ appears twice in the disjoint union, with two different labels.

A disjoint union of an indexed family of sets $$(A_i : i\in I)$$ is a set $$A,$$ often denoted by $\bigsqcup_{i \in I} A_i,$ with an injection of each $$A_i$$ into $$A,$$ such that the images of these injections form a partition of $$A$$ (that is, each element of $$A$$ belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union.

In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection. In this context, the notation $\coprod_{i\in I} A_i$ is often used.

The disjoint union of two sets $$A$$ and $$B$$ is written with infix notation as $$A \sqcup B$$. Some authors use the alternative notation $$A \uplus B$$ or $$A \operatorname{{\cup}\!\!\!{\cdot}\,} B$$ (along with the corresponding $\biguplus_{i\in I} A_i$ or $\operatorname{{\bigcup}\!\!\!{\cdot}\,}_{i\in I} A_i$ ).

A standard way for building the disjoint union is to define $$A$$ as the set of ordered pairs $$(x, i)$$ such that $$x \in A_i,$$ and the injection $$A_i \to A$$ as $$x \mapsto (x, i).$$

Example
Consider the sets $$A_0 = \{5, 6, 7\}$$ and $$A_1 = \{5, 6\}.$$ It is possible to index the set elements according to set origin by forming the associated sets $$\begin{align} A^*_0 & = \{(5, 0), (6, 0), (7, 0)\} \\ A^*_1 & = \{(5, 1), (6, 1)\}, \\ \end{align} $$

where the second element in each pair matches the subscript of the origin set (for example, the $$0$$ in $$(5, 0)$$ matches the subscript in $$A_0,$$ etc.). The disjoint union $$A_0 \sqcup A_1$$ can then be calculated as follows: $$A_0 \sqcup A_1 = A^*_0 \cup A^*_1 = \{(5, 0), (6, 0), (7, 0), (5, 1), (6, 1)\}.$$

Set theory definition
Formally, let $$\left(A_i : i \in I\right)$$ be an indexed family of sets indexed by $$I.$$ The disjoint union of this family is the set $$\bigsqcup_{i \in I} A_i = \bigcup_{i \in I} \left\{(x, i) : x \in A_i\right\}.$$ The elements of the disjoint union are ordered pairs $$(x, i).$$ Here $$i$$ serves as an auxiliary index that indicates which $$A_i$$ the element $$x$$ came from.

Each of the sets $$A_i$$ is canonically isomorphic to the set $$A_i^* = \left\{(x,i) : x \in A_i\right\}.$$ Through this isomorphism, one may consider that $$A_i$$ is canonically embedded in the disjoint union. For $$i \neq j,$$ the sets $$A_i^*$$ and $$A_j^*$$ are disjoint even if the sets $$A_i$$ and $$A_j$$ are not.

In the extreme case where each of the $$A_i$$ is equal to some fixed set $$A$$ for each $$i \in I,$$ the disjoint union is the Cartesian product of $$A$$ and $$I$$: $$\bigsqcup_{i \in I} A_i = A \times I.$$

Occasionally, the notation $$\sum_{i \in I} A_i$$ is used for the disjoint union of a family of sets, or the notation $$A + B$$ for the disjoint union of two sets. This notation is meant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family. Compare this to the notation for the Cartesian product of a family of sets.

In the language of category theory, the disjoint union is the coproduct in the category of sets. It therefore satisfies the associated universal property. This also means that the disjoint union is the categorical dual of the Cartesian product construction. See Coproduct for more details.

For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying abuse of notation, the indexed family can be treated simply as a collection of sets. In this case $$A_i^*$$ is referred to as a of $$A_i$$ and the notation $$\underset{A \in C}{\,\,\bigcup\nolimits^{*}\!} A$$ is sometimes used.

Category theory point of view
In category theory the disjoint union is defined as a coproduct in the category of sets.

As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct. This justifies the second definition in the lead.

This categorical aspect of the disjoint union explains why $$\coprod$$ is frequently used, instead of $$\bigsqcup,$$ to denote coproduct.

title: Disjoint union


text:
Talk:Gerhard Schröder

Biggest achievement - Nord Stream?
I don't think it makes a lot of sense to reduce Schröder's chancellorship to Nord Stream. In Germany, he is primarily known for two things: Not participating in the invasion of Iraq in 2003 and implementing the labor market reform "Agenda 2010". — Preceding unsigned comment added by 169.231.55.152 (talk) 20:51, 7 March 2022 (UTC)

Addition to Controversies
With "Hartz IV", Schröder has openly betrayed the guidelines and ideals of the SPD. He had also instigated a hunt against long-time unemployed together with Clement in the media. Like Clement, he was never prosecuted for slandering.

--2A02:560:4272:1200:8021:39B3:A088:AA56 (talk) 16:46, 28 September 2020 (UTC)

 * These reforms and the "Schwarze Null"-fiscal policy weakened the social fabric of the German society and the employees' side in pay negotiations and the public infrastructure. That enabled the rise of far-right AfD and was co-responsible for the economic crises in the Euro-zone. But the current leader of the SPD faction in German federal parliament still praises these "reforms": SPD-Fraktionschef nimmt Altbundeskanzler Schröder in Schutz. --Myosci (talk) 20:41, 21 February 2022 (UTC)

Current Role - Gazprom
I have changed the lead from

... is a German retired politician, consultant and lobbyist, who served as the chancellor of Germany from 1998 to 2005 ... TO

... is a member of the board of the Russian state owned company Gazprom, he is also a retired German politician, consultant and lobbyist, who served as the chancellor of Germany from 1998 to 2005.....

This is NOT a disruptive edit. Unless your viewpoint is from the Kremlin. Its a fact, a highly relevant fact - his current role, and the role he is most known for throughout the world. 79.155.94.28 (talk) 17:23, 27 February 2022 (UTC)

 * This is most certainly a disruptive edit because it violates the MOS:LEADSENTENCE editing guideline. Schröder is most notable as a German politician. You are welcome to insert the Gazprom info further down. I will note that this is already stated as Schröder joined the board of directors of the Nord Stream joint venture  in the Gerhard Schröder Relations with Russia section. Peaceray (talk) 17:49, 27 February 2022 (UTC)

Recent developments are in the lead summary instead of in the article where they aren't at all
No comment. — Preceding unsigned comment added by 5.173.104.89 (talk) 01:57, 19 April 2022 (UTC)

 * This is correct. The events discussed in the lead summary seem significant enough in the context of his life that that maybe a sentence summarising them is justified, with the full content placed in the article body.  Could somebody with more knowledge of German current affairs than I please review the content in the lead and see if it's suitable to be moved as is, or whether edits are required for completeness or NPOV? --23:29, 4 May 2022 (UTC) Robert Merkel (talk) 23:29, 4 May 2022 (UTC)

Kim So-Yeon
Did someone vandalise this page? No citations on the supposed marriage and a photoshopped image under personal life. 2603:8001:C00:9659:31CE:1B1B:1230:8BE0 (talk) 04:30, 19 April 2022 (UTC)

 * I would think so, yes. In the German press his wife's name is cited as "Kim Yo-Jong". 84.229.94.108 (talk) 84.229.94.108 (talk) 05:50, 23 May 2022 (UTC)

Lead
Considering that he was Chancellor for 7 years, the lead should be expanded to reflect on that. As it currently stands there's nothing in the lead on what he actually did in office. Rousillon (talk) 01:20, 2 June 2022 (UTC)

title: Talk:Gerhard Schröder