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Bretwalda

Bretwalda (also brytenwalda and bretenanwealda, sometimes capitalised) is an Old English word. The first record comes from the late 9th-century Anglo-Saxon Chronicle. It is given to some of the rulers of Anglo-Saxon kingdoms from the 5th century onwards who had achieved overlordship of some or all of the other Anglo-Saxon kingdoms. It is unclear whether the word dates back to the 5th century and was used by the kings themselves or whether it is a later, 9th-century, invention. The term bretwalda also appears in a 10th-century charter of Æthelstan. The literal meaning of the word is disputed and may translate to either 'wide-ruler' or 'Britain-ruler'.

The rulers of Mercia were generally the most powerful of the Anglo-Saxon kings from the mid 7th century to the early 9th century but are not accorded the title of bretwalda by the Chronicle, which had an anti-Mercian bias. The Annals of Wales continued to recognise the kings of Northumbria as "Kings of the Saxons" until the death of Osred I of Northumbria in 716.

Listed by the Anglo-Saxon Chronicle

 * Ælle of Sussex (488–c. 514)
 * Ceawlin of Wessex (560–592, died 593)
 * Æthelberht of Kent (590–616)
 * Rædwald of East Anglia (c. 600–around 624)
 * Edwin of Deira (616–633)
 * Oswald of Northumbria (633–642)
 * Oswiu of Northumbria (642–670)
 * Egbert of Wessex (829–839)
 * Alfred of Wessex (871–899)

Mercian rulers with similar or greater authority

 * Penda of Mercia (626/633–655)
 * Wulfhere of Mercia (658–675)
 * Æthelred of Mercia (675–704, died 716)
 * Æthelbald of Mercia (716–757)
 * Offa of Mercia (757–796)
 * Cœnwulf of Mercia (796–821)

Other claimants

 * Æthelstan of Wessex (927–939)

Etymology
The first syllable of the term bretwalda may be related to Briton or Britain. The second element is taken to mean 'ruler' or 'sovereign'. Thus, one interpretation might be 'sovereign of Britain'. Otherwise, the word may be a compound containing the Old English adjective brytten ('broad', from the verb breotan meaning 'to break' or 'to disperse'), an element also found in the terms bryten rice ('kingdom'), bryten-grund ('the wide expanse of the earth') and bryten cyning ('king whose authority was widely extended'). Though the origin is ambiguous, the draughtsman of the charter issued by Æthelstan used the term in a way that can only mean 'wide-ruler'.

The latter etymology was first suggested by John Mitchell Kemble who alluded that "of six manuscripts in which this passage occurs, one only reads Bretwalda: of the remaining five, four have Bryten-walda or -wealda, and one Breten-anweald, which is precisely synonymous with Brytenwealda"; that Æthelstan was called brytenwealda ealles ðyses ealondes, which Kemble translates as 'ruler of all these islands'; and that bryten- is a common prefix to words meaning 'wide or general dispersion' and that the similarity to the word bretwealh ('Briton') is "merely accidental".

Contemporary use
The first recorded use of the term Bretwalda comes from a West Saxon chronicle of the late 9th century that applied the term to Ecgberht, who ruled Wessex from 802 to 839. The chronicler also wrote down the names of seven kings that Bede listed in his Historia ecclesiastica gentis Anglorum in 731. All subsequent manuscripts of the Chronicle use the term Brytenwalda, which may have represented the original term or derived from a common error.

There is no evidence that the term was a title that had any practical use, with implications of formal rights, powers and office, or even that it had any existence before the 9th-century. Bede wrote in Latin and never used the term and his list of kings holding imperium should be treated with caution, not least in that he overlooks kings such as Penda of Mercia, who clearly held some kind of dominance during his reign. Similarly, in his list of bretwaldas, the West Saxon chronicler ignored such Mercian kings as Offa.

The use of the term Bretwalda was the attempt by a West Saxon chronicler to make some claim of West Saxon kings to the whole of Great Britain. The concept of the overlordship of the whole of Britain was at least recognised in the period, whatever was meant by the term. Quite possibly it was a survival of a Roman concept of "Britain": it is significant that, while the hyperbolic inscriptions on coins and titles in charters often included the title rex Britanniae, when England was unified the title used was rex Angulsaxonum, ('king of the Anglo-Saxons'.)

Modern interpretation by historians
For some time, the existence of the word bretwalda in the Anglo-Saxon Chronicle, which was based in part on the list given by Bede in his Historia Ecclesiastica, led historians to think that there was perhaps a "title" held by Anglo-Saxon overlords. This was particularly attractive as it would lay the foundations for the establishment of an English monarchy. The 20th-century historian Frank Stenton said of the Anglo-Saxon chronicler that "his inaccuracy is more than compensated by his preservation of the English title applied to these outstanding kings". He argued that the term bretwalda "falls into line with the other evidence which points to the Germanic origin of the earliest English institutions".

Over the later 20th century, this assumption was increasingly challenged. Patrick Wormald interpreted it as "less an objectively realized office than a subjectively perceived status" and emphasised the partiality of its usage in favour of Southumbrian rulers. In 1991, Steven Fanning argued that "it is unlikely that the term ever existed as a title or was in common usage in Anglo-Saxon England". The fact that Bede never mentioned a special title for the kings in his list implies that he was unaware of one. In 1995, Simon Keynes observed that "if Bede's concept of the Southumbrian overlord, and the chronicler's concept of the 'Bretwalda', are to be regarded as artificial constructs, which have no validity outside the context of the literary works in which they appear, we are released from the assumptions about political development which they seem to involve... we might ask whether kings in the eighth and ninth centuries were quite so obsessed with the establishment of a pan-Southumbrian state".

Modern interpretations view the concept of bretwalda overlordship as complex and an important indicator of how a 9th-century chronicler interpreted history and attempted to insert the increasingly powerful Saxon kings into that history.

Overlordship
A complex array of dominance and subservience existed during the Anglo-Saxon period. A king who used charters to grant land in another kingdom indicated such a relationship. If the other kingdom were fairly large, as when the Mercians dominated the East Anglians, the relationship would have been more equal than in the case of the Mercian dominance of the Hwicce, which was a comparatively small kingdom. Mercia was arguably the most powerful Anglo-Saxon kingdom for much of the late 7th though 8th centuries, though Mercian kings are missing from the two main "lists". For Bede, Mercia was a traditional enemy of his native Northumbria and he regarded powerful kings such as the pagan Penda as standing in the way of the Christian conversion of the Anglo-Saxons. Bede omits them from his list, even though it is evident that Penda held a considerable degree of power. Similarly powerful Mercia kings such as Offa are missed out of the West Saxon Anglo-Saxon Chronicle, which sought to demonstrate the legitimacy of their kings to rule over other Anglo-Saxon peoples.

title: Bretwalda


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Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function $$f$$ mapping a nonempty compact convex set to itself, there is a point $$x_0$$ such that $$f(x_0)=x_0$$. The simplest forms of Brouwer's theorem are for continuous functions $$f$$ from a closed interval $$I$$ in the real numbers to itself or from a closed disk $$D$$ to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset $$K $$ of Euclidean space to itself.

Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.

The theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The case of differentiable mappings of the $n$-dimensional closed ball was first proved in 1910 by Jacques Hadamard and the general case for continuous mappings by Brouwer in 1911.

Statement
The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows:

 * In the plane: Every continuous function from a closed disk to itself has at least one fixed point.

This can be generalized to an arbitrary finite dimension:

 * In Euclidean space:Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.

A slightly more general version is as follows:

 * Convex compact set:Every continuous function from a nonempty convex compact subset K of a Euclidean space to K itself has a fixed point.

An even more general form is better known under a different name:

 * Schauder fixed point theorem:Every continuous function from a nonempty convex compact subset K of a Banach space to K itself has a fixed point.

Importance of the pre-conditions
The theorem holds only for functions that are endomorphisms (functions that have the same set as the domain and codomain) and for nonempty sets that are compact (thus, in particular, bounded and closed) and convex (or homeomorphic to convex). The following examples show why the pre-conditions are important.

The function f as an endomorphism
Consider the function
 * $$f(x) = x+1$$

with domain [-1,1]. The range of the function is [0,2]. Thus, f is not an endomorphism.

Boundedness
Consider the function
 * $$f(x) = x+1,$$

which is a continuous function from $$\mathbb{R}$$ to itself. As it shifts every point to the right, it cannot have a fixed point. The space $$\mathbb{R}$$ is convex and closed, but not bounded.

Closedness
Consider the function
 * $$f(x) = \frac{x+1}{2},$$

which is a continuous function from the open interval (−1,1) to itself. Since x = 1 is not part of the interval, there is not a fixed point of f(x) = x. The space (−1,1) is convex and bounded, but not closed. On the other hand, the function f have a fixed point for the closed interval [−1,1], namely f(1) = 1.

Convexity
Convexity is not strictly necessary for Brouwer's fixed-point theorem. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, Brouwer's fixed-point theorem is equivalent to forms in which the domain is required to be a closed unit ball $$D^n$$. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also closed, bounded, connected, without holes, etc.).

The following example shows that Brouwer's fixed-point theorem does not work for domains with holes. Consider the function $$f(x)=-x$$, which is a continuous function from the unit circle to itself. Since -x≠x holds for any point of the unit circle, f has no fixed point. The analogous example works for the n-dimensional sphere (or any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it has a hole (and so it is not convex). The function f have a fixed point for the unit disc, since it takes the origin to itself.

A formal generalization of Brouwer's fixed-point theorem for "hole-free" domains can be derived from the Lefschetz fixed-point theorem.

Illustrations
The theorem has several "real world" illustrations. Here are some examples.

 * 1) Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at least one point of the crumpled sheet that lies directly above its corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it.
 * 2) Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country.  There will always be a "You are Here" point on the map which represents that same point in the country.
 * 3) In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a delicious cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, that the liquid after stirring is contained within the space originally taken up by it, and that the glass (and stirred surface shape) maintain a convex volume.  Ordering a cocktail shaken, not stirred defeats the convexity condition ("shaking" being defined as a dynamic series of non-convex inertial containment states in the vacant headspace under a lid).  In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state.

Explanations attributed to Brouwer
The theorem is supposed to have originated from Brouwer's observation of a cup of gourmet coffee. If one stirs to dissolve a lump of sugar, it appears there is always a point without motion. He drew the conclusion that at any moment, there is a point on the surface that is not moving. The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit. The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.

Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet." Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such as Stefan Banach's, that guarantee uniqueness.

One-dimensional case
In one dimension, the result is intuitive and easy to prove. The continuous function f is defined on a closed interval [a, b] and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval [a, b] which maps x to x (light green).

Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the function g which maps x to f(x) − x. It is ≥ 0 on a and ≤ 0 on b. By the intermediate value theorem, g has a zero in [a, b]; this zero is a fixed point.

Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."

History
The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis. The case n = 3 first was proved by Piers Bohl in 1904 (published in Journal für die reine und angewandte Mathematik). It was later proved by L. E. J. Brouwer in 1909. Jacques Hadamard proved the general case in 1910, and Brouwer found a different proof in the same year. Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's intuitionist ideals. Although the existence of a fixed point is not constructive in the sense of constructivism in mathematics, methods to approximate fixed points guaranteed by Brouwer's theorem are now known.

Before discovery
At the end of the 19th century, the old problem of the stability of the solar system returned into the focus of the mathematical community. Its solution required new methods. As noted by Henri Poincaré, who worked on the three-body problem, there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge." He also noted that the search for an approximate solution is no more efficient: "the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision".

He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant flow? Poincaré discovered that the answer can be found in what we now call the topological properties in the area containing the trajectory. If this area is compact, i.e. both closed and bounded, then the trajectory either becomes stationary, or it approaches a limit cycle. Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval t. If the area is a circular band, or if it is not closed, then this is not necessarily the case.

To understand differential equations better, a new branch of mathematics was born. Poincaré called it analysis situs. The French Encyclopædia Universalis defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing". In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem, although the connection with the subject of this article was not yet apparent. A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the fundamental group or sometimes the Poincaré group. This method can be used for a very compact proof of the theorem under discussion.

Poincaré's method was analogous to that of Émile Picard, a contemporary mathematician who generalized the Cauchy–Lipschitz theorem. Picard's approach is based on a result that would later be formalised by another fixed-point theorem, named after Banach. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a contraction.

First proofs
At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. Piers Bohl, a Latvian mathematician, applied topological methods to the study of differential equations. In 1904 he proved the three-dimensional case of our theorem, but his publication was not noticed.

It was Brouwer, finally, who gave the theorem its first patent of nobility. His goals were different from those of Poincaré. This mathematician was inspired by the foundations of mathematics, especially mathematical logic and topology. His initial interest lay in an attempt to solve Hilbert's fifth problem. In 1909, during a voyage to Paris, he met Henri Poincaré, Jacques Hadamard, and Émile Borel. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were the origin of a fruitful exchange of letters with Hadamard. For the next four years, he concentrated on the proof of certain great theorems on this question. In 1912 he proved the hairy ball theorem for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point. These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem. The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such as homotopy, the underlying concept of the Poincaré group. In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods. Hans Freudenthal comments on the respective roles as follows: "Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator."

Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension, as well as other key theorems such as the invariance of dimension. In the context of this work, Brouwer also generalized the Jordan curve theorem to arbitrary dimension and established the properties connected with the degree of a continuous mapping. This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became algebraic topology.

Reception
The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called fixed-point theory. Brouwer's theorem is probably the most important. It is also among the foundational theorems on the topology of topological manifolds and is often used to prove other important results such as the Jordan curve theorem.

Besides the fixed-point theorems for more or less contracting functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the Borsuk–Ulam theorem says that a continuous map from the n-dimensional sphere to Rn has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, the Lefschetz fixed-point theorem provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to Banach spaces. This generalization is known as Schauder's fixed-point theorem, a result generalized further by S. Kakutani to set-valued functions. One also meets the theorem and its variants outside topology. It can be used to prove the Hartman-Grobman theorem, which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the Central Limit Theorem. The theorem can also be found in existence proofs for the solutions of certain partial differential equations.

Other areas are also touched. In game theory, John Nash used the theorem to prove that in the game of Hex there is a winning strategy for white. In economics, P. Bich explains that certain generalizations of the theorem show that its use is helpful for certain classical problems in game theory and generally for equilibria (Hotelling's law), financial equilibria and incomplete markets.

Brouwer's celebrity is not exclusively due to his topological work. The proofs of his great topological theorems are not constructive, and Brouwer's dissatisfaction with this is partly what led him to articulate the idea of constructivity. He became the originator and zealous defender of a way of formalising mathematics that is known as intuitionism, which at the time made a stand against set theory. Brouwer disavowed his original proof of the fixed-point theorem.

A proof using degree
Brouwer's original 1911 proof relied on the notion of the degree of a continuous mapping, stemming from ideas in differential topology. Several modern accounts of the proof can be found in the literature, notably.

Let $$K=\overline{B(0)}$$ denote the closed unit ball in $$\mathbb R^n$$ centered at the origin. Suppose for simplicity that $$f:K\to K$$ is continuously differentiable. A regular value of $$f$$ is a point $$p\in B(0)$$ such that the Jacobian of $$f$$ is non-singular at every point of the preimage of $$p$$. In particular, by the inverse function theorem, every point of the preimage of $$f$$ lies in $$B(0)$$ (the interior of $$K$$). The degree of $$f$$ at a regular value $$p\in B(0)$$ is defined as the sum of the signs of the Jacobian determinant of $$f$$ over the preimages of $$p$$ under $$f$$:

 * $$\operatorname{deg}_p(f) = \sum_{x\in f^{-1}(p)} \operatorname{sign}\,\det (df_x).$$

The degree is, roughly speaking, the number of "sheets" of the preimage f lying over a small open set around p, with sheets counted oppositely if they are oppositely oriented. This is thus a generalization of winding number to higher dimensions.

The degree satisfies the property of homotopy invariance: let $$f$$ and $$g$$ be two continuously differentiable functions, and $$H_t(x)=tf+(1-t)g$$ for $$0\le t\le 1$$. Suppose that the point $$p$$ is a regular value of $$H_t$$ for all t. Then $$\deg_p f = \deg_p g$$.

If there is no fixed point of the boundary of $$K$$, then the function
 * $$g(x)=\frac{x-f(x)}{\sup_{y\in K}\left|y-f(y)\right|}$$

is well-defined, and

$$H(t,x) = \frac{x-tf(x)}{\sup_{y\in K}\left|y-tf(y)\right|}$$

defines a homotopy from the identity function to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so $$g$$ also has degree one at the origin. As a consequence, the preimage $$g^{-1}(0)$$ is not empty. The elements of $$g^{-1}(0)$$ are precisely the fixed points of the original function f.

This requires some work to make fully general. The definition of degree must be extended to singular values of f, and then to continuous functions. The more modern advent of homology theory simplifies the construction of the degree, and so has become a standard proof in the literature.

A proof using the hairy ball theorem
The hairy ball theorem states that on the unit sphere $1⁄2$ in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector field $S$ on $w$. (The tangency condition means that $S$ = 0 for every unit vector $w(x) ⋅ x$.) Sometimes the theorem is expressed by the statement that "there is always a place on the globe with no wind". An elementary proof of the hairy ball theorem can be found in.

In fact, suppose first that $x$ is continuously differentiable. By scaling, it can be assumed that $w$ is a continuously differentiable unit tangent vector on $w$. It can be extended radially to a small spherical shell $S$ of $A$. For $S$ sufficiently small, a routine computation shows that the mapping $t$($f_{t}$) = $x$ + $x$ is a contraction mapping on $t w(x)$ and that the volume of its image is a polynomial in $A$. On the other hand, as a contraction mapping, $t$ must restrict to a homeomorphism of $f_{t}$ onto (1 + $S$)$t^{2}$ $1⁄2$ and $S$ onto (1 + $A$)$t^{2}$ $1⁄2$. This gives a contradiction, because, if the dimension $A$ of the Euclidean space is odd, (1 + $n$)$t^{2}$/2 is not a polynomial.

If $n$ is only a continuous unit tangent vector on $w$, by the Weierstrass approximation theorem, it can be uniformly approximated by a polynomial map $S$ of $u$ into Euclidean space. The orthogonal projection on to the tangent space is given by $A$($v$) = $x$($u$) - $x$($u$) ⋅ $x$. Thus $x$ is polynomial and nowhere vanishing on $v$; by construction $A$/||$v$|| is a smooth unit tangent vector field on $v$, a contradiction.

The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that $S$ is even. If there were a fixed-point-free continuous self-mapping $n$ of the closed unit ball $f$ of the $B$-dimensional Euclidean space $n$, set

 * $${\mathbf w}({\mathbf x}) = (1 - {\mathbf x}\cdot {\mathbf f}({\mathbf x}))\, {\mathbf x} - (1 - {\mathbf x}\cdot {\mathbf x})\, {\mathbf f}({\mathbf x}).$$

Since $V$ has no fixed points, it follows that, for $f$ in the interior of $x$, the vector $B$($w$) is non-zero; and for $x$ in $x$, the scalar  product $S$ ⋅ $x$($w$) = 1 – $x$ ⋅ $x$($f$) is strictly positive. From the original $x$-dimensional space Euclidean space $n$, construct a new auxiliary ($V$)-dimensional space $n + 1$ = $W$ x R, with coordinates $V$ = ($y$, $x$). Set

 * $${\mathbf X}({\mathbf x},t)=(-t\,{\mathbf w}({\mathbf x}), {\mathbf x}\cdot {\mathbf w}({\mathbf x})).$$

By construction $t$ is a continuous vector field on the unit sphere of $X$, satisfying the tangency condition $W$ ⋅ $y$($X$) = 0. Moreover, $y$($X$) is nowhere vanishing (because, if x has norm 1, then $y$ ⋅ $x$($w$) is non-zero; while if $x$ has norm strictly less than 1, then $x$ and $t$($w$) are both non-zero). This contradiction proves the fixed point theorem when $x$ is even. For $n$ odd, one can apply the fixed point theorem to the closed unit ball $n$ in $B$ dimensions and the mapping $n + 1$($F$,$x$) = ($y$($f$),0). The advantage of this proof is that it uses only elementary techniques; more general results like the Borsuk-Ulam theorem require tools from algebraic topology.

A proof using homology or cohomology
The proof uses the observation that the boundary of the n-disk Dn is Sn−1, the (n − 1)-sphere.

Suppose, for contradiction, that a continuous function f : Dn → Dn has no fixed point. This means that, for every point x in Dn, the points x and f(x) are distinct. Because they are distinct, for every point x in Dn, we can construct a unique ray from f(x) to x and follow the ray until it intersects the boundary Sn−1 (see illustration). By calling this intersection point F(x), we define a function F : Dn → Sn−1 sending each point in the disk to its corresponding intersection point on the boundary. As a special case, whenever x itself is on the boundary, then the intersection point F(x) must be x.

Consequently, F is a special type of continuous function known as a retraction: every point of the codomain (in this case Sn−1) is a fixed point of F.

Intuitively it seems unlikely that there could be a retraction of Dn onto Sn−1, and in the case n = 1, the impossibility is more basic, because S0 (i.e., the endpoints of the closed interval D1) is not even connected. The case n = 2 is less obvious, but can be proven by using basic arguments involving the fundamental groups of the respective spaces: the retraction would induce a surjective group homomorphism from the fundamental group of D2 to that of S1, but the latter group is isomorphic to Z while the first group is trivial, so this is impossible. The case n = 2 can also be proven by contradiction based on a theorem about non-vanishing vector fields.

For n > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of homology groups: the homology Hn−1(Dn) is trivial, while Hn−1(Sn−1) is infinite cyclic. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group.

The impossibility of a retraction can also be shown using the de Rham cohomology of open subsets of Euclidean space En. For n ≥ 2, the de Rham cohomology of U = En – (0) is one-dimensional in degree 0 and n - 1, and vanishes otherwise. If a retraction existed, then U would have to be contractible and its de Rham cohomology in degree n - 1 would have to vanish, a contradiction.

A proof using Stokes' theorem
As in the proof of Brouwer's fixed-point theorem for continuous maps using homology, it is reduced to proving that there is no continuous retraction $x$ from the ball $F$ onto its boundary ∂$B$. In that case it can be assumed that $B$ is smooth, since it can be approximated using the Weierstrass approximation theorem or by convolving with non-negative smooth bump functions of sufficiently small support and integral one (i.e. mollifying). If $F$ is a volume form on the boundary then by Stokes' theorem,
 * $$0<\int_{\partial B}\omega = \int_{\partial B}F^*(\omega) = \int_BdF^*(\omega)= \int_BF^*(d\omega)=\int_BF^*(0) = 0,$$

giving a contradiction.

More generally, this shows that there is no smooth retraction from any non-empty smooth oriented compact manifold $ω$ onto its boundary. The proof using Stokes' theorem is closely related to the proof using homology, because the form $M$ generates the de Rham cohomology group $ω$(∂$H^{n-1}$) which is isomorphic to the homology group $M$(∂$H_{n-1}$) by de Rham's theorem.

A combinatorial proof
The BFPT can be proved using Sperner's lemma. We now give an outline of the proof for the special case in which f is a function from the standard n-simplex, $$\Delta^n,$$ to itself, where

 * $$\Delta^n = \left\{P\in\mathbb{R}^{n+1}\mid\sum_{i = 0}^{n}{P_i} = 1 \text{ and } P_i \ge 0 \text{ for all } i\right\}.$$

For every point $$P\in \Delta^n,$$ also $$f(P)\in \Delta^n.$$ Hence the sum of their coordinates is equal:

 * $$\sum_{i = 0}^{n}{P_i} = 1 = \sum_{i = 0}^{n}{f(P)_i}$$

Hence, by the pigeonhole principle, for every $$P\in \Delta^n,$$ there must be an index $$j \in \{0, \ldots, n\}$$ such that the $$j$$th coordinate of $$P$$ is greater than or equal to the $$j$$th coordinate of its image under f:

 * $$P_j \geq f(P)_j.$$

Moreover, if $$P$$ lies on a k-dimensional sub-face of $$\Delta^n,$$ then by the same argument, the index $$j$$ can be selected from among the k + 1 coordinates which are not zero on this sub-face.

We now use this fact to construct a Sperner coloring. For every triangulation of $$\Delta^n,$$ the color of every vertex $$P$$ is an index $$j$$ such that $$f(P)_j \leq P_j.$$

By construction, this is a Sperner coloring. Hence, by Sperner's lemma, there is an n-dimensional simplex whose vertices are colored with the entire set of n + 1 available colors.

Because f is continuous, this simplex can be made arbitrarily small by choosing an arbitrarily fine triangulation. Hence, there must be a point $$P$$ which satisfies the labeling condition in all coordinates: $$f(P)_j \leq P_j$$ for all $$j.$$

Because the sum of the coordinates of $$P$$ and $$f(P)$$ must be equal, all these inequalities must actually be equalities. But this means that:

 * $$f(P) = P.$$

That is, $$P$$ is a fixed point of $$f.$$

A proof by Hirsch
There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by noting that the map f can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the Weierstrass approximation theorem or by convolving with smooth bump functions. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by Sard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction.

R. Bruce Kellogg, Tien-Yien Li, and James A. Yorke turned Hirsch's proof into a computable proof by observing that the retract is in fact defined everywhere except at the fixed points. For almost any point, q, on the boundary, (assuming it is not a fixed point) the one manifold with boundary mentioned above does exist and the only possibility is that it leads from q to a fixed point. It is an easy numerical task to follow such a path from q to the fixed point so the method is essentially computable. gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.

A proof using oriented area
A variation of the preceding proof does not employ the Sard's theorem, and goes as follows. If $$r\colon B\to \partial B$$ is a smooth retraction, one considers the smooth deformation $$g^t(x):=t r(x)+(1-t)x,$$ and the smooth function
 * $$\varphi(t):=\int_B \det D g^t(x) \, dx.$$

Differentiating under the sign of integral it is not difficult to check that (t) = 0 for all t, so φ is a constant function, which is a contradiction because φ(0) is the n-dimensional volume of the ball, while φ(1) is zero. The geometric idea is that φ(t) is the oriented area of gt(B) (that is, the Lebesgue measure of the image of the ball via gt, taking into account multiplicity and orientation), and should remain constant (as it is very clear in the one-dimensional case). On the other hand, as the parameter t passes from 0 to 1 the map gt transforms continuously from the identity map of the ball, to the retraction r, which is a contradiction since the oriented area of the identity coincides with the volume of the ball, while the oriented area of r is necessarily 0, as its image is the boundary of the ball, a set of null measure.

A proof using the game Hex
A quite different proof given by David Gale is based on the game of Hex. The basic theorem regarding Hex, first proven by John Nash, is that no game of Hex can end in a draw; the first player always has a winning strategy (although this theorem is nonconstructive, and explicit strategies have not been fully developed for board sizes of dimensions 10 x 10 or greater). This turns out to be equivalent to the Brouwer fixed-point theorem for dimension 2. By considering n-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the determinacy theorem for Hex.

A proof using the Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem says that if a continuous map f from a finite simplicial complex B to itself has only isolated fixed points, then the number of fixed points counted with multiplicities (which may be negative) is equal to the Lefschetz number
 * $$\displaystyle \sum_n(-1)^n\operatorname{Tr}(f|H_n(B))$$

and in particular if the Lefschetz number is nonzero then f must have a fixed point. If B is a ball (or more generally is contractible) then the Lefschetz number is one because the only non-zero simplicial homology group is: $$H_0(B)$$ and f acts as the identity on this group, so f has a fixed point.

A proof in a weak logical system
In reverse mathematics, Brouwer's theorem can be proved in the system WKL0, and conversely over the base system RCA0 Brouwer's theorem for a square implies the weak Kőnig's lemma, so this gives a precise description of the strength of Brouwer's theorem.

Generalizations
The Brouwer fixed-point theorem forms the starting point of a number of more general fixed-point theorems.

The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary Hilbert space instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not compact. For example, in the Hilbert space ℓ2 of square-summable real (or complex) sequences, consider the map f : ℓ2 → ℓ2 which sends a sequence (xn) from the closed unit ball of ℓ2 to the sequence (yn) defined by
 * $$y_0 = \sqrt{1 - \|x\|_2^2}\quad\text{ and}\quad y_n = x_{n-1} \text{ for } n \geq 1.$$

It is not difficult to check that this map is continuous, has its image in the unit sphere of ℓ2, but does not have a fixed point.

The generalizations of the Brouwer fixed-point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and also often an assumption of convexity. See fixed-point theorems in infinite-dimensional spaces for a discussion of these theorems.

There is also finite-dimensional generalization to a larger class of spaces: If $$X$$ is a product of finitely many chainable continua, then every continuous function $$f:X\rightarrow X$$ has a fixed point, where a chainable continuum is a (usually but in this case not necessarily metric) compact Hausdorff space of which every open cover has a finite open refinement $$\{U_1,\ldots,U_m\}$$, such that $$U_i \cap U_j \neq \emptyset$$ if and only if $$|i-j| \leq 1$$. Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers.

The Kakutani fixed point theorem generalizes the Brouwer fixed-point theorem in a different direction: it stays in Rn, but considers upper hemi-continuous set-valued functions (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.

The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of singular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of Dn.

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

title: Brouwer fixed-point theorem


text:
Talk:Brouwer fixed-point theorem

Accessible proof
Courant and Robbins provide an accessible proof. —Preceding unsigned comment added by 198.144.199.xxx (talk • contribs) 30 August 2001

Citation style
This article mixes parenthetical referencing with footnoted references. The parenthetical ones were there first, so according to WP:CITEVAR we'd have to use that until explicit consensus. However, it would be significantly easier to turn the couple of parenthetical ones into footnotes than about 50 footnotes into parentheticals. Can we form consensus to continued using footnoted references? – Finnusertop (talk ⋅ contribs) 19:59, 24 February 2019 (UTC)

 * I'm sure that'd be okay here. –Deacon Vorbis (carbon • videos) 00:04, 25 February 2019 (UTC)
 * Great. I've turned the remaining parentheticals into footnotes. – Finnusertop (talk ⋅ contribs) 00:10, 25 February 2019 (UTC)

Function mapping in closedness section
It is stated that the function f(x) = (x+1)/2 is a continous function from the open interval (-1,1) to itself. Is it not the case that the function maps from (-1,1) to (0,1)? Salomonaber (talk) 00:13, 11 March 2020 (UTC)

 * It doesn't claim (nor is it required) that the function is surjective, so what's there is correct and appropriate. The example could have even arranged for a bijection, but I don't think it matters much either way. –Deacon Vorbis (carbon • videos) 00:32, 11 March 2020 (UTC)

Highly skeptical that the remarks "said to have [been] added" by Brouwer are actually due to him
"Brouwer is said to have added: 'I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet.'"

The citation is apparently from a French-language educational TV show (https://archive.is/20130113210953/http://archives.arte.tv/hebdo/archimed/19990921/ftext/sujet5.html). The remarks appear to be spoken by a fictional Brouwer trying to explain his result. The web page that this refers to gives no citation.

I would like to know who originally came up with the "crumpled paper theorem" explanation of the BFPT. It could have been Brouwer himself, but my guess is it was not. — Preceding unsigned comment added by Natkuhn (talk • contribs) 01:06, 8 May 2020 (UTC)

 * Oops, yeah, that's a good catch. This probably deserves some looking into. –Deacon Vorbis (carbon • videos) 01:19, 8 May 2020 (UTC)

Did Brouwer offer the first proof for continuous functions?
In the book History of Topology by James on pages 273-274: "Bohl's theorem is also equivalent to the Brouwer theorem. Bohl's theorem was published in 1904, with a proof that required that f be differentiable. Brouwer published his fixed point theorem, for continuous functions on the 3-ball, in 1909. When the first proof for the n-ball, with f differentiable, appeared in print a year later, in an appendix by J. Hadamard to a text by Tannery, the theorem was called the 'Brouwer Fixed Point Theorem', which suggests that the result was already famous by that time. It is not known in what year Brouwer made his discovery and, apparently, communicated it to other mathematicians in an informal manner. The first published proof of the general case, that is, for continuous functions on the n-ball, was by Brouwer himself in 1912."

In the book ''Brouwer Degree. The Core of Nonlinear Analysis'' by Mawhin on page 393: "In 1910, Jules Tannery published the second volume of the second edition of his book ‘Introduction à la théorie des fonctions de variables réelles’, for the time and still now a very modern presentation of analysis, introducing Weierstrass’ rigor in France. This volume two ended with a Note of Jacques Hadamard, connected in the following way to Tannery’s book material: 'The proof, following M. Ames, of Jordan’s theorem on closed curves without double point is based upon the concept of order of a point or, equivalently on the consideration of the variation of the argument. The generalization to the case where the dimension is larger than two is given by the Kronecker index. It is a now classical notion, mainly since the publication of the Traité d’Analyse of Mr. Picard (T. I, p. 123; T. II, p. 193). It has received new applications in various recent works. My aim is to present here some of them. All the following reasonings [...] only use the continuity of the considered functions.'"

The previous quote from Mawhin's book has a strange omission ([...]) that may indicate that Hadamard did not write down a proof for the continuous case. Can someone please check out if Hadamard proved the general case for differentiable functions and not for continuous functions?

 * Comments. Please sign your posts with ~ and, per WP:TPG, add comments at the end of the talk page. Please stop edit-warring to insert your own point of view. There is already a discussion of the history of the FPT in the section Brouwer fixed-point theorem. Bohl's proof applied to three dimensions. There are several historical accounts of the FPT, notably "A history of algebraic and differential topology, 1900–1960" by Jean Dieudonné. Mathsci (talk) 17:43, 21 February 2022 (UTC)

title: Talk:Brouwer fixed-point theorem


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Talk:Biochemistry

Reorganising the article
In my opinion the article in its present form does not possess sufficient structure, covering biomolecules in too much detail and in an illogical way - e.g. 2 separate sections on proteins, while only treating metabolism in a small section under carbohydrates, and with very little on gene expression/regulation etc.. In addition it largely lacks any in-line referencing.

I've copied the article to a user page and am currently experimenting with a fairly substantial reorganisation, please feel free to check it out and discuss your thoughts on it. PiFanatic (talk) 19:05, 16 January 2011 (UTC)

I think you are absolutely right. The big concern with the article in general is for me that there is actually no clear definition of biochemistry itself. As a scientific field it mostly comprises the study of all biological systems on a molecular level. This means that almost any life-science discipline has something to do with biochemistry, because every biological process has a biochemical cause. Therefore I would suggest that this article should differentiate between the "textbook" knowledge (e.g. Stryer) and the real science. Maybe this could be achieved by introducing the basic concepts of biochemistry and then give some further link to the different sub-disciplines (e.g structural biology, metabolomics, chemical biology) and/or related disciplines (molecular biology, genetics, physiology, etc.) --InvertedRepeat (talk) 22:17, 11 February 2017 (UTC)

The overview has some grammatical issues and a lot of redundant information. It is also not very clear or precise in language in the explanation making it confusing for the readers.The detailed description in the overview can be written more effectively .Faizanwar08 (talk) 16:08, 9 October 2018 (UTC)

 * I agree, some of the wording is redundant. For example, when mentioning that biochemistry involves the study of the structure, function, and interaction of biomolecules, I don't think it is necessary to add "bonding" because it implies interaction between molecules. Olgaliobimova (talk) 02:10, 8 December 2022 (UTC)

History Refference
alchemy —Preceding unsigned comment added by Drrake (talk • contribs) 09:22, 4 August 2010 (UTC)

MCOTF:Suggestions here
Looking over the article, here are some of my suggestions:
 * the first line- biochemistry is the chemistry of life - a bit vague, isnt it? How about Biochemistry is the name given to a hybrid branch of chemistry which specialises in the chemical processes in living organisms....
 * subdisciplines section- I dont think it is just these topics that are covered - biochemistry is divided either into many topics such as carbs, proteins etc ~(+MANY more such as hormonal), or into cellular and molecular biochemistry. There is significant overlap with physiological disciplines.
 * More relevance to medical and zoological topics - introduction of topics like "pharmacology" or "toxicology".

What do you guys think?

PhatRita 15:18, 31 August 2005 (UTC)
 * Merriam-Webster defines it as "chemistry that deals with the chemical compounds and processes occurring in organisms". How about something like "Biochemistry is the chemistry of life, a bridge between biology and chemistry that studies how complex chemical reactions give rise to life. It is a hybrid branch of chemistry which specialises in the chemical processes in living organisms." Just writing that off the top of my head in the middle of the night. — Knowledge Seeker দ 05:26, September 1, 2005 (UTC)
 * Yes! That is a much more captivating introduction. Cybergoth 04:16, 2 September 2005 (UTC)

Regarding the Relationship to other "molecular-scale" biological sciences section:
 * I would suggust using the term molecular genetics rather than "genetics". It is more specific to the "molecular scale" of biochemisty. Cybergoth 04:21, 2 September 2005 (UTC)
 * I wasn't sure where you wanted this change, Cybergoth; why don't you go ahead and change it yourself—I agree with your suggestion. — Knowledge Seeker দ 04:17, September 5, 2005 (UTC)

I've started to expand this article, but there are a couple areas I'll need help on. I'll try to do some reading, but my memory of non-human biochemistry is pretty shaky—I barely remember the Calvin cycle and light and dark reactions and such; my microbial biochemistry is very weak too. Also, I may need some help with lipids—we'll see. Feel free to rewrite the sections I'm doing. Am I putting too much/too little in? — Knowledge Seeker দ 06:25, September 5, 2005 (UTC)

Micheal E. Sugar ?
is this a real person or just vandalism? Today, there are three main types of biochemistry as established by Michael E. Sugar. mathew (talk) 01:03, 3 May 2009 (UTC)

I concur. I'm removing the reference, unless anyone has any serious objections. Jhay116 (talk) 20:13, 15 June 2009 (UTC)

Training
It would be interesting to have a section on training, similar to what's been added to articles about medical subspecialties. Jack Daw 18:37, 13 March 2007 (UTC)

Metabolism
The article metabolism is to a large extent an introduction to biochemistry and contains much more info on biochemistry than the present article does. therefore I think it should be more clearly highlighted here. --Etxrge (talk) 16:45, 5 December 2007 (UTC)

Animal Biochemistry
The article Animal Biochemistry could become a redirect or even deleted or what ever, but it is giving noting new! Somebody should get ride of it!--Stone (talk) 21:26, 8 January 2008 (UTC)

Neurobiochemistry
There should be some reference to the subfields of Biochemistry e.g. neurobiochemistry, clinical biochemistry and others. Those with knowledge in these fields could contribute. Also since the biochemistry page must be considered the "main page" more links to biochemistry pages would be nice. Magnusig (talk) 11:09, 11 May 2008 (UTC)

Expansion on Lipids Functions in Article
The article mentions a fair amount of detail involving the roles of carbohydrates and proteins in biochemistry. Although lipids are defined, the article has no mention of their importance in biological systems. Extracting some of the information from the “Lipids” main-page would give a more complete overview as far as the macromolecules relevant to biochemistry. Including the below information would achieve this objective: The main biological functions of lipids include energy storage, acting as structural components of cell membranes, and participating as important signaling molecules. A biological membrane is a form of lipid bilayer, as is a liposome. The formation of lipid bilayers is an energetically-preferred process when glycerophospholipids are in an aqueous environment. In an aqueous system, the polar heads of lipids orientate towards the polar, aqueous environment, while the hydrophobic tails minimize their contact with water. Triacylglycerols, stored in adipose tissue, are a major form of energy storage in animals. Animals use triglycerides for energy storage because of its high caloric content (9 KCal/g), whereas plants, which do not require energy for movement, can afford to store food for energy in a less compact but more easily accessible form, such as starch (carbohydrate). Recently, evidence has arisen supporting the idea that lipid signaling is essential to cell signaling. Lipid signaling may occur via activation of GPCR's or nuclear receptors, and members of several different lipid categories have been identified as signaling molecules and cellular messengers. —Preceding unsigned comment added by Kjmusic13 (talk • contribs) 16:23, 24 May 2008 (UTC)

Current Topics in Biochemistry
The article doesn’t address any current issues dealt with in the field of biochemistry. I feel as though it would be greatly enhanced if some of the big issues could be added to it in order to give an understanding of the impact of biochemistry on the world. These wouldn’t necessarily need to be very in depth, so long as a link was provided to another Wikipedia page discussing the topic. Taking information directly from the Wikipedia main pages and having them listed here would help alert the reader to the topics. Some important ones may include:

Stem Cells (Main page: Stem Cells)

Stem cells are cells found in most, if not all, multi-cellular organisms. They are characterized by the ability to renew themselves through mitotic cell division and differentiating into a diverse range of specialized cell types. Stem cell therapy is thought to be a way to help repair damaged tissues. However, there exists a widespread controversy over human embryonic stem cell research that emanates from the techniques used in the creation and usage of stem cells. Opponents of the research argue that embryonic stem cell technologies are a slippery slope to reproductive cloning and can fundamentally devalue human life. Those in the pro-life movement argue that a human embryo is a human life and is therefore entitled to protection. Contrarily, supporters of embryonic stem cell research argue that such research should be pursued because the resultant treatments could have significant medical potential. It is also noted that excess embryos created for in vitro fertilization could be donated with consent and used for the research.

Anabolic Steroid Use (Main page: Anabolic Steroid)

Performance enhancing substances have been used for thousands of years in traditional medicine by societies around the world, with the aim of promoting vitality and strength. Anabolic steroids have been used by men and women in many different kinds of professional sports to attain a competitive edge or to assist in recovery from injury. Such use is prohibited by the rules of the governing bodies of many sports. Anabolic steroid use occurs among adolescents, especially by those participating in competitive sports. It has been suggested that the prevalence of use among high-school students in the U.S. may be as high as 2.7%. Male students used anabolic steroids more frequently than female students and, on average, those who participated in sports used steroids more often than those who did not.

Sammich117 (talk) 05:59, 3 June 2008 (UTC)

Spam link
The external link reference to the Protein Zone seems to be a site selling books on proteins.65.4.1.40 (talk) 00:04, 26 January 2009 (UTC)

 * Removed. Tim Vickers (talk) 00:32, 26 January 2009 (UTC)

Suggestions
I didn't want to unceremoniously trash Suzanne's text for this article (taken from BioChemistry), so I'm putting it here so that she or others may incorporate it in this page as appropriate --dja

Biochemistry considers the chemical properties of biological systems. Formally, biochemistry includes a description of the macromolecules, including proteins, lipids, polysaccharides, and nucleic acids, and small molecules present in cells. The bulk of biochemical investigation focuses on the properties of proteins, many of which are enzymes. For historical reasons, the biochemistry of metabolism has been one of the most extensively described aspect of the cell.

have hopefully incorporated above material appropriatelly -- sodium

Wilhelm Schuessler
''There seems to be no reference whatsoever to Dr. Wilhelm Schuessler, who discovered the 12 essential mineral elements of which the body is both composed and requires for life. Schuessler was a German Scientist, homoeopath and Medical Doctor who lived in Oldenburg Germany ( 1821 - 1898 )and considered the father of Biochemistry having worked with many such as Rudolf Virchow (cellular pathology) The first biochemic association was founded in Oldenburg on July 17th 1885.The Institute of Biochemistry (Schuessler) Biochemischer Bund Deutschland (Ev) is still "alive and well" with web page at www.biochemie-online.de Further information may be obtained in the English language by contacting The Institute of Biochemic medicine (Asia Pacific) A branch of the Biochemischer Bund Deutschland - which is located in Australia biochemics@bigpond.com.au''
 * (removed from article --jag123 10:42, 28 Apr 2005 (UTC))

This article lacks direction
I think this article would give non-biochemists a slightly misleading notion of what the subject is about. The article jumps straight into detail about lipids and carbohydrates and fails to give a clear outline of what modern biochemistry as opposed to old school biochemistry is really about. Lets not forget that the focus of the subject has changed markedly over its history.

I think it would be really important to point out that biochemistry now underpins all life sciences from plant science to medicine and therefore all life science subjects are now effectively branches of biochemistry. Thus biochemistry has become a victim of its own success with all life science subjects involved in their own biochemical research. Biochemistry's own area of investigation now mainly lies with the processes that go on within living cells and is essentially the same as molecular cell biology.Tom the biochemist (talk) 18:31, 3 June 2010 (UTC)

I think the article definitely needs some work as regards structure and content. I don't think 'monomers and polymers' really does justice to the depth of the subject. A fairly substantial rewrite might be necessary for some parts. The article should ideally give an overview of what is covered by the subject and how it links in to other disciplines, while leaving the fine detail to more specific pages. PiFanatic (talk) 18:24, 16 January 2011 (UTC)

Seemingly Magical?
I think the 'seemingly magical phenomenon of life' in the first paragraph sounds a bit woolly, perhaps something like 'incredible complexity of life' or similar would be better?

PiFanatic (talk) 18:44, 18 November 2010 (UTC)

this is more like a C-class article...no discussion of experiments and methods
These sections are relevant to the topic of biochemistry, but biochemistry is also a discipline, i.e. a profession. How the Krebs cycle was elucidated or how the Hershey-Chase experiment was carried out, and basically the tools available to a biochemist should take up half of this article. 216.197.66.61 (talk) 03:04, 22 March 2012 (UTC)

Needs rewriting
As others noted, this article is poor. Biochemistry is not just describing carbohydrates or proteins, most of what's written here can go into their own individual pages (and why would there be two separate sections each for protein, carbohydrate, lipids and nucleic acids anyway?) What's needed here is a broad view of biochemistry as a discipline, its concepts, practices and applications. It would need extensive rewrite, I'm not sure if I can do it, but I'll probably give it a go some time, and work through it slowly. A lot of what's currently here will be deleted as they can be linked to their own pages. Hzh (talk) 20:17, 28 July 2015 (UTC)
 * I've started by merging duplicate sections, the rest of the rewrite will come later. Those who want to contribute are welcome to do so.Hzh (talk) 13:12, 29 July 2015 (UTC)

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Citations needed
The section - Starting Materials: the chemical elements of life - and onward need citations. "In agriculture, biochemists investigate soil and fertilizers, and try to discover ways to improve crop cultivation, crop storage and pest control." in the first section needs a citation also The history section on the beginning of biochemistry is slightly out of order - needs to be reorganized, mostly the last part about Lavoisier.

Ashbyaa (talk) 17:31, 18 March 2017 (UTC)

Broader Information
This article generally could benefit from adding more subsections, such as on the relationship of biochemistry to other fields, or common biochemical lab techniques. Some of the sections, such as "Metabolism" needs an introductory sentence or two, additional subheadings on metabolic processes, and perhaps a subsection on regulation. Other sections, such as "biomolecules" have way too much detail, and could stand to be substantially condensed.

392MetabolicEditor (talk) 06:16, 20 March 2017 (UTC)

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Article improvement assignment
I will be working on improving this article with two others for an assignment for my English class, one of my group members usernames is yaxyu143. Currently we will be monitoring for grammatical errors as best we can. In addition we are working on sources to include and cite in this article. Any other suggestions or feedback would be helpful! (my username is samhendrickson135) — Preceding unsigned comment added by Samhendrickson135 (talk • contribs) 05:03, 16 November 2018 (UTC)

Extremophiles?
So, extremophiles are really cool organisms and all, and obviously are important in the life sciences, but do they merit a subsection in this specific article? The text of the section doesn't make much of an effort to connect its content to the broader concept of biochemistry. Would appreciate the input of more experienced editors - I don't really know what I'm doing!

Yogurt Tub (talk) 05:14, 30 March 2021 (UTC)Yogurt_Tub

Relationship to other "molecular-scale" biological sciences section
I put a multiple-issues template in the final section of the article. It doesn't cite any sources and appears to be WP:OR, as it draws fine lines between subdisciplines that are not all that distinct. If we are going to precisely define the difference between biochemistry and molecular biology, then we need a good, authoritative source to do that. The graphic is helpful but, again, without a source it is just someone's personal mental model, i.e. OR. Do any of the major biochemistry textbooks have a description of the relationships between these disciplines? I'll get on it. WeirdNAnnoyed (talk) 02:01, 1 August 2023 (UTC)

title: Talk:Biochemistry


text:
Benzoic acid

Benzoic acid is a white (or colorless) solid organic compound with the formula C6H5COOH, whose structure consists of a benzene ring (C6H6) with a carboxyl (\sC(\dO)OH) substituent. The benzoyl group is often abbreviated "Bz" (not to be confused with "Bn" which is used for benzyl), thus benzoic acid is also denoted as BzOH, since the benzoyl group has the formula –C6H5CO. It is the simplest aromatic carboxylic acid. The name is derived from gum benzoin, which was for a long time its only source.

Benzoic acid occurs naturally in many plants and serves as an intermediate in the biosynthesis of many secondary metabolites. Salts of benzoic acid are used as food preservatives. Benzoic acid is an important precursor for the industrial synthesis of many other organic substances. The salts and esters of benzoic acid are known as benzoates.

History
Benzoic acid was discovered in the sixteenth century. The dry distillation of gum benzoin was first described by Nostradamus (1556), and then by Alexius Pedemontanus (1560) and Blaise de Vigenère (1596).

Justus von Liebig and Friedrich Wöhler determined the composition of benzoic acid. These latter also investigated how hippuric acid is related to benzoic acid.

In 1875 Salkowski discovered the antifungal properties of benzoic acid, which was used for a long time in the preservation of benzoate-containing cloudberry fruits.

Industrial preparations
Benzoic acid is produced commercially by partial oxidation of toluene with oxygen. The process is catalyzed by cobalt or manganese naphthenates. The process uses abundant materials, and proceeds in high yield.
 * [[Image:Benzoic acid-chemical-synthesis-1.svg|center|toluene oxidation]]

The first industrial process involved the reaction of benzotrichloride (trichloromethyl benzene) with calcium hydroxide in water, using iron or iron salts as catalyst. The resulting calcium benzoate is converted to benzoic acid with hydrochloric acid. The product contains significant amounts of chlorinated benzoic acid derivatives. For this reason, benzoic acid for human consumption was obtained by dry distillation of gum benzoin. Food-grade benzoic acid is now produced synthetically.

Laboratory synthesis
Benzoic acid is cheap and readily available, so the laboratory synthesis of benzoic acid is mainly practiced for its pedagogical value. It is a common undergraduate preparation.

Benzoic acid can be purified by recrystallization from water because of its high solubility in hot water and poor solubility in cold water. The avoidance of organic solvents for the recrystallization makes this experiment particularly safe. This process usually gives a yield of around 65%.

By hydrolysis
Like other nitriles and amides, benzonitrile and benzamide can be hydrolyzed to benzoic acid or its conjugate base in acid or basic conditions.

From Grignard reagent
Bromobenzene can be converted to benzoic acid by "carboxylation" of the intermediate phenylmagnesium bromide. This synthesis offers a convenient exercise for students to carry out a Grignard reaction, an important class of carbon–carbon bond forming reaction in organic chemistry.

Oxidation of benzyl compounds
Benzyl alcohol and benzyl chloride and virtually all benzyl derivatives are readily oxidized to benzoic acid.

Uses
Benzoic acid is mainly consumed in the production of phenol by oxidative decarboxylation at 300−400 °C:

The temperature required can be lowered to 200 °C by the addition of catalytic amounts of copper(II) salts. The phenol can be converted to cyclohexanol, which is a starting material for nylon synthesis.

Precursor to plasticizers
Benzoate plasticizers, such as the glycol-, diethyleneglycol-, and triethyleneglycol esters, are obtained by transesterification of methyl benzoate with the corresponding diol. These plasticizers, which are used similarly to those derived from terephthalic acid ester, represent alternatives to phthalates.

Precursor to sodium benzoate and related preservatives
Benzoic acid and its salts are used as food preservatives, represented by the E numbers E210, E211, E212, and E213. Benzoic acid inhibits the growth of mold, yeast and some bacteria. It is either added directly or created from reactions with its sodium, potassium, or calcium salt. The mechanism starts with the absorption of benzoic acid into the cell. If the intracellular pH changes to 5 or lower, the anaerobic fermentation of glucose through phosphofructokinase is decreased by 95%. The efficacy of benzoic acid and benzoate is thus dependent on the pH of the food. Benzoic acid, benzoates and their derivatives are used as preservatives for acidic foods and beverages such as citrus fruit juices (citric acid), sparkling drinks (carbon dioxide), soft drinks (phosphoric acid), pickles (vinegar) and other acidified foods.

Typical concentrations of benzoic acid as a preservative in food are between 0.05 and 0.1%. Foods in which benzoic acid may be used and maximum levels for its application are controlled by local food laws.

Concern has been expressed that benzoic acid and its salts may react with ascorbic acid (vitamin C) in some soft drinks, forming small quantities of carcinogenic benzene.

Medicinal
Benzoic acid is a constituent of Whitfield's ointment which is used for the treatment of fungal skin diseases such as ringworm and athlete's foot. As the principal component of gum benzoin, benzoic acid is also a major ingredient in both tincture of benzoin and Friar's balsam. Such products have a long history of use as topical antiseptics and inhalant decongestants.

Benzoic acid was used as an expectorant, analgesic, and antiseptic in the early 20th century.

Niche and laboratory uses
In teaching laboratories, benzoic acid is a common standard for calibrating a bomb calorimeter.

Biology and health effects
Benzoic acid occurs naturally as do its esters in many plant and animal species. Appreciable amounts are found in most berries (around 0.05%). Ripe fruits of several Vaccinium species (e.g., cranberry, V. vitis macrocarpon; bilberry, V. myrtillus) contain as much as 0.03–0.13% free benzoic acid. Benzoic acid is also formed in apples after infection with the fungus Nectria galligena. Among animals, benzoic acid has been identified primarily in omnivorous or phytophageous species, e.g., in viscera and muscles of the rock ptarmigan (Lagopus muta) as well as in gland secretions of male muskoxen (Ovibos moschatus) or Asian bull elephants (Elephas maximus). Gum benzoin contains up to 20% of benzoic acid and 40% benzoic acid esters.

In terms of its biosynthesis, benzoate is produced in plants from cinnamic acid. A pathway has been identified from phenol via 4-hydroxybenzoate.

Reactions
Reactions of benzoic acid can occur at either the aromatic ring or at the carboxyl group.

Aromatic ring

 * [[Image:Benzoic acid-chemical-reaction-1.svg|center|benzoic acid aromatic ring reactions]]

Electrophilic aromatic substitution reaction will take place mainly in 3-position due to the electron-withdrawing carboxylic group; i.e. benzoic acid is meta directing.

Carboxyl group
Reactions typical for carboxylic acids apply also to benzoic acid.

 * Benzoate esters are the product of the acid catalysed reaction with alcohols.
 * Benzoic acid amides are usually prepared from benzoyl chloride.
 * Dehydration to benzoic anhydride is induced with acetic anhydride or phosphorus pentoxide.
 * Highly reactive acid derivatives such as acid halides are easily obtained by mixing with halogenation agents like phosphorus chlorides or thionyl chloride.
 * Orthoesters can be obtained by the reaction of alcohols under acidic water free conditions with benzonitrile.
 * Reduction to benzaldehyde and benzyl alcohol is possible using DIBAL-H, LiAlH4 or sodium borohydride.
 * Decarboxylation to benzene may be effected by heating in quinoline in the presence of copper salts. Hunsdiecker decarboxylation can be achieved by heating the silver salt.
 * [[Image:Benzoic acid-chemical-reaction-2.svg|center|400px|benzoic acid group reactions]]

Safety and mammalian metabolism
It is excreted as hippuric acid. Benzoic acid is metabolized by butyrate-CoA ligase into an intermediate product, benzoyl-CoA, which is then metabolized by glycine N-acyltransferase into hippuric acid. Humans metabolize toluene which is also excreted as hippuric acid.

For humans, the World Health Organization's International Programme on Chemical Safety (IPCS) suggests a provisional tolerable intake would be 5 mg/kg body weight per day. Cats have a significantly lower tolerance against benzoic acid and its salts than rats and mice. Lethal dose for cats can be as low as 300 mg/kg body weight. The oral for rats is 3040 mg/kg, for mice it is 1940–2263 mg/kg.

In Taipei, Taiwan, a city health survey in 2010 found that 30% of dried and pickled food products had benzoic acid.

title: Benzoic acid


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Boltzmann distribution

In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution ) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form:

 * $$p_i \propto \exp\left(- \frac{\varepsilon_i}{kT} \right)$$

where $\tfrac{p_i}{p_j}$ is the probability of the system being in state $T$, $ε_{i} − ε_{j}$ is the exponential function, $p_{i}$ is the energy of that state, and a constant $i$ of the distribution is the product of the Boltzmann constant $ε_{i}$ and thermodynamic temperature $kT$. The symbol $\propto$ denotes proportionality (see  for the proportionality constant).

The term system here has a wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore the Boltzmann distribution can be used to solve a wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied.

The ratio of probabilities of two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference:

 * $$\frac{p_i}{p_j} = \exp\left( \frac{\varepsilon_j - \varepsilon_i}{kT} \right)$$

The Boltzmann distribution is named after Ludwig Boltzmann who first formulated it in 1868 during his studies of the statistical mechanics of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium" The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.

The Boltzmann distribution should not be confused with the Maxwell–Boltzmann distribution or Maxwell-Boltzmann statistics. The Boltzmann distribution gives the probability that a system will be in a certain state as a function of that state's energy, while the Maxwell-Boltzmann distributions give the probabilities of particle speeds or energies in ideal gases. The distribution of energies in a one-dimensional gas however, does follow the Boltzmann distribution.

The distribution
The Boltzmann distribution is a probability distribution that gives the probability of a certain state as a function of that state's energy and temperature of the system to which the distribution is applied. It is given as $$ p_i=\frac{1}{Q} \exp\left(- \frac{\varepsilon_i}{kT} \right) = \frac{ \exp\left(- \tfrac{\varepsilon_i}{kT} \right) }{ \displaystyle \sum_{j=1}^{M} \exp\left(- \tfrac{\varepsilon_j}{kT} \right) } $$

where: Q = \sum_{j=1}^{M} \exp\left(- \tfrac{\varepsilon_j}{kT} \right) $$ It results from the constraint that the probabilities of all accessible states must add up to 1.
 * $exp$ is the exponential function,
 * $k$ is the probability of state $T$,
 * $p_{i}$ is the energy of state $i$,
 * $ε_{i}$ is the Boltzmann constant,
 * $i$ is the absolute temperature of the system,
 * $k$ is the number of all states accessible to the system of interest,
 * $T$ (denoted by some authors by $M$) is the normalization denominator, which is the canonical partition function$$

Using Lagrange multipliers, one can prove that the Boltzmann distribution is the distribution that maximizes the entropy $$S(p_1,p_2,\cdots,p_M) = -\sum_{i=1}^{M} p_i\log_2 p_i$$

subject to the normalization constraint that $\sum p_i=1$ and the constraint that $\sum {p_i {\varepsilon}_i}$  equals a particular mean energy value, except for two special cases. (These special cases occur when the mean value is either the minimum or maximum of the energies $Q$. In these cases, the entropy maximizing distribution is a limit of Boltzmann distributions where $Z$ approaches zero from above or below, respectively.)

The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the NIST Atomic Spectra Database.

The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states $ε_{i}$ and $T$ is given as $$\frac{p_i}{p_j} = \exp\left( \frac{\varepsilon_j - \varepsilon_i}{kT} \right)$$

where:
 * $i$ is the probability of state $j$,
 * $p_{i}$ the probability of state $i$,
 * $p_{j}$ is the energy of state $j$,
 * $ε_{i}$ is the energy of state $i$.

The corresponding ratio of populations of energy levels must also take their degeneracies into account.

The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over bound states accessible to them. If we have a system consisting of many particles, the probability of a particle being in state $ε_{j}$ is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state $j$. This probability is equal to the number of particles in state $i$ divided by the total number of particles in the system, that is the fraction of particles that occupy state $i$.

 * $$p_i = \frac{N_i}{N}$$

where $i$ is the number of particles in state $i$ and $N_{i}$ is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state $i$ as a function of the energy of that state is $$ \frac{N_i}{N} = \frac{ \exp\left(- \frac{\varepsilon_i}{kT} \right) }{ \displaystyle \sum_{j=1}^{M} \exp\left(- \tfrac{\varepsilon_j}{kT} \right) } $$

This equation is of great importance to spectroscopy. In spectroscopy we observe a spectral line of atoms or molecules undergoing transitions from one state to another. In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state. This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a forbidden transition.

The softmax function commonly used in machine learning is related to the Boltzmann distribution:

(p_1, \ldots, p_M) = \operatorname{softmax} \left[- \frac{\varepsilon_1}{kT}, \ldots, - \frac{\varepsilon_M}{kT} \right] $$

Generalized Boltzmann distribution
Distribution of the form
 * $$\Pr\left(\omega\right)\propto\exp\left[\sum_{\eta=1}^{n}\frac{X_{\eta}x_{\eta}^{\left(\omega\right)}}{k_{B}T}-\frac{E^{\left(\omega\right)}}{k_{B}T}\right]$$

is called generalized Boltzmann distribution by some authors.

The Boltzmann distribution is a special case of the generalized Boltzmann distribution. The generalized Boltzmann distribution is used in statistical mechanics to describe canonical ensemble, grand canonical ensemble and isothermal–isobaric ensemble. The generalized Boltzmann distribution is usually derived from the principle of maximum entropy, but there are other derivations.

The generalized Boltzmann distribution has the following properties:
 * It is the only distribution for which the entropy as defined by Gibbs entropy formula matches with the entropy as defined in classical thermodynamics.
 * It is the only distribution that is mathematically consistent with the fundamental thermodynamic relation where state functions are described by ensemble average.

In statistical mechanics
The Boltzmann distribution appears in statistical mechanics when considering closed systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble. Some special cases (derivable from the canonical ensemble) show the Boltzmann distribution in different aspects:

 * Canonical ensemble (general case)
 * The canonical ensemble gives the probabilities of the various possible states of a closed system of fixed volume, in thermal equilibrium with a heat bath. The canonical ensemble has a state probability distribution with the Boltzmann form.

 * Statistical frequencies of subsystems' states (in a non-interacting collection)
 * When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the statistical frequency of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the expected statistical frequency distribution of subsystem states has the Boltzmann form.

 * Maxwell–Boltzmann statistics of classical gases (systems of non-interacting particles)
 * In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. Maxwell–Boltzmann statistics give the expected number of particles found in a given single-particle state, in a classical gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.

Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:
 * When a system is in thermodynamic equilibrium with respect to both energy exchange and particle exchange, the requirement of fixed composition is relaxed and a grand canonical ensemble is obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a microcanonical ensemble applies instead.
 * If the subsystems within a collection do interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an analytical solution. The canonical ensemble can however still be applied to the collective states of the entire system considered as a whole, provided the entire system is in thermal equilibrium.
 * With quantum gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by Fermi–Dirac statistics or Bose–Einstein statistics, depending on whether the particles are fermions or bosons, respectively.

In mathematics

 * In more general mathematical settings, the Boltzmann distribution is also known as the Gibbs measure.
 * In statistics and machine learning, it is called a log-linear model.
 * In deep learning, the Boltzmann distribution is used in the sampling distribution of stochastic neural networks such as the Boltzmann machine, restricted Boltzmann machine, energy-based models and deep Boltzmann machine. In deep learning, the Boltzmann machine is considered to be one of the unsupervised learning models. In the design of Boltzmann machine in deep learning, as the number of nodes are increased the difficulty of implementing in real time applications becomes critical, so a different type of architecture named Restricted Boltzmann machine is introduced.

In economics
The Boltzmann distribution can be introduced to allocate permits in emissions trading. The new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries.

The Boltzmann distribution has the same form as the multinomial logit model. As a discrete choice model, this is very well known in economics since Daniel McFadden made the connection to random utility maximization.

title: Boltzmann distribution


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Leg theory

Leg theory is a bowling tactic in the sport of cricket. The term leg theory is somewhat archaic, but the basic tactic remains a play in modern cricket.

Simply put, leg theory involves concentrating the bowling attack at or near the line of leg stump. This may or may not be accompanied by a concentration of fielders on the leg side. The line of attack aims to cramp the batsman, making him play the ball with the bat close to the body. This makes it difficult to hit the ball freely and score runs, especially on the off side. Since a leg theory attack means the batsman is more likely to hit the ball on the leg side, additional fielders on that side of the field can be effective in preventing runs and taking catches.

Stifling the batsman in this manner can lead to impatience and frustration, resulting in rash play by the batsman which in turn can lead to a quick dismissal. Concentrating attack on the leg stump is considered by many cricket fans and commentators to lead to boring play, as it stifles run scoring and encourages batsmen to play conservatively.

Leg theory can be a moderately successful tactic when used with both fast bowling and spin bowling, particularly leg spin to right-handed batsmen or off spin to left-handed batsmen. However, because it relies on lack of concentration or discipline by the batsman, it can be risky against patient and skilled players, especially batsmen who are strong on the leg side. The English opening bowlers Sydney Barnes and Frank Foster used leg theory with some success in Australia in 1911–12. In England, at around the same time, Fred Root was one of the main proponents of the same tactic.

Fast leg theory
In 1930, England captain Douglas Jardine, together with Nottinghamshire's captain Arthur Carr and his bowlers Harold Larwood and Bill Voce, developed a variant of leg theory in which the bowlers bowled fast, short-pitched balls that would rise into the batsman's body, together with a heavily stacked ring of close fielders on the leg side. The idea was that when the batsman defended against the ball, he would be likely to deflect the ball into the air for a catch.

Jardine called this modified form of the tactic fast leg theory. On the 1932–33 English tour of Australia, Larwood and Voce bowled fast leg theory at the Australian batsmen. It turned out to be extremely dangerous, and most Australian players sustained injuries from being hit by the ball. Wicket-keeper Bert Oldfield's skull was fractured by a ball hitting his head (although the ball had first glanced off the bat and Larwood had an orthodox field), almost precipitating a riot by the Australian crowd.

The Australian press dubbed the tactic Bodyline, and claimed it was a deliberate attempt by the English team to intimidate and injure the Australian players. Reports of the controversy reaching England at the time described the bowling as fast leg theory, which sounded to many people to be a harmless and well-established tactic. This led to a serious misunderstanding amongst the English public and the Marylebone Cricket Club – the administrators of English cricket – of the dangers posed by Bodyline. The English press and cricket authorities declared the Australian protests to be a case of sore losing and "squealing".

It was only with the return of the English team and the subsequent use of Bodyline against English players in England by the touring West Indian cricket team in 1933 that demonstrated to the country the dangers it posed. The MCC subsequently revised the Laws of Cricket to prevent the use of "fast leg theory" tactics in future, also limiting the traditional tactic.

title: Leg theory


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Blythe Danner

Blythe Katherine Danner (born February 3, 1943) is an American actress. Accolades she has received include two Primetime Emmy Awards for Best Supporting Actress in a Drama Series for her role as Izzy Huffstodt on Huff (2004–2006), and a Tony Award for Best Featured Actress for her performance in Butterflies Are Free on Broadway (1969–1972). Danner was twice nominated for the Primetime Emmy for Outstanding Guest Actress in a Comedy Series for portraying Marilyn Truman on Will & Grace (2001–06; 2018–20), and the Primetime Emmy for Outstanding Lead Actress in a Miniseries or Movie for her roles in We Were the Mulvaneys (2002) and Back When We Were Grownups (2004). For the latter, she also received a Golden Globe Award nomination.

Danner played Dina Byrnes in Meet the Parents (2000) and its sequels Meet the Fockers (2004) and Little Fockers (2010). She has collaborated on several occasions with Woody Allen, appearing in three of his films: Another Woman (1988), Alice (1990), and Husbands and Wives (1992). Her other notable film credits include 1776 (1972), Hearts of the West (1975), The Great Santini (1979), Mr. & Mrs. Bridge (1990), The Prince of Tides (1991), To Wong Foo, Thanks for Everything! Julie Newmar (1995), The Myth of Fingerprints (1997), The X-Files (1998), Forces of Nature (1999), The Love Letter (1999), The Last Kiss (2006), Paul (2011), Hello I Must Be Going (2012), I'll See You in My Dreams (2015), and What They Had (2018).

Danner is the sister of Harry Danner and the widow of Bruce Paltrow. She is the mother of actress Gwyneth Paltrow and director Jake Paltrow.

Early life
Danner was born in Philadelphia, Pennsylvania, the daughter of Katharine (née Kile) and Harry Earl Danner, a bank executive. She has a brother, opera singer and actor Harry Danner, a sister and a maternal half-brother. Danner has Pennsylvania Dutch, some English and Irish ancestry; her maternal grandmother was a German immigrant, and one of her paternal great-grandmothers was born in Barbados to a family of European descent.

Danner graduated from George School, a Quaker high school located near Newtown, Bucks County, Pennsylvania, in 1960.

Career
A graduate of Bard College, Danner's first roles included the 1967 musical Mata Hari and the 1968 Off-Broadway production of Summertree. Her early Broadway appearances included Cyrano de Bergerac (1968) and her Theatre World Award-winning performance in The Miser (1969). She won the Tony Award for Best Featured Actress in a Play for portraying a free-spirited divorcée in Butterflies Are Free (1970). In 1972, Danner portrayed Martha Jefferson in the film version of 1776. That same year, she played the unknowing wife of a husband who committed murder, opposite Peter Falk and John Cassavetes, in the Columbo episode "Etude in Black".

Her earliest starring film role was opposite Alan Alda in To Kill a Clown (1972). Danner appeared in the episode of M*A*S*H entitled "The More I See You", playing the love interest of Alda's character Hawkeye Pierce. She played lawyer Amanda Bonner in television's Adam's Rib, opposite Ken Howard as Adam Bonner. She played Zelda Fitzgerald in F. Scott Fitzgerald and 'The Last of the Belles' (1974). She was the eponymous heroine in the film Lovin' Molly (1974) (directed by Sidney Lumet). She appeared in Futureworld, playing Tracy Ballard with co-star Peter Fonda (1976). In the 1982 TV movie Inside the Third Reich, she played the wife of Albert Speer. In the film version of Neil Simon's semi-autobiographical play Brighton Beach Memoirs (1986), she portrayed a middle-aged Jewish mother. She has appeared in two films based on the novels of Pat Conroy, The Great Santini (1979) and The Prince of Tides (1991), as well as two television movies adapted from books by Anne Tyler, Saint Maybe and Back When We Were Grownups, both for the Hallmark Hall of Fame.

Danner appeared opposite Robert De Niro in the 2000 comedy hit Meet the Parents, and its sequels, Meet the Fockers (2004) and Little Fockers (2010).

From 2001 to 2006, she regularly appeared on NBC's sitcom Will & Grace as Will Truman's mother Marilyn. From 2004 to 2006, she starred in the main cast of the comedy-drama series Huff. In 2005, she was nominated for three Primetime Emmy Awards for her work on Will & Grace, Huff, and the television film Back When We Were Grownups, winning for her role in Huff. The following year, she won a second consecutive Emmy Award for Huff. For 25 years, she has been a regular performer at the Williamstown Summer Theater Festival, where she also serves on the board of directors.

In 2006, Danner was awarded an inaugural Katharine Hepburn Medal by Bryn Mawr College's Katharine Houghton Hepburn Center. In 2015, Danner was inducted into the American Theater Hall of Fame.

Environmental activism
Danner has been involved in environmental issues such as recycling and conservation for over 30 years. She has been active with INFORM, Inc., is on the Board of Environmental Advocates of New York and the board of directors of the Environmental Media Association, and won the 2002 EMA Board of Directors Ongoing Commitment Award. In 2011, Danner joined Moms Clean Air Force, to help call on parents to join in the fight against toxic air pollution.

Health care activism
After the death of her husband Bruce Paltrow from oral cancer, she became involved with the nonprofit Oral Cancer Foundation. In 2005, she filmed a public service announcement to raise public awareness of the disease and the need for early detection. She has since appeared on morning talk shows and given interviews in such magazines as People. The Bruce Paltrow Oral Cancer Fund, administered by the Oral Cancer Foundation, raises funding for oral cancer research and treatment, with a particular focus on those communities in which healthcare disparities exist.

She has also appeared in commercials for Prolia, a brand of denosumab used in the treatment of osteoporosis.

Personal life
Danner was married to producer and director Bruce Paltrow, who died of oral cancer in 2002. She and Paltrow had two children together, actress Gwyneth Paltrow and director Jake Paltrow.

Danner's niece is the actress Katherine Moennig, the daughter of her maternal half-brother William.

Danner co-starred with her daughter in the 1992 television film Cruel Doubt and again in the 2003 film Sylvia, in which she portrayed Aurelia Plath, mother to Gwyneth's title role of Sylvia Plath.

Danner is a practitioner of transcendental meditation, which she has described as "very helpful and comforting".

title: Blythe Danner