Jan A Bergstra
email: j.a.bergstra@uva.nl

Division by Zero: A Survey of Options

The idea that, as opposed to the conventional viewpoint, division by zero may produce a meaningful result, is long standing and has attracted interest from many sides. We provide a survey of some options for defining an outcome for the application of division in case the second argument equals zero. The survey is limited by a combination of simplifying assumptions which are grouped together in the idea of a premeadow, which generalises the notion of an associative transfield.

Introduction

The number of options available for assigning a meaning to the expression 1/0 is remarkably large. In order to provide an informative survey of such options some conditions may be imposed, thereby reducing the number of options. I will understand an option for division by zero as an arithmetical datatype, i.e. an algebra, with the following signature:

• a single sort with name V ,

• constants 0 (zero) and 1 (one) for sort V ,

• 2-place functions • (multiplication) and + (addition),

• unary functions -(additive inverse, also called opposite) and -1 (multiplicative inverse),

• 2 place functions -(subtraction) and / (division).

Decimal notations like 2, 17, -8 are used as abbreviations, e.g. 2 = 1 + 1, and -3 = -((1 + 1) + 1). With inverse the multiplicative inverse is meant, while the additive inverse is referred to as opposite.

This signature is referred to as the signature of meadows Σ Md in [START_REF] Bergstra | Meadows and the equational specification of division[END_REF], with the understanding that both inverse and division (and both opposite and subtraction) are present.

A number of assumptions will be made with the intention to limit the scope of the discussion to manageable proportions. We consider an arithmetic datatype with domain |V | for sort V , and we assume that the following conditions are met.

Nontriviality. 0 = 1.

Totality. Addition, opposite, subtraction, multiplication, inverse, and division are total functions.

Additive monoid. (|V |, + , 0) is a commutative monoid (for all x ∈ |V |, x + 0 = 0 + x = x, and addition is commutative and associative).

Multiplicative monoid. (|V |, • , 1) is a commutative monoid (for all x ∈ |V |, x • 1 = 1 • x = x, and multiplication is commutative and associative).

Zero is idempotent. 0 • 0 = 0 (that 1 is idempotent follows from the additive monoid requirement.)

Opposite. x+y = 0 → y = -x (opposite produces all proper additive inverses).

The opposite of one is a proper additive inverse. 1 + (-1) = 0.

Inverse. x • y = 1 → y = x -1 (inverse produces all proper multiplicative inverses).

Subtraction.

x-y = x+(-y) (and therefore in combination with the additive monoidal structure: -x = 0 -x.)

Division. x/y = x • x -1 (and therefore in combination with the multiplicative monoidal structure: x -1 = 1/x.)

Arithmetical datatypes which meet these requirements may have many unusual properties. Below we will find that plausible equations like x -x = 0, x • x -1 = 1, x•(y+z) = x•y+x•z, (1/x)•(1/y) = 1/(x•y) are refuted in some of the options for arithmetic that are considered. A (nontrivial) premeadow is a datatype (algebra) which satisfies each of the above requirements. We notice that unicity of proper additive inverses follows from the additive monoidal structure: if x+y = 0 = x+z, then y = 0+y = (x+z)+y = (x+y)+z = 0+z = z. Similarly the multiplicative monoidal structure implies that proper multiplicative inverses are unique. The notion of a premeadow is very close to but slightly less demanding than the notion of a transfield as introduced in [START_REF] Reis | Construction of the transreal numbers and algebraic transfields[END_REF]. More precisely, all associative transfields are premeadows (a transfield may have a nonassicative addition). While a transfield must have a field as a substructure of its reduct to the signature of rings, a premeadow need not necessarily include a field. Every transfield is a premeadow, however, and every premeadow for which the reduct upon forgetting the inverse function and division contains a field as a substructure is a transfield. In particular every meadow is a premeadow, but not every meadow is a transfield. We refer to [START_REF] Bethke | The initial meadows[END_REF] for examples of meadows that are not transfields.

The constraints of a premeadow rule out certain mechanisms from floating point arithmetic. For instance if one assumes the existence of a positive zero, say 0 + (which may or may not be identified with the unsigned zero 0) and a negative zero, say 0 -, where the negative zero satisfies 1 + 0 -= 1 and (-1) + 0 -= -1 then it follows from the requirements of a premeadow that 0 -= 0 -+ 0 = 0 -+ (1 + (-1)) = (0 -+ 1) + (-1) = (1 + 0 -) + (-1) = 1 + (-1) = 0 and formalisation in terms of a premeadow cannot be used to explain how and why 0 and 0 -differ. The question to which extent existing as well as conceivable floating point models can be faithfully and productively modelled by means of algebras and abstract datatypes seems not to have received much attention and it seems fair to say that floating point arithmetic cannot be used as an argument in favour of any design of a specific arithmetical datatype involving division by zero.

Our discussion will be confined to premeadows with characteristic 0. Generalisation of the definitions below to the case of characteristic p > 0 is unproblematic, however.

Because subtraction and division are found by means of explicit definition from opposite and inverse, the presence of these 2 place operations is merely a matter of notational convenience below. The notion of a fraction, however, depends on the presence of a division operation in the signature. Following [START_REF] Bergstra | Transformation of fractions into simple fractions in divisive meadows[END_REF] a fraction is defined as expression over the signature Σ M d, or an extension thereof, with division as the leading function symbol. For a fraction P/Q, P is called its numerator and Q is called its denominator. It follows from these conventions that both the numerator and the denominator of a fraction are expressions.

Definition 1.1. A premeadow is an algebra with signature Σ Md which satisfies the above 10 requirements with the possible exception of nontriviality. A nontrivial premeadow is a premeadow in which 0 = 1.

Premeadows constitute a quasi-variety because all of the listed requirements take the form of conditional equations.

On the significance of signatures for "division by zero"

The importance of being explicit about signatures is twofold. First of all with a signature comes a natural notion of syntax, and as a consequence a distinction between syntax and semantics. Secondly, upon having adopted the signature of meadows, the status of an expression 1/0 is entirely unproblematic, and for that reason the question about its meaning is not only a reasonable issue, but it has even become an unavoidable matter. Working with signatures and in particular working with the signature of meadows creates a context in which the topic of division by zero arises in a natural manner. Questions about division by zero can only arise if inverse and/or division are understood as operations for which the interpretation is not fixed for once and for all. The situation for opposite and for inverse is comparable: the opposite of a value may not be a proper additive inverse of it, and the inverse of a value may not be a proper multiplicative inverse of it. In both cases we choose not to have a function symbol for "the proper inverse of" in the signature, and this choice is primarily motivated by the advantages of working with total algebras.

It is left up to the design of a model, that is the choice of semantics, to determine the meaning of any expression, in particular of expressions like 1/0. Σ Md extends the signature of rings with inverse and division. The signature of fields, however, if one wishes to think in those terms, is the same as the signature of rings, and does not include either inverse or division. As a consequence, in the setting of fields, it is implausible to view, say the equation x -1 • y -1 = (x • y) -1 as a potential axiom about the inverse function. Assuming in the context of fields that one considers inverse to be a partial function, then the meaning of this equation is quite sensitive to the logic of partial functions which one prefers to adopt. Indeed, adopting the axiom may have an impact on the domain of the inverse function, which introduces a risk of circular reasoning.

In the context of premeadows, however, such an equation is perfectly plausible as an axiom. The limitation to commutative multiplication is a matter of convenience, and most of the topics discussed in the paper admit generalisation to the non-commutative (also called skew) case.

The terminology of inverse functions merits some further attention. In the following definitions φ may be a partial function.

Definition 1.2. A total or partial function φ( ) captures multiplicative inverses if for all x and y in |V |, if x • y = 1 then in φ(x) is defined and x • φ(x) = 1. Definition 1.3. A unary operator φ( ) is a multiplicative inverse operator if it captures multiplicative inverses.

These definitions motivate the terminology used above when referring to -1 as a multiplicative inverse on the basis of the requirement x•y = 1 → y = x -1 . It is possible that besides -1 other functions, which may have an explicit definition over the signature of meadows, also capture multiplicative inverses. With inverse we will refer to the specific (total) multiplicative inverse -1 . Definition 1.4. A function φ( ) produces multiplicative inverses if for all x in its domain, x • φ(x) = 1.

From these assumptions an important conclusion can be derived: Proposition 1.1. The inverse operator in a non-trivial premeadow does not produce a multiplicative inverse for 0.

Proof. Assume that -1 produces a multiplicative inverse q = 0 -1 on 0, then:

0 = 0 • 1 = 0 • (0 • q) = (0 • 0) • q = 0 • q = 1
, which contradicts the non-triviality assumption on the premeadow at hand.

In a similar manner one finds that if an inverse operator produces multiplicative inverses it cannot have 0 as a result. Many authors experiment with 1/∞ = 0 or variations thereof and the following result applies to such proposals. Proposition 1.2. If the inverse operator in a non-trivial premeadow produces proper multiplicative inverses on all nonzero arguments it does not produce value 0 on any nonzero argument.

Proof. Assume that -1 is an operator which produces proper multiplicative inverses and which takes value 0 on argument q = 0, i.e. q -1 = 0 and q •q -1 = 1.

Then 1 = 1 • 1 = (q • q -1 ) • (q • q -1 ) = (q • 0) • (q • 0) = q • (0 • (q • 0)) = q • ((q • 0) • 0) = q • (q • (0 • 0)) = (q • q) • (0 • 0) = (q • q) • 0 = q • (q • 0) = q • (0 • q) = q • 1 = q.
Now combining 1 = q with the assumption q • q -1 = 1 yields 1 = q • q -1 = 1 • 0 = 0 which contradicts the non-triviality requirement.

The subject of division by zero, under the constraints mentioned above, is about the design of versions of arithmetic which include multiplication and an inverse function which produces proper inverses when these exist and which is defined on 0.

It is an implicit assumption that arithmetic is not fixed for once and for all, in other words that different designs of arithmetic are possible and are worth investigation. At the same time arithmetic is merely an intuitive notion which grasps systems of numbers equipped with constants and operations from various signatures.

Options outside the quasivariety of premeadows

Several authors have proposed a perspective on division by zero which allows for 0 • 0 -1 = 1. For instance in [START_REF] Czajko | On Cantorian spacetime over number systems with division by zero[END_REF] 1/0 = ∞ is adopted in combination with 1 • ∞ = 1. Such proposals are significant, but seem to be technically less accessible than proposals which are compatible with the requirements of premeadows. With Proposition 1.1 the existence of a proper inverse of zero is ruled out in premeadows, however. As stated above the survey of this paper will stay within the confinements of premeadows and therefore none of the options which are discussed below satisfy the identity 0 • 0 -1 = 1.

2 From the conventional approach to 1/0 to motivating alternatives

It is often assumed that the ordinary conventions for dealing with the partial multiplicative inverse operator are well-known and self-evident. That may be the case but writing explicitly about such conventions is not entirely straightforward. It seems to be the case that notations like 1/0 have no place in professional mathematics. Not only do such notations not occur, such notations do not occur for a good reason which is understood to be based on common background knowledge. It is the nature of this background knowledge which must be made explicit, and then upon having been made explicit it may or may not require further substantiation. Now the mere fact that 1/0 has no meaning is no principled reason for the strong preference for not using 1/0 as a notation in a professional text.

Consider the infinite sums P k = Σ ∞ n=k 1/n for k a nonzero natural number. It is reasonable to state that P 1 = 1 + P 2 and it is also reasonable to say that P 1 diverges, which means that it is not defined and therefore that it has no numerical value. The idea that P 1 diverges is compatible with

P 1 = ∞. With Q k = Σ ∞ n=k (-1) n • n, however, it is still reasonable to state that Q 1 = -1 + Q 2 while the Q k exist even less than the P k as the Q k represent sums which don't even converge to ∞ or to -∞.
Now consider the assertion that 1/0 is undefined. It seems to be the case that the latter assertion is not acceptable as a constituent of a competent mathematical text. One may conclude that 1/0 is to a lesser extent a fraction than that Σ ∞ n=k 1/n is a sum.

Principled (conventional) interpretation of the division sign

Mathematical conventions allow remarkable flexibility. Working in binary notation it is common to read 10 as the number two and the equation 1 + 1 = 10 makes sense. And working modulo 3 it is known that 1 + 2 = 0. Working in a non-standard model of the Peano axioms of arithmetic addition and multiplication work in unexpected ways. However, when it comes to 1/0 there seems to be no semantic flexibility left as 1/0 = q can only hold if 0 • q = 1, which is problematic indeed. Suppose one introduces an absorptive error value ⊥ such that 1/0 = ⊥ and such that the error propagates through all operations. Then ⊥ + (-⊥) = ⊥. It follows that the introduction of an error value calls into question the use of the opposite, for which one may expect that adopting the rule x + (-x) = 0 constitutes a very firm commitment, a commitment the deviation of which is just as problematic, or even more problematic that deviating from a commitment to 0 • 0 -1 = 1. It follows from this observation that already adopting the existence of an error value as the result of division by zero would be incompatible with deeply rooted intuitions and conventions on the use of mathematical notations.

It may be the case that the conception that 1/0 does not exist is the most rigidly held convention of all conventions regarding the use of elementary mathematical notation. Not only is 1/0 non-existent but as an idea it embodies the very concept of mathematical non-existence, which might even be prior to the concept of existence in the same way as the idea of inconsistency may be prior to the idea of consistency. It is intuitively obvious that the concept of the non-existence of a conceivable entity is hardly communicated in a best possible manner by having a preferred notation for precisely that entity.

Context dependence of occurrences of division

For an expression P of the form Q/R to occur in a text, in a specific position, it is conventionally required that the text embeds said (specific) occurrence of P in a context which is such that from the assumptions imposed on the parameters that are available in the given context it can be inferred, either formally or informally, but in any case to the reader's satisfaction, that R is nonzero.

For instance the assertion "for all x = 0 it is the case that x • (1/x) = 1" complies with the mentioned requirement on contexts: 1/x occurs in a context in which it has been assumed that x is nonzero. The mentioned assertion is therefore regarded as unproblematic form the perspective of division by zero.

In contrast with these observations the first order sentence χ defined by

χ ≡ ∀x ∈ Q (x = 0 → x • (1/x) = 1
) is somehow troublesome. For χ to be true it must be the case that substituting 0 for x produces a true statement [0/x]χ ≡ 0 = 0 → 0 • (1/0) = 1. Now for [0/x]χ to be considered a valid assertion each of its components must have some truth value, at least when adopting a classical two-valued logic. Thus either 0 • (1/0) = 1 is considered true or 0 • (1/0) = 1 is considered true but conventional mathematics commits to neither of these as 0

• (1/0) is considered undefined. For Φ ≡ ∀x ∈ Q(x = 0 → x•(1/x) = 1
) to be considered true some additional assumptions are required. Indeed a plurality of disparate strategies can be used to that end. We list some options:

1. a logic of partial functions may be used (there are many options for logics of partial functions) which incorporates a three valued logic, or 2. the occurrence of 1/0 is considered to denote an error value in an ordinary error algebra so that 0 • (1/0) = 1 is satisfied, or 3. the universal quantifier is understood as an infinitary conjunction by reading Φ as q∈Q,q =0 (q • (1/q) = 1), with the effect that no subformula involving 1/0 appears, or 4. the implication connective in [0/x]χ is understood in terms of a short circuit logic, in which the conclusion of the implication is only evaluated once (i.e. after) the condition has been found valid.

Syntax versus semantics: not a conventional intuition

Returning to the legitimacy of writing expressions like 1/0 in conventional mathematical practice we notice that in conventional mathematics it is not even plausible to ask for the value of 1/0 because there is no notion of syntax which entitles 1/0 to the status of being asked questions about. In other words, an expression is a tool for denoting an entity, and whoever makes use of the expression is supposed to have sorted out in advance why doing so is realistic. But conventional mathematics is also pragmatic and liberal to some extent. When stating that the fraction 4/6 is not simplified, reference is made to the fraction as an expression, rather than to the rational number which it denotes. Asking if the fraction 1/0 has a value is likely to be countered with the objection that 1/0 is not a fraction, just because it has no value. The idea of a contrast between syntax and semantics, where syntax consists of expressions, to the extent that one may always ask whether or not an expression is legal, and do so in advance of the determination of its meaning, at least in principle, is not supported by conventional mathematical practice. It is commonplace, however, to ask for an expression that signifies a semantic entity, for instance the non-negative square root of 2 once its existence has been made plausible. Thus, to the extent that conventional mathematical practice supports a distinction between syntax and semantics, the latter serves as a tool whereas the former takes priority. In the case of the inverse of zero, this convention implies that by the mere use of an expression 1/0 it is already acknowledged that, at least in principle, the author (speaker) knows what this expression may mean and is able to provide an explanation of that meaning.

As a consequence of the limited separation of syntax and semantics the use of axioms for the specification of operations is somehow limited too. However attractive it may seem to be from a formal perspective the idea that inverse or division is merely specified by one or more first order axioms is not very appealing to most mathematicians. As a consequence of this bias against the use of syntactic methods it is conceivable that less than optimal use has been made of specification techniques and term rewriting techniques in the context of arithmetic.

Motivation(s) for having inverse defined on zero

various approaches to the inverse of zero are motivated in different ways. Four motives for investigating a specific approach can be distinguished at least: Curiosity driven motivation. One may investigate a certain approach without any claim that is is of either practical or philosophical value. For instance:

• In [START_REF] Bergstra | Division by zero in non-involutive meadows[END_REF] the option that 1/0 = 1 is investigated without any regard to whether or not there is an advantage in adopting that option.

• Incorporating the idea of the Riemann sphere into arithmetic. (The theory of wheels: [START_REF] Setzer | Wheels[END_REF][START_REF] Carlström | Wheels-on division by zero[END_REF][START_REF] Carlström | Partiality and choice, foundational contributions[END_REF], with subsequent work in [START_REF] Santangelo | The axiomatic construction of a new algebraic structure in order to extend a field and define division by zero[END_REF].)

• Making progress on the first order theory of fields. (E.g. in [START_REF] Komori | Free algebras over all fields and pseudo-fields[END_REF][START_REF] Ono | Equational theories and universal theories of fields[END_REF].)

• In [START_REF] Ufuoma | On the operation of division by zero in Bhaskara's framework: criticisms, modifications, and justifications[END_REF] the idea is explored that 1/0 = (-1)!. The paper contains a significant attempt to survey the history of division by zero, which, however contradicts the claim of [START_REF] Boyer | An ancient reference to division by zero[END_REF] that the history of division starts with the ancient Greeks.

Instrumental motivation for specific objectives. A certain approach may be considered useful for pursuing specific objectives which are at first sight unrelated to division by zero. Among such objectives we mention:

• Exploiting calculational correspondence with formal reasoning patterns in geometry and analysis (E.g [START_REF] Michiwaki | Reality of the division by zero z/0 = 0[END_REF] upon adopting 1/0 = 0.)

• Simplification of semantics and proof theory. (1/0 = 0 as assumed in various proof checking systems.)

• Making use of the framework of equational abstract datatype specifications for the investigation of specific arithmetical datatypes. For instance upon adopting 1/0 = 0, work towards a theory of fractions is reported in [START_REF] Bergstra | Transformation of fractions into simple fractions in divisive meadows[END_REF]). For algebraic datatype specification we mention e.g. [START_REF] Ehrich | Specification of Abstract Data Types[END_REF].

Moderate preference for a specific design. The idea that a certain approach to division by zero is preferable to all competing approaches, including the conventional idea of having division by zero undefined, serves as a motivation for some work. The development of Transmathematics (see e.g [START_REF] Reis | Construction of the transreal numbers and algebraic transfields[END_REF]) incorporates an attempt to rework parts of mathematics in a preferred design involving a notion of division by zero based on signed infinities.

Exposing a strong and definite preference. An author or group of authors may be convinced that adopting a certain approach to division by zero will be beneficial for the development of mathematics at large. This conviction comes with the idea that the approach at hand provides a superior perspective in comparison to other approaches, including conventional mathematics.

For instance in [START_REF] Michiwaki | Reality of the division by zero z/0 = 0[END_REF] and work cited in that paper, the respective authors express a strong preference for 1/0 = 0 = 0/0. Remarkably, authors from the Institute of Reproducing Kernels claim that setting 0 = 0/0 = 1/0 is a discovery which dates back to 2014 only ( [START_REF]Announcement 412: The 4th birthday of the division by zero z/0 = 0[END_REF][START_REF] Saitoh | Who did derive First the Division by Zero 1/0 and the Division by Zero Calculus tan(π/2) = 0, log 0 = 0 as the outputs of a computer[END_REF]). This claim is to our perception refuted by the existence of [START_REF] Ono | Equational theories and universal theories of fields[END_REF]. The latter paper claims no priority for the idea that 1/0 = 0 while making much progress regarding the logical analysis of that idea, to such an extent that its place in the history of the subject seems to be undisputable.

Admittedly in [START_REF] Ono | Equational theories and universal theories of fields[END_REF] no strong preference for 1/0 = 0 is formulated. The history of expressing a strong preference for calculating with 1/0 = 0 throughout mathematics may well have started in 2014 just as claimed in [START_REF]Announcement 412: The 4th birthday of the division by zero z/0 = 0[END_REF][START_REF] Saitoh | Who did derive First the Division by Zero 1/0 and the Division by Zero Calculus tan(π/2) = 0, log 0 = 0 as the outputs of a computer[END_REF].

A survey of minimal premeadows of characteristic zero

A datatype is minimal if it has no proper subalgebras. Below we will consider minimal arithmetical datatypes only, thereby leaving the reals for later work. We assume that the set Q of rational numbers is given and that 0, 1, addition, opposite, subtraction, and multiplication on Q are known. We refrain from adopting any particular construction of the rational numbers, we merely assume that these are given as a set with known interpretations for each closed expression 1/q where q denotes a nonzero integer. We will consider arithmetical datatypes with a domain that includes Q and may extend it with at most three pairwise different values outside Q taken from ⊥, +∞, -∞. All functions from Σ Md except inverse extend the interpretations of these functions on Q while inverse and division are constrained by finding a proper inverse or quotient if one exits in Q. With Q we refer to the arithmetical datatype of rationals in which inverse and division are partial functions. Q is not a premeadow but it can be easily turned into a premeadow Q p by taking for 0 -1 any rational number p as a value. Below we will discuss in some detail Q 0 and Q 1 .

Totalising division with an absorptive element

The simplest idea for developing a premeadow which includes the rational numbers is to introduce ⊥ outside Q as a new value which serves as an absorptive element. We use ⊥ which is the customary notation in the theory of abstract datatypes for such an element where it is often referred to as an error element.

In transmathematics (see e.g. [START_REF] Anderson | Perspecx Machine VIII, axioms of transreal arithmetic[END_REF]) the notation Φ is preferred for an absorptive element, and the negative connotation of an error is deliberately avoided. This strategy may be applied to any partial algebra. The value ⊥ is supposed to propagate through all functions and to serve as the inverse of 0: 0 -1 = ⊥ and for all x, x

+ ⊥ = ⊥ + x = -⊥ = x -⊥ = ⊥ -x = x • ⊥ = ⊥ • x = ⊥ -1 = ⊥/x = x/⊥ = ⊥.
The structure thus obtained is denoted with Q ⊥ . From the perspective of abstract datatypes Q ⊥ is a very plausible structure, but, we found almost no literature about this particular structure. In [START_REF] Bergstra | Division by zero in common meadows[END_REF], Q ⊥ is referred to as the common meadow of rational numbers, thereby emphasizing its proximity to a common understanding of partiality of division, while writing, a for additional element instead of ⊥ in order to avoid any distraction potentially caused by the traditional negative connotations of ⊥. In [START_REF] Bergstra | Division by zero in common meadows[END_REF] the the equational theory of Q ⊥ is studied in considerable detail, resulting in a completeness theorem restricted to the case of characteristic zero. In [START_REF] Bergstra | Fracpairs and fractions over a reduced commutative ring[END_REF] the same structure appears as the initial algebra of so-called fracpairs. In [START_REF] Reis | Construction of the transreal numbers and algebraic transfields[END_REF] the real number version of Q ⊥ occurs as example 19.

Semantic justification

If for a closed expression t Q ⊥ |= t = 0 then this identity is entirely trustworthy for any mathematician. The fact that inverse was made total can only make calculations deviate in the direction of non-rational values, i.e in the direction of ⊥.

One may criticise Q ⊥ for being pessimistic about the information which may be obtained from expressions involving division by zero.

Practical justification

We have no information about the practical use which has been made of this particular totalised arithmetic. The equational theory of Q ⊥ is somewhat cumbersome and seems not to provide an attractive platform for the systematic development of further equational specifications, either initial or loose, of datatypes which incorporate functions and sorts tailored to specific applications. It is an advantage of Q ⊥ that the unconditional fraction addition rule (UFAR) holds:

x y + u v = x • v + y • u y • v

Zero as the inverse of zero

Instead of taking a new value ⊥ as the outcome of 0 -1 one may take an existing value for instance 0 -1 = 0, thus obtaining the arithmetical; datatype Q 0 . This structure, and its first order and equational theory has been first studied in significant detail in [START_REF] Komori | Free algebras over all fields and pseudo-fields[END_REF] and [START_REF] Ono | Equational theories and universal theories of fields[END_REF]. An equational initial algebra specification for Q 0 is given in [START_REF] Bergstra | The rational numbers as an abstract data type[END_REF]. For further work on the equational theory of Q 0 and its counterparts for real and complex numbers we refer to [START_REF] Bergstra | Subvarieties of the variety of meadows[END_REF][START_REF] Bergstra | Equations for formally real meadows[END_REF][START_REF] Bergstra | Meadows and the equational specification of division[END_REF][START_REF] Bergstra | Division by zero in non-involutive meadows[END_REF]. Adopting 0 -1 = 0 underlies [START_REF] Michiwaki | Reality of the division by zero z/0 = 0[END_REF] and many related papers. The same design decision has been adopted in various theorem provers.

Semantic justification

From a point of view of symmetry setting 0 -1 to 0 is a reasonable, and even elegant way of completing the graph of the partial inverse function to a total one. We see no principled justification for, say, the following equation 1 (1/0)-(1/0) = 0 in terms of the concept of division or with the help of analytic or asymptotic methods. However, one may easily get used to 1/0 = 0 and the consequences thereof and avoid negative consequences of this assumption. The fact that a systematic axiomatic approach becomes feasible serves as a justification for adopting a seemingly arbitrary value of inverse on 0.

In Q 0 UFAR fails: 1 = 1/0 + 1/1 = (1 • 1 + 0 • 1)/(0 • 1) = 1/0 = 0 and as a consequence the theory of fractions for Q 0 is more involved than the fraction theory of Q ⊥ . Instead of UFAR the conditional fraction addition rule (CFAR) is valid in Q 0 :

x = 0 ∧ y = 0 → x y + u v = x • v + y • u y • v 3.2.

Practical justification

Four lines of practical justification can currently be distinguished: (i) based on the assumption that 0 -1 = 0 a rich meta-theory can be developed, (ii) we found that the equational theory of arithmetic with inverse is made total by adopting 0 -1 = 0 is attractive and allows for extension in several useful directions, (iii) by adopting 0 -1 = 0 the logic of proof checkers may be simplified in a useful manner, while preventing errors of calculation and proof is not made much harder, (iv) Some authors claim that adopting 0 -1 = 0 works unexpectedly well in various different branches of mathematics (see [START_REF] Michiwaki | Reality of the division by zero z/0 = 0[END_REF] and related papers).

One as the inverse of zero

Taking 0 -1 = 1 the structure Q 1 is obtained. This option has been investigated in detail in [START_REF] Bergstra | Division by zero in non-involutive meadows[END_REF]. We notice that Q 1 satisfiers CFAR but not UFAR:

1 + 1 = 1 0 + 1 0 = 1 • 0 + 0 • 1 0 • 0 = 0 + 0 0 = 0 0 = 0 • 1 = 0 3.

Justification

A semantic justification of this arithmetical datatype is comparable to the justification given for Q 0 though somewhat less convincing because there is less symmetry, as inverse ceases to be an involution, and the equational theory becomes more involved and less attractive without any perceivable gains. Apart from some spurious mentioning of the idea that 0 -1 = 1 in the educational literature we have not seen any systematic work on this basis. At this stage we are unaware of any practical justification for this design. One might just as well consider 0 -1 = 731 giving Q 731 and agree in advance that no definition or theorem is made dependent of the parameter 731 which could just as well have been replaced by 732, or any other arbitrary rational number. Perhaps it is an advantage of working with 0 -1 = 731 in Q 731 that it is easy to detect any unwanted dependency of a further result or development from the ad hoc number 731.

Unsigned ∞ as the inverse of zero: uncommon wheels

A different idea is to let 0 -1 take a new value q outside Q (just as ⊥) but in such a manner that q -1 = 0. It is customary to denote q with ∞ in this case. The simplest design of an arithmetical datatype of this kind, denoted with Q ∞ is as follows:

Q ∞ has domain Q ∪ {∞} and the operations are extended as follows (assuming that the resulting structure will be a premeadow):

• 0 -1 = ∞ and ∞ -1 = 0, • ∞ • x = ∞ + x = -∞ = ∞.

Semantic justification

A semantic justification for Q ∪ {∞} may be as follows: some closed expressions which evaluate to ⊥ in Q ⊥ are now evaluated to a meaningful value, for instance:

1/(1/0) = lim x↓0 1/(1/x) = lim x↑0 1/(1/x) = lim x→0 x = 0
However a similar motivation can not in each case be given, for instance let

P (x, y) = 1 (1/x) -(1/y)
then Q ∞ |= P (0, 0) = 0. Indeed lim x↓0 (lim y↓0 P (x, y)) = 0 but consider the following path through this two dimensional limit: lim x↓0 P (x, x • (1 + x)) = 1. It follows that the limit lim x↓0,y↓0 P (x, y) is not well-defined. Next consider

Q(x, y) = 1 (1/x) • (1/(1/y))
We find Q ∞ |= Q(0, 0) = 0 but lim x↓0 (lim y↓0 Q(x, y)) = ∞, whence the 2dimensional limit lim x↓0,y↓0 Q(x, y) does not exist. One may conclude that justification of outcomes of evaluation in Q ∞ on the basis of asymptotic considerations is problematic. This observation motivates the design of wheels. In addition we notice both UFAR and CFAR fail in

Q ∞ : 0 = 0 + 0 = 1 ∞ + 1 ∞ = 1 • ∞ + ∞ • 1 ∞ • ∞ = ∞ • ∞ ∞ = ∞ • 1 ∞ = ∞ • 0 = ∞
To the best of our knowledge Q ∞ has not been investigated, or even proposed, in the literature on division by zero. Nevertheless it is a very plausible structure for which we suggest the following name: the uncommon wheel of rationals. The idea is that, in line with [START_REF] Setzer | Wheels[END_REF], an arithmetical datatype with a single non-finite value for the inverse of zero, i.e. a non-rational element e so that e = 0 -1 , and e -1 = 0 is referred to as a wheel. Now the common wheels (simply referred to as wheels), as discussed below, are equipped with an absorbtive element ⊥ while the uncommon wheels are not.

3.5

The wheel of rationals: combining ∞ and ⊥ Q ∞ can be adapted in such a manner that problematic outcomes are avoided by incorporating ⊥. This construction leads to the wheel of rationals (here denoted as Q ∞,⊥ ), as first specified in [START_REF] Setzer | Wheels[END_REF].

• |Q ∞,⊥ | = Q ∪ {∞, ⊥}, • 0 -1 = ∞, • ∞ -1 = 0, • ∞ • 0 = ⊥, • ∞ + ∞ = ⊥, • ∞ • ∞ = ∞, • -∞ = ∞, • for x ∈ Q, x = 0: ∞ • x = ∞, • for x ∈ Q: ∞ + x = ∞, • x + ⊥ = ⊥ + x = -⊥ = x -⊥ = ⊥ -x = x • ⊥ = ⊥ • x = ⊥ -1 = ⊥/x = x/⊥ = ⊥
These definitions suffice in combination with the requirement that Q ∞,⊥ is a premeadow which extends Q and which extends the graphs of partial inverse and division to total functions. It is worth mentioning that various familiar equations are invalid in the wheel of rationals, e.g. x • 0 = 0, an the so-called quasi-cardinality rule (x/y) + (z/y) = (x + z)/y (the name of this identity comes from [1]). We also note that UFAR and CFAR both fail in Q ∞,⊥ :

∞ = 0 • 0 + 0 • 0 = 0 ∞ + 0 ∞ = 0 • ∞ + ∞ • 0 ∞ • ∞ = ⊥ + ⊥ ∞ = ⊥ ∞ = ⊥
another example indicates that an occurrence of the constant 0 plays no role in this fact:

0 = 0 + 0 = 1 ∞ + 1 ∞ = 1 • ∞ + ∞ • 1 ∞ • ∞ = ∞ + ∞ ∞ = ⊥ ∞ = ⊥

Semantic justification

The wheel of rationals seems to be justified in the following sense: it produces proper rational outcomes only in cases where limits are taken in arbitrary order and from an arbitrary direction, but we have no proof this somewhat informal intuition.

The metatheory of wheels is attractive and has been developed to a significant extent in [START_REF] Carlström | Wheels-on division by zero[END_REF][START_REF] Carlström | Partiality and choice, foundational contributions[END_REF]. However, we are unaware of the existence of applications of wheels.

Transrationals: signed infinities combined with Φ

In some cases the wheel of rationals fails to provide a proper value (that is, a value in Q) while it would be justified to do so. For instance the following identity can be justified if one thinks of ∞ as a positive infinite value which satisfies ∞ + ∞ = ∞:

1 (1/0) + (1/0) = 0
On the other hand there is less justification for having 1 (1/0) + ((-1)/0) = 0

By distinguishing positive and negative infinity both cases can be separated. If 1/0 is positive infinity then (-1)/0 can be considered negative infinity, and while ∞+∞ = ∞ is plausible there is no plausible value for ∞+(-∞) except an error value. These considerations constitute the recent introduction of transrationals.

Proposition 3.1. For each closed expression t not involving Φ and ⊥: if

Q ±∞,Φ |= t = Φ then Q inf,⊥ |= t = ⊥.
Proof. Straightforward with induction on the length of a reduction path to normal form in the rewrite system for transrationals.

Semantic justification

Providing a semantic justification for Q ±∞,Φ is not entirely straightforward. For instance consider the expression Q(x, y) = (x -1 + (((-1)/y) -1 ) -1 ) -1 . Evaluation of Q(0, 0) in the premeadow of transrationals yields:

Q(0, 0) = 1 0 -1 + (((-1)/0) -1 ) -1 = 1 ∞ + ((-∞) -1 ) -1 = 1 ∞ + ∞ = 1 ∞ = 0
On the other hand lim x↓0,y↓0 R(x, y) = lim x↓0,y↓0

x -1 -y -1 and the latter limit does not exist. It appears that for explaining the results of evaluation in the arithmetical datatype of transrationals it is important one must have an inside out evaluation strategy in mind. Moreover a preference for innermost evaluation takes priority over considerations in terms of limits of related rational functions. Innermost evaluation requires that an expression is only evaluated after each of its subexpressions has been evaluated.

Each of the identities listed in the description can be given an explanation in terms of limits and then by adopting the principle of innermost evaluation one obtains a justification for each derivable identity between closed terms in Q ±∞,Φ .

Beyond transrationals? Incorporating a negative zero

Consider the fraction

P (x, y) = 1 ( 1 x + -1 ( 1 ((-1)/y) ) )
Evaluation of the closed fraction P (0, 0) in the premeadow of transrationals yields:

P (0, 0) = 1 1 0 + -1 ( 1 ((-1)/0) ) = 1 ∞ + -1 ( 1 -∞ ) = 1 ∞ + -1 0 = 1 ∞ + (-∞) = 1 Φ = Φ
Now lim x↓0,y↓0 P (x, y) = lim x↑0,y↑0 P (x, y) = 0 from which it may be inferred that there is ample asymptotic justification for having P (0, 0) = 0, and that a reasonable case can be made to look for a further refinement or adaptation of transrational arithmetic in which P (0, 0) evaluates to 0 rather than to Φ. We notice that Q 0 |= P (0, 0) = 0 but one may claim with ample justification that Q 0 allows too many identities, so that contemplating Q 0 does not provide a useful solution to this question, and neither does working in Q ∞,⊥ where P (0, 0) evaluates to ⊥, because Q ∞,⊥ fails to differentiate between the two expressions discussed in the beginning of Paragraph 3.6.

There is room for modification of the transrational arithmetic as defined in [START_REF] Anderson | Perspecx Machine VIII, axioms of transreal arithmetic[END_REF] in order to take alternative requirements regarding the evaluation of closed expressions into account. Problem 3.1. Does there exist an arithmetical datatype which satisfies P (0, 0) = 0 and in which all or most expressions which evaluate in transrational arithmetic to a finite non-error result evaluate to the same result, while still sufficiently many expressions evaluate to an error (i.e. to Φ).

At first sight the introduction of a negative zero, as promoted for instance in the IEEE-754 floating point standard, may easily lead to a solution to this question, but the details involved are not straightforward. We refer to the filter calculus of [START_REF] Beeson | The meaning of infinity in calculus and computer algebra systems[END_REF] as work which is relatively close to this topic.

Transrational arithmetic with a second zero

As an experiment we introduce the additional constant which represents negative zero, or rather the inverse of negative infinity, and which we will call "second zero", in order to avoid suggesting the plausibility of certain equations, such as for instance -= 0 or (-) -1 = ∞. Because the axioms below imply that = (-(0 -1 ) -1 this constant logically serves as an abbreviation. Its role in term rewriting is different and more important as it may serve as a normal form in circumstances where the expression (-(0 -1 ) -1 of which it serves as an abbreviation is not considered a normal form. The rewrite system is adapted by including in addition the following 18 rewrite rules:

• ∞ • = Φ, • ∞ = Φ, -∞ • = Φ, • -∞ = Φ, • ∞ + = ∞, + ∞ = ∞, • (-∞) + = -∞, + (-∞) = -∞, • -1 = -∞,
• ∞ -1 = (0, 1), (-∞) -1 = ,

• -= ,

• + (n, m) = , (n, m) + = ,

• + = , • • = , • • (n, m) = , (n, m) • = ,
Together these rules will be called TM . The rewrite system is terminating and confluent. The arithmetic datatype thus obtained, will be denoted with Q ±∞, ,Φ is a premeadow. Q ±∞, ,Φ satisfies P (0, 0) = 0. However, in contrast with transrational arithmetic Q ±∞, ,Φ |= Q(0, 0) = Φ (with Q(x, y) as in Paragraph 3.6.1). Compared with Q ±∞,Φ , Q ±∞, ,Φ is a less natural arithmetical datatype because of the asymmetry between ∞ -1 and (-∞) -1 . For instance in Q ±∞, ,Φ , 1 + ∞ -1 = 1 and 1 + (-∞) -1 = (-∞) -1 , with the latter equation having no asymptotic justification at all. Q ±∞, ,Φ qualifies as a solution of Problem 3.1 but it is probably too asymmetric to be useful.

Finding adaptations of transrational arithmetic in which there is a meaningful role for a second zero, which may behave more like a negative zero than , is at the time of writing this survey an open topic for further research, which in all likelihood allows for a plurality of different designs.

The arithmetical datatype Q ±∞, ,Φ , which we will call transrationals (transrational arithmetic) with a second zero is merely one option indicating such possibilities, and no claim is made that Q ±∞, ,Φ will constitute a prominent outcome of such investigations.

Concluding remarks

The theme of division by zero is introduced and the conventional approach to this matter is sketched. The central role of the appreciation of a contrast between syntax and semantics is emphasised and an attempt is made to formulate in what sense elaborating on the theme of division by zero is reasonably possible.

Using the notion of a premeadow, which generalises the specailization to the assiciative case of the notion of a transfield as proposed in [START_REF] Reis | Construction of the transreal numbers and algebraic transfields[END_REF], a brief survey is given of options for defining arithmetical datatypes providing a total extension of the division operator. The survey is restricted to characteristic zero and to datatypes which extend the rational numbers by one or finitely more additional elements.

Our survey of options is incomplete, first of all because the literature on division by zero is already hard to grasp and quite diverse, but more importantly because some possible avenues have yet been left unexplored, as is exemplified by the role of signed zeroes, which feature in practice more than in theory. An example is given of an arithmetical datatype which contains a negative zero.

In the tradition of transrationals ⊥ is denoted Φ, and so we may have both constants and require that ⊥ = Φ.

We will denote the arithmetical datatype of transrationals with Q ±∞,Φ . The domain includes on top of the known rationals three additional elements: |Q ±∞,Φ | = Q ∪ {∞, -∞, Φ}. We assume that the rational numbers are given as (n, m) with n ∈ Z, m ∈ N, m > 0 and gcd(m, n) = 1. Thus 0 is represented by (0, 1) and 1 is represented by (1, 1). S(n, m) = (n\ gcd(n, m), m\ gcd(n, m)) denotes simplification of the pair (n, m) with \ denoting integer division. The operations of Q ±∞,Φ are determined by the following 43 rewrite rules. In these rules the pairs (n, m) are considered auxiliary constants.

• 0 = (0, 1), 1 = (1, 1), ⊥ = Φ,

Here it is understood that say (n, m)•(k, l) = S(n•k, m•l) represents a single step for each pair (n, m) with m = 0. It is easy to see that Q ±∞,Φ is a premeadow. The rewrite system is confluent and terminating.


MELKANA Vincent Alberge
email: valberge@fordham.edu

Melkana Brakalova-Trevithick

Athanase Papadopoulos

Melkana Brakalova
email: brakalova@fordham.edu

A COMMENTARY ON TEICHM ÜLLER'S PAPER UNTERSUCHUNGEN ÜBER KONFORME UND QUASIKONFORME ABBILDUNGEN (INVESTIGATIONS ON CONFORMAL AND QUASICONFORMAL MAPPINGS)

Keywords: AMS classification: 30F60, 32G15, 30C62, 30C75, 30C70 Conformal invariants, conformal radius, module of a ring domain, extremal domain, reduced module, quasiconformal mapping, Modulsatz, type problem, line complex

Introduction

The paper Untersuchungen über konforme und quasikonforme Abbildungen (Investigations of conformal and quasiconformal mappings) [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF] is Teichmüller's Habilitationsschrift, which he presented in Berlin in 1938, under the supervision of Ludwig Bieberbach. 1 In 1936, Teichmüller attended lectures on function theory by Rolf Nevanlinna who was teaching at the University of Göttingen as a visiting professor for Date: December 23, 2019. [START_REF] Abikoff | Oswald Teichmüller[END_REF] Teichmüller had obtained his doctorate in 1935 in Göttingen under the supervision of Helmut Hasse. The subject was linear operators on Hilbert spaces over the quaternions [START_REF] Teichmüller | Operatoren im Wachsschen Raum[END_REF]. In the meanwhile, he shifted his topics of interest. For details about Teichmüller's life, we refer the reader to [START_REF] Abikoff | Oswald Teichmüller[END_REF] and [START_REF] Schappacher | Oswald Teichmüller -Leben und Werk[END_REF].

the academic year. These lectures had a major influence on him. Teichmüller remained interested in this topic until the end of his (short) life. His Habilitationsschrift is one of his important papers on complex analysis. He wrote several other papers related to this subject, see [START_REF] Teichmüller | Eine Umkehrung des zweiten Hauptsatzes der Wertverteilungslehre[END_REF][START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF][START_REF] Teichmüller | Eine Verschärfung des Dreikreisesatzes[END_REF][START_REF] Teichmüller | Vermutungen und Sätze über die Werverteilung gebrochener Funktionen endlicher Ordnung[END_REF][START_REF] Teichmüller | Einfache Beispiele zur Wertverteilungslehre[END_REF].

The paper we are commenting on here is difficult to read. Like several other of Teichmüller's paper, while the mathematical ideas it contains are very dense, and the proofs are concise, sometimes sketchy, Cazacu observes in [START_REF] Cazacu | Foundations of quasiconformal mapping[END_REF]: "Teichmüller's paper(s) are written in a warm, direct style. He emphasizes aims, ideas, difficulties, clearly explains notations, methods, and compares his results with others concerning similar problems." All throughout the paper, Teichmüller gives credit to the works of other mathematicians which he uses, improves or has been motivated by. Another characteristic of this paper is that its spirit and the proofs it contains are purely geometric. We may quote here a passage from the paper, during a computation he makes of the module function Φ(P ). Teichmüller writes ( §2.1):

[. . . ] Explicitly calculating this requires the introduction of elliptic functions and one can observe a connection between the function Φ(P ) and the values of the elliptic modular function for purely imaginary periodic ratios. In the interest of preserving the purity of the method used and to avoid tedious calculations, we will not make use of this connection at all; rather, we derive everything we wish to know about the function Φ(P ) from its geometric definition.

The paper has seven sections. The first six are concerned with the conformal geometry of doubly-and simply-connected domains (the latter with marked points), extremal problems for conformal invariants, and quasiconformal mappings. Teichmüller studies properties of modules of quadrilaterals and ring domains and reduced modules of simplyconnected domains. He often relies on applications of the length-area method. One of the main motivations for his investigations is to give a geometric proof of what he calls the Main Lemma (see Item [START_REF] Ahlfors | Über eine in der neueren Wertverteilungslehre betrachtete Klasse transzendenter Funktionen[END_REF] and Theorem 5.2 below), stated at the beginning of the paper, of which an improved version is now known as the Teichmüller-Wittich-Belinskiȋ theorem. This result was motivated by the developments of (at that time) new value distribution theory. Indeed, Teichmüller says that he reached this statement while we was working with Hans Wittich on a class of Riemann surfaces introduced by Egon Ullrich. (Such surfaces are now called surfaces with finitely periodic ends). Teichmüller also writes that he first tried to prove this lemma without success by using analytical methods developed by Ahlfors in [START_REF] Ahlfors | Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen[END_REF] and [START_REF] Ahlfors | Über eine in der neueren Wertverteilungslehre betrachtete Klasse transzendenter Funktionen[END_REF]. Throughout Teichmüller's paper, there is no explanation about how this result could be used and we have to quote another paper by Teichmüller. In [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF], Teichmüller writes:

In order to study a conformal mapping from a Riemann surface A to another Riemann surface B, one constructs an explicitly known quasiconformal mapping from A to some Riemann surface C [...] so that B and C are closely related and so that the best possible estimate from above for the dilatation quotients viewed as a function of points of C is known. [...] One then determines the properties attached to the mappings from C to B whose dilatation quotients satisfy this estimate. Thanks to the known mapping A → C, each of these properties gives a statement about the mapping A → B to be studied. This method and therefore the Main Lemma has been first applied by Lê Vȃn [START_REF] Vȃn | Über das Umkehrproblem der Wertverteilungstheorie[END_REF] to solve a particular case of the so-called Nevanlinna inverse problem. We refer to the paper [START_REF] Papadopoulos | Value distribution theory and Teichmüller's paper Einfache Beispiele zur Wertverteilungslehre[END_REF] in the present volume and to the paper [START_REF] Drasin | Quasiconformal mappings in value-distribution theory[END_REF] for an exposition of this problem, to which, later on, Drasin provided a full solution in [START_REF] Drasin | The inverse problem of the Nevanlinna theory[END_REF]; see also the monograph [START_REF] Goldberg | Value Distribution of Meromorphic Functions[END_REF]. The last section of Teichmüller's paper [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF] is a contribution to the type problem, in which the author gives a negative answer to a conjecture made by Nevanlinna. The methods rely to some extent on the quasiconformal techniques developed in the previous sections, but are not based on the Main Lemma.

Let us add that Teichmüller's main interest in the paper we are concerned with here is the question of uniformization and conformal representation. He writes ( §5.1): " [. . . ] it seems necessary to stress that I do not investigate the concept of module of a ring domain for its own sake, but that the focus is on an investigation of the problem of conformal mappings and uniformization."

Let us give a list of other main results and tools that are introduced in this paper.

(1) A study of the conformal geometry of annuli and quadrilaterals: monotonicity of modules, regions that are solutions of extremal problems, the introduction of new conformal objects: conformal radius, reduced module, reduced logarithmic area, and others. (2) A distortion theorem which is an improvement of the main distortion lemma in Ahlfors' thesis, obtained (unlike Ahlfors' proof) without using differential equations but using only geometric methods. (3) An important lemma known today as Teichmüller's Modulsatz which is used in his proof of the Main Lemma (see [START_REF] Ahlfors | Über eine in der neueren Wertverteilungslehre betrachtete Klasse transzendenter Funktionen[END_REF] or 5.2). The lemma gives a sufficient condition for a set that lies between two doubly-connected domains in the complex plane to be close to a circle. (4) The use of a class of quasiconformal mappings w = w(z) of the complex plane which is more general than the class used today:

the dilatation quotient of such a map at each point (that is, the quotient of the great to the small axis of the infinitesimal ellipse whic is the image of an infinitesimal circle) is not required to be bounded by a uniform constant, but it is assumed that it is bounded by some specific function of |z| with controlled growth. (5) An asymptotic result for quasiconformal mappings, namely, a bound as in [START_REF] Ahlfors | Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen[END_REF], that insures that |w| ∼ const.|z| as z → ∞ (a property we call "circularity at infinity" of the image of circles by such quasiconformal mappings). Following Teichmüller, we shall refer to this result as the Main Lemma. (6) A contribution to the type problem, and more precisely, a condition for a Riemann surface to be of hyperbolic type. Teichmüller uses for this purpose the notion of line complex, a combinatorial device to encode a Riemann surface that is a branched cover of the sphere. At the same time, he settles (by the negative) a question posed by Nevanlinna on the type problem. We have divided this commentary into sections which highlight the main results we stated above. More precisely, in Section 2 we will recall the geometric tools used by Teichmüller, such as the module of a ring domain, the reduced module of a simply-connected domain and the module of a quadrilateral. In Section 3, we will comment on the improvement made by Teichmüller to the Ahlfors distortion theorem. In Section 4, we will present the main ideas of the proof of the Modulsatz theorem along with some applications. In the last section, Section 5, we will recall the notion of quasiconformal mappings that Teichmüller uses and we will explain the Main Lemma and its improvements. Let us point out that we have decided to comment on the last section of [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF], that is the contribution to the type problem, in another chapter of the present volume, [START_REF] Alberge | Teichmüller's work on the type problem[END_REF].

Module theorems

In the first two sections of his paper, Teichmüller develops from first principles properties of modules of ring domains using logarithmic area and logarithmic length and modules of quadrilaterals, and he defines and studies properties of reduced modules of simply-connected domains. The references to his paper for what concerns reduced module, in the monographs by Wittich [START_REF] Wittich | Neuere Untersuchungen über eindeutige analytische Funktionen[END_REF] and Jenkins [START_REF] Jenkins | Univalent Functions and Conformal Mappings[END_REF], suggest that he was the first to develop this kind of results. We can also mention the book by Künzi [41] which contains the main results of Teichmüller from [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF] that we are commenting on here. In his study of moduli of ring domains, Teichmüller, as he mentions in the introduction, was motivated by an estimate of Ahlfors, which he uses after applying a suitable auxiliary conformal map. Since this involves only considerations of the point ∞, he was led to study conformal mappings of neighborhoods of this point, excluding this point itself, that is, of annuli.

Let us recall in detail the notions introduced and used in Teichmüller's paper [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF].

Teichmüller starts in §1.1 by recalling the definition of a ring domain, that is, a doubly-connected domain bounded by two continua.

In §1.2, he shows that any ring domain G can be mapped conformally onto an annulus r < |w| < R, where 0 < r < R < ∞. The proof uses the uniformization theorem (Teichmüller refers to Koebe). He then defines the module M of G as log R r . This is a conformal invariant and the definition is equivalent to a well-known classical definition based on extremal length, see [START_REF] Ahlfors | Conformal invariants: Topics in geometric function theory[END_REF][START_REF] Ahlfors | Quasiconformal mappings and their applications[END_REF]. Let us point out that Teichmüller's definition of module differs by a multiplicative ratio of 2π from the definition of the module that involves the notion of extremal length. We also note that conformal invariants of ring domains were already studied by Schottky. The latter proved in his doctoral dissertation [START_REF] Schottky | Über konforme Abbildung von mehrfach zusammenhängenden Fläche[END_REF] (1877) that two annuli {r 1 < |z| < r 2 } and {r ′ 1 < |z| < r ′ 2 } are conformally equivalent if and only if we have

r 2 r 1 = r ′ 2 r ′ 1 .
In §1.3, Teichmüller introduces the logarithmic length of a curve γ in C \ {0}, defined as

γ |d log z| = γ |dz| |z| 2 ,
and the logarithmic area F of a domain G ⊂ C \ {0}, defined as

G d log z = G dz |z| 2 ,
where, dz = dxdy. These two quantities represent the Euclidean length and area of the transformed, to the log z-plane, original curve and domain, where the log z-plane is the rectangle in the Euclidean plane which is the image of the annulus r < |w| < R by the logarithm map, after the annulus has been cut along the interval [-R, -r]. If M and F denote respectively the module and the logarithmic area of G, Teichmüller shows that [START_REF] Abikoff | Oswald Teichmüller[END_REF] 2πM ≤ F.

Let us say a few words about Teichmüller's proof of (1). This proof uses the so-called length-area method, a method that he refers to in the introduction as the "Grötzsch's and Ahlfors' methods" Let us recall that the length-area method involves comparison of Euclidean lengths of curves and area in the logarithmic plane w = log z-plane, followed by a repeated integration and use of the Cauchy-Schwarz inequality. The concept of extremal length takes root in the use of this method which was initiated by Beurling in his thesis [START_REF] Beurling | Étude sur un problème de majoration[END_REF] (see also the end of this section), and further developed by Beurling and Ahlfors2 in the 1940s by working not only with the Euclidean metric but with a whole class of conformal metrics (see [START_REF] Ahlfors | Invariants conformes et problèmes extrémaux[END_REF] and [8, p. 50]). According to Ahlfors,3 his first used of the length-area method appears in his thesis (see also [START_REF] Ahlfors | Sur le nombre des valeurs asymptotiques d'une fonction entières d'ordre fini[END_REF][START_REF] Ahlfors | Über die asymptotischen Werte der ganzen Funktionen endlicher Ordnung[END_REF][START_REF] Ahlfors | Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen[END_REF]) and was inspired by the book Funktionentheorie (1922) by Hurwitz and Courant [START_REF] Hurwitz | Funktionentheorie[END_REF]. In §1.4 and §1.5, Teichmüller presents an elegant proof of the following two properties of the module, and, at the same time, a solution of two extremal problems:

Proposition 2.1. If a ring domain G ′ with module M ′ is a subset of a ring domain G with module M such that G ′ separates the two complementary continua of G, then M ′ ≤ M. Equality holds if and only if G ′ = G.
Proposition 2.2. If a ring domain G contains two disjoint ring domains G ′ and G ′′ each of which separates the complementary continua of G and has modules M ′ and M ′′ respectively, then M ′ + M ′′ ≤ M.

Equality holds if and only if

G = {r < |z| < R}, G ′ = {r < |z| < ρ} and G ′′ = {ρ < |z| < R}.
This kind of module inequalities and of solutions of extremal problems involving ring domains were first studied by Grötzsch in [START_REF] Grötzsch | A'Campo-Neuen, On some extremal problems of conformal mappings[END_REF]. The reader may also refer to the chapter [START_REF] Alberge | On five papers by Herbert Grötzsch[END_REF] in the present volume containing an exposition of these results.

In §1.6, Teichmüller starts by recalling the notion of conformal radius, which he calls "mapping radius." We recall that the conformal radius of a simply-connected domain G of C containing 0 is the positive number R for which G is mapped conformally onto the unit disc such that 0 is mapped onto 0 and the derivative at 0 is equal to 1 R . He continues by defining the notion of reduced module of a simply-connected domain G containing z = 0 as follows. Let ρ > 0 be small enough in order to have the disc of radius ρ centered at z = 0 contained in G and let M ρ be the module of the ring domain obtained from G by removing such a disc. By means of what Teichmüller calls a "distortion theorem" (a term which refers to one of Koebe's theorems), he obtains the following estimate:

(2)

|M ρ + log ρ -log R| ≤ 2ρ R -4ρ .
Then, the reduced module M of G at z = 0 is defined as lim

ρ→0 (M ρ +log ρ)
and by means of ( 2) is equal to the logarithm of the conformal radius of G. The definition is then extended to the reduced module at any point of G, and by the transformation

z → 1 z , if ∞ ∈ G, it is also extended at z = ∞.
In the special case where

G ′ = {0 < |z| < R} and G ′′ = {|z| > R}, we have M ′ = log R and M ′′ = -log R, respectively, thus M ′ + M ′′ = 0.
In §1.7, Teichmüller defines the reduced logarithmic area of a simplyconnected domain G containing z = 0 but not z = ∞ by the formula

F = F ρ + 2π log ρ,
where ρ is small enough so that the disc D ρ of radius ρ centered at z = 0 is contained in G, and F ρ denotes the logarithmic area of the ring domain G \ D ρ . He then obtains an inequality similar to (1), namely, he proves that for a simply-connected domain G containing z = 0 of reduced module M and of reduced logarithmic area F , we have

(3) 2π M ≤ F ,

and this equality holds if and only if

G = |z| < e M .
Teichmüller's proof involves the length-area method but he points out that it can be done by what he calls "Bieberbach's method." Teichmüller justifies such a preference by saying that he only wants to apply geometric methods.

In §1.8, as a conclusion of its first section, Teichmüller deduces two results on simply-connected domains which are analoguous to Propositions 2.1 and 2.2. He also obtains the following result which is of special importance since it is used in the proof of the special Modulsatz, see §4 below.

Proposition 2.3. Let G ′ and G ′′ be two disjoint simply-connected domains with G ′ containing z = 0 and G ′′ containing z = ∞ and having reduced modules M ′ and M ′′ respectively. Then

M ′ + M ′′ ≤ 0.

Equality holds only when

G ′ = |z| < e M ′ and G ′′ = |z| > e M ′ .
We now review extremal domains studied by Teichmüller in §3 and §4.

In §2.1- §2.3, Teichmüller works with a particular class of ring domains, namely, for P > 1, the class of ring domains that separate the unit disc from z = ∞ such that the distance of the unbounded component of their corresponding complementary region from the origin is at most P .

Let G P be the exterior of the unit disc cut along the real axis from z = P to z = ∞. Teichmüller proves several properties of G P , including the following: Proposition 2.4. Let log Φ(P ) be the module of G P . Then, (1) Φ(P ) > P .

(2) Φ(P ) is a continuous strictly monotone increasing function of P , and Φ(P ) takes all values between 1 and ∞.

(3) Φ(P )/4P → 1 in an increasing manner as P → ∞.

(4) Suppose that for P > 1 the ring domain G belongs to the above class. Let M be the module of this domain. Then,

M ≤ log Φ(P )
and equality holds if and only if G is equal to some G P up to a rotation about the origin.

Let us make a few remarks on these results.

Remarks 2.5. 1.-Items (1), ( 2) and ( 3) are consequences of Inequality (2) above, with R = 4.

2.-The value log 4 is the reduced module of the whole plane cut along the real axis from 1 to ∞.

3.-Teichmüller's proofs of these results do not involve use of elliptic integrals and rely only on geometric methods. The reason, as he says, is that he wanted to preserve "purity" and "avoid tedious calculations."

4.

-Proofs of such results by means of elliptic integrals can be found in Hersch's thesis [START_REF] Hersch | Longueurs extrémales et théories des fonctions[END_REF] and also in the book by Lehto and Virtanen [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF].

5.-The result in item ( 4) is referred to in [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF] as the Grötzsch module theorem-the extremal domain G P is called the Grötzsch extremal domain-and Teichmüller proves it in two different geometric ways. He also explains how one of them can be used to prove "Koebe's theorem with the exact constant 1 4 ." Teichmüller notes at the beginning of §2.1 that this extremal problem has been already solved by Grötzsch in [START_REF] Grötzsch | A'Campo-Neuen, On some extremal problems of conformal mappings[END_REF], using the so-called Grötzsch method of strips. A proof based on the extremal length is contained in [START_REF] Ahlfors | Lectures on quasiconformal mappings. Second Version[END_REF] and in [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF].

6.-Teichmüller uses these results in [64] in order to prove that "a quasiconformal mapping of the unit circle |z| < 1 onto itself, where all boundary points remain fixed, shifts the center z = 0 by at most 2 (sup D(z)) In §2.4, Teichmüller solves another extremal problem, namely, he proves that for 1 < P 1 < P 2 , the annulus 1 < |z| < P 2 with a slit along the segment [P 1 , P 2 ] has the largest module amongst all ring domains separating their |z| = 1 boundary component from ∞ and such that the outer boundary component lies in the annulus r < |z| < P 2 , for some 1 < r ≤ P 1 . After proving this extremal property, Teichmüller concludes this subsection by a result which is a weak version of the Modulsatz (see Theorem 4.1 below for the statement).

In §2.5, Teichmüller leaves the investigation of extremal problems in order to study normal families between doubly-connected domains, where the base domain is the annulus 1 < |z| < R.

He then returns in §2.6 (and until the end of §2) to the study of another extremal problem and proves what is often referred to as Teichmüller's module theorem. (This is a terminology used in Lehto and Virtanen's monograph [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF].) We recall that this result says that among all doubly-connected domains that separate the points 0 and ρe iϕ from P e iθ and ∞, the one with the largest module is (up to a rotation) the so-called Teichmüller extremal domain, that is, the complex plane cut along the real axis from -ρ to 0 and from P to ∞. Teichmüller denotes the module of the latter by log Ψ P ρ . The proof of this property is sketchy (it is not as detailed as the one for the corresponding result on Grötzsch's module). A more detailed proof can be found for instance in [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF], [START_REF] Ahlfors | Lectures on quasiconformal mappings. Second Version[END_REF] and [START_REF] Ahlfors | Conformal invariants: Topics in geometric function theory[END_REF].

Teichmüller then compares the functions Ψ and Φ. He obtains, using conformal mappings, In the last subsection, §2.8, as a consequence of the extremal problem, Teichmüller deduces that any doubly-connected domain of the plane separating 0 from ∞ whose module is at least equal to e π always contains a circle of center 0.

Ψ P ρ = Φ 1 + 2 P ρ 1 +
We conclude our comments on §2 by adding that, as Teichmüller writes, the study of extremal domain and its module are related to Ahlfors' proof of Denjoy's conjecture. As we shall see, this relation leads to an improvement of the distortion lemma of Ahlfors.

Teichmüller starts §3 by recalling different conformal invariants associated with a quadrilateral, namely, the cross-ratio, the harmonic measure and the module. He concludes §3.1 by explaining how these notions are related to each other. Let us recall here that the module of a quadrilateral is the ratio of the sides of a Euclidean rectangle that is conformally equivalent to it.

Teichmüller gives in §3.2 an estimate of this module that he qualifies as "simple" and which is at the basis of the length-area method. We give below the statement, following Teichmüller's notation.

Proposition 2.6. Let V be a quadrilateral with sides a, b, c, d and let it be mapped conformally onto the rectangle 0 < u < a, 0 < v < b in the w = u + iv-plane, with a, b, c, d mapped respectively to v = 0, u = a, v = a, u = 0. Let β be the infimum of the lengths of all curves in V joining a and c, and let F be the area of V. Then

a b ≤ F β 2 .
Equality holds if and only if V is a rectangle with sides a, b, c, d.

In §3.2, Teichmüller derives some consequences from this result. For instance, using the same notation as above, he proves that [START_REF] Ahlfors | Quasiconformal mappings and their applications[END_REF] αβ ≤ F with equality only for rectangles. He also proves that such an inequality is also satisfied for quadrilaterals on a Riemannian surface. He then observes that [START_REF] Ahlfors | Lectures on quasiconformal mappings. Second Version[END_REF] min {α, β} 2 ≤ F, with equality only for squares. 4 Teichmüller concludes this subsection by generalizing the notion of extremal distance introduced by Beurling in his thesis, [START_REF] Beurling | Étude sur un problème de majoration[END_REF], for a pair of disjoint boundary arcs, and proves by means of Proposition 2.6 that in the case of a quadrilateral, the extremal length coincides with the reciprocal of the associated module.

One has to note here that Teichmüller's notion of extremal distance is different from the notion of extremal distance of a pair of disjoint boundary curves introduced later on by Ahlfors and Beurling. 5 We are now ready to talk about Ahlfors' distortion result and its improvement by Teichmüller. 4 These two results were proved later by Besicovitch in [START_REF] Besicovitch | On two problems of Löwner[END_REF] in a manner which according to Jenkins in [START_REF] Jenkins | Univalent Functions and Conformal Mappings[END_REF] is a "length-area proof of primitive type." Besicovitch in his paper does not mention Teichmüller though he refers to Löwner. Let us add that using the same strategy as Teichmüller, it is not difficult to prove that one obtains the same inequalities as ( 6) and ( 7) by replacing the Euclidean lengths α and β by appropriate extremal lengths. The inequalities so obtained are attributed by Hersch in [START_REF] Hersch | Longueurs extrémales et géométrie globale[END_REF] and Jenkins in [START_REF] Jenkins | Univalent Functions and Conformal Mappings[END_REF] to Teichmüller. Indeed, it is not difficult to see that Teichmüller only deals with conformal metrics that are pullback of the Euclidean metric by conformal mappings.

Ahlfors' distortion theorem

Ahlfors proves the Denjoy conjecture in his doctoral dissertation, written under the supervision of Nevanlinna and published in [START_REF] Ahlfors | Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen[END_REF]. The result is announced in [START_REF] Ahlfors | Sur le nombre des valeurs asymptotiques d'une fonction entières d'ordre fini[END_REF], and it also appears in [START_REF] Ahlfors | Über die asymptotischen Werte der ganzen Funktionen endlicher Ordnung[END_REF]. This conjecture says that the number of finite asymptotic values of an entire function of order k is at most 2k. Making this statement precise needs the definition of an appropriate notion of "order" and "asymptotic value" of an entire function, and this is done in the setting of Nevanlinna's theory (or value distribution theory); see the paper [START_REF] Papadopoulos | Value distribution theory and Teichmüller's paper Einfache Beispiele zur Wertverteilungslehre[END_REF] in the present volume. As mentionned earlier in the paper, Ahlfors' approach to the Denjoy conjecture was completely new and based on the length-area method. From such a method he obtains two "main inequalities," one of which is called the "Erste Hauptungleichung" (first main inequality) and is the key result for the proof of the Denjoy conjecture. It is now known as the Ahlfors distortion theorem. Regarding this result, Ahlfors writes in [10, Vol. 1, p.1]: [...] In my thesis [START_REF] Ahlfors | Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen[END_REF] the lemma on conformal mapping has become the main theorem in the form of a strong and explicit inequality or distortion theorem for the conformal mapping from a general strip domain to a parallel strip, together with a weaker inequality in the opposite direction. [...] A more precise form of the first inequality was later given by O. Teichmüller.

One can also point out that Jenkins and Oikawa in [START_REF] Jenkins | On results of Ahlfors and Hayman[END_REF] gave another proof of the two Ahlfors inequalities by means of the extremal length method. Their proof of the first one was described by Ahlfors as a "virtually trivial proof." Following Teichmüller's exposition (from §3.5 to §3.7), we now present Ahlfors' distortion theorem along with the main ideas of its proof given by Teichmüller. For this purpose, we first need to set the framework.

Let G be a strip domain, that is, a simply-connected domain of the z = (x, y)-plane with two marked accessible boundary points (that is, points that can be joined by a continuous curve to an interior point) r 1 and r 2 satisfying ℜr 1 < ℜr 2 .

For every x in the open interval ℜr 1 < x < ℜr 2 , the strip domain G is divided by the line ℜz = x into at most countably many subsets of which one, denoted by G x , contains r 1 on its boundary. This set G x has also the property that the last piece of every curve in G ending at r 1 lies in G x . The interior points of G which are boundary points of G x satisfy ℜz = x and are divided into countably many cross-cuts S i of G, that is, simple arcs whose endpoints lie in the boundary of G and whose interior points lie in the interior of G. Each of these cross-cuts S i partitions G into two pieces, one containing G x and another one which we will call for now G i . The strip G is then the disjoint union of G x , the S i , and G i . Note that all cross-cuts ℜz = x of G that do not lie on the boundary of G x are erased, so that G i may contain points with ℜz < x. Whichever G i contains r 2 as boundary point, and also contains the end piece of each curve in G terminating in r 2 , will be separated from G x by the cross-cut S i = S x .

Let Θ(x) be the length of S x . As Nevanlinna points out in [49, p. 93], such a function "[...] is in general of a complicated nature," but Teichmüller proves that it is lower semi-continuous and therefore measurable. By the uniformization theorem, one maps the domain G onto the parallel strip {w = u + iv | 0 < v < B} such that the boundary points r 1 and r 2 are sent respectively to -∞ and ∞. For ℜr 1 < x < ℜr 2 , the cross-cut S x is thus mapped onto an arc L x that connects the two boundary straight lines of the strip.

We set

u 1 (x) = min {ℜw | w ∈ L x } and u 2 (x) = max {ℜw | w ∈ L x } .
Then, as recalled by Teichmüller, Ahlfors' distortion theorem says that (8)

u 1 (x ′′ ) -u 2 (x ′ ) B > x ′′ x ′ dx Θ(x) -4 whenever for ℜr 1 < x ′ < x ′′ < ℜr 2 , x ′′ x ′ dx Θ(x) > 2.
In order to prove this, Teichmüller first proves by means of the length-area method that (9) ∀x ∈ (ℜr 1 , ℜr 2 ) ,

x ′′ x ′ dx Θ(x) ≤ a b ,
where a and b are the lengths of the sides of the rectangle R(a, b) that is biholomorphic to the quadrilateral Q(x ′ , x ′′ ) bounded by G and the cross-cuts S x ′ and S x ′′ and for which the sides of length a are images of the cross-cuts by such a biholomorphism. Then, he continues by applying suitable conformal mappings as follows. First, he maps the rectangle R(a, b) conformally onto the upper half of an annulus of module π a b . Then he maps the quadrilateral Q(x ′ , x ′′ ) onto the upper half-part of a symmetric doubly-connected domain that separates the points 0 and 1 from e π u 1 (x ′′ )-u 2 (x ′ ) B and ∞. From the extremal problem he proves earlier, namely, the so-called Teichmüller module theorem, he obtains [START_REF] Ahlfors | Collected works, in 2 volumes, Series: Contemporary Mathematicians Ser[END_REF]

a b ≤ 1 π log Ψ e π u 1 (x ′′ )-u 2 (x ′ ) B .
By using the inverse function of 1 π log Ψ(e πx ), inequalities ( 5), ( 9) and ( 10), Teichmüller finally obtains

(11) u 1 (x ′′ ) -u 2 (x ′ ) B > x ′′ x ′ dx Θ(x) - 4 log 2 π - 1 π log 1 1 -8e -π x ′′ x ′ dx Θ(x)
.

It now becomes elementary to verify that such an inequality implies Ahlfors' distortion theorem. Let us conclude this section by adding that the main steps of Teichmüller's proof can be found in [49, p. 97] and [8, p. 76].

The Modulsatz

We now comment on the Modulsatz and its consequences. At the beginning of §4, Teichmüller explains that he is interested in estimating the location of the points that lie between two disjoint ring subdomains of a given annulus G = {z | r < |z| < R}. For this purpose, he states what he calls the Modulsatz and which says the following:

Theorem 4.1 (The Modulsatz). For every ε > 0, there exists δ > 0 such that for any two disjoint ring subdomains G ′ and G ′′ of G where G ′ separates 0 from G ′′ and with modules M and M ′ respectively, if

M ′ + M ′′ ≥ log R r -δ,
then each point separated from 0 by G ′ and from ∞ by G ′′ belongs to the annulus

(12) log r + M ′ -ε ≤ log |z| ≤ log R -M ′′ + ε.
As he explains, such a result is a consequence of the following: Theorem 4.2 (Special Modulsatz). For every ε > 0, there exists δ = δ(ε) such that for any two disjoint simply connected domains of the Riemann sphere G ′ and G ′′ containing respectively 0 and ∞ and of reduced modules respectively M ′ and M ′′ , if

M ′ + M ′′ ≥ -δ, then the complement of G ′ ∪ G ′′ is contained in the circular ring M ′ -ε ≤ log |z| ≤ -M ′′ + ε.
Indeed, one has just to apply Theorem 4.2 to the regions obtained from G ′ and G ′′ by adjoining to them their corresponding complements containing 0 and ∞ respectively.

Since the Modulsatz is a consequence of the special Modulsatz, Section 4 of Teichmüller's paper is mainly devoted to the proof of the latter.

Teichmüller gives two proofs. The first one is a proof by contradiction which is based on a normal family argument and the Rouché theorem but as Teichmüller points out it is "purely existential," in the sense that it only assures the existence of δ = δ(ε) without specifying how it depends on ε. His second proof gives the relationship between δ and ε and is among the major geometric and more sophisticated achievements of the paper. Following Teichmüller's notation, we now present the main ideas of this proof.

Let G ′ and G ′′ be two disjoint simply connected domains containing respectively 0 and ∞ and of reduced modules respectively M ′ and M ′′ . By means of a homothety of center the origin, one can assume that z 0 = -1 does not belong to G ′ ∪ G ′′ . Therefore, proving the so-called Special Modulastz is equivalent to finding δ such that if M ′ + M ′′ ≥ -δ then M ′ ≤ ε and M ′′ ≤ ε. With this in mind, Teichmüller constructs two disjoint simply connected domains that solve an extremal problem. The construction is as follows. Let q > 1 be a parameter. Let I and A be respectively the interior and exterior of the unit circle in the wplane. By multiplying the argument of a complex number by q one identifies the boundary arc w = e iθ , |θ| ≤ π q of I with the boundary curve of A. Furthermore, one glues the remaining boundary arc of I by identifying each element with its complex conjugate. Therefore, one obtains a closed Riemann surface of genus 0 and one can prove (Teichmüller does!) that it can be conformally mapped onto the whole z-plane where w = 0 and w = ∞ correspond to respectively z = 0 and z = ∞, the boundary arc w = e iθ , |θ| ≤ π q of I corresponds to a closed curve which starts and ends at z = -q 2 and encloses the origin, and the remaining boundary arc of I corresponds to the line segment that connects z = -1 to z = -q 2 . The domains I and A are then carried to disjoint simply connected domains denoted respectively by S ′ q and S ′′ q . By setting M ′ q for the reduced module of S ′ q and M ′′ q for the reduced module of S ′′ q , Teichmüller proves the following extremal property:

Proposition 4.3.
Let G ′ and G ′′ be two disjoint simply connected domains containing respectively 0 and ∞ and both not containing z = -1.

Let M ′ and M ′′ be their corresponding reduced modules. Then,

∀q > 1, q 2 M ′ + M ′′ ≤ q 2 M ′ q + M ′′ q .
Moreover, the equality holds only when G ′ = G ′ q and G ′′ = G ′ q . Teichmüller's proof of such a result uses the same idea as the one of his second proof of the so-called Grötzsch module theorem. After showing this result, Teichmüller gives a planar description of the set B of all possible pairs of reduced modules associated with two disjoint simply connected domains defined as in the statement of Proposition 4.3. More precisely, he proves that B is a convex region that is bounded by the curve formed by the points M ′ q , M ′′ q , (0, 0) and M ′′ q , M ′ q and when the parameter q varies over the set of real numbers greater than 1. He continues by determining the exact values of M ′ q and M ′′ q and thus he proves that the boundary curve of B is twice differentiable but not thrice differentiable at (0, 0). He concludes his second proof of Theorem 4.2 by setting

ε = M ′ q δ = -M ′ q -M ′′ q
and he obtains the following estimates:

(13) δ ∼ ε 2 log 1 ε
, as ε → 0. §4 ends by discussion of the possibilities of improving δ(ε) (making it larger) in the case of the Modulsatz. Teichmüller suggests a methodsimilar to the one of the proof of the special Modulsatz-for finding the best pair (M ′ , M ′′ ) of modules of subdomains G ′ and G ′′ of G = {z | r < |z| < R} that would determine such a δ. He does not execute the argument, as it leads to methods which involve elliptic functions, methods that he does not want to use, as he states at several places in his paper. 6Teichmüller derives in §5 a few consequences of the Modulsatz. In particular, he studies the behaviour of a family Γ = {C λ } λ∈I of disjoint simple closed curves indexed by a subset I of R that accumulates at +∞ satisfying the following two properties:

(i) for any pair (λ, µ) ∈ I 2 such that λ < µ, the closed curve C λ separates 0 from C µ ; (ii) the curve C λ shrinks to ∞ as λ → +∞. For such a family of curves, if λ < µ then because of (i). one has a ring domain bounded by C λ and C µ . If its module is denoted by M(λ, µ) then (ii). implies that, for λ fixed, M(λ, µ) → ∞ as µ → ∞. Teichmüller examines conditions on M(λ, µ) under which the family Γ is almost circular at ∞, that is, when C µ approaches a circle as µ → ∞. Following Teichmüller, we recall that C µ approaches a circle as µ → ∞ if its so-called logarithmic oscillation ω(µ) approaches 0 as µ → ∞. The logarithmic oscillation ω(µ) of C µ is defined as

ω(µ) = log r 2 (µ) r 1 (µ)
where r 1 (µ) = min By observing that for any λ < µ, the doubly-connected domain bounded by C λ and C µ lies between the annuli {z | r 2 (λ) < |z| < r 1 (µ)} and {z | r 1 (λ) < |z| < r 2 (µ)} and using ( 14), Teichmüller proves the sufficient condition for almost circularity. His proof of the necessary condition is more complicated and is based on the Modulsatz, the normal family argument he obtained in his §2.5 and the Koebe distortion theorem. In addition to finding conditions for the almost circularity of Γ at ∞, Teichmüller is interested in the asymptotic behaviour of r i (µ) (i=1,2) as µ → ∞. By means of ( 14) and properties of modules of ring domains he proves the following: As we shall see in the next section, Teichmüller uses Lemma 4.4 in order to prove the Main Lemma. Moreover, as he emphasizes, these investigations on the asymptotic behaviour of the functions ω, r 1 and r 2 are useful in value distribution theory since they allow to estimate the number of solutions in a disc |z| < r of an equation of the type f (z) = a where f is a meromorphic function. Teichmüller continues by proving in §5 a result similar to Lemma 4.4 that gives estimates at infinity of the functions r i (i = 1, 2). His proof is mainly based on the relation [START_REF] Alberge | Teichmüller's work on the type problem[END_REF] and, as he points out, the estimates so obtained are actually an improvement of estimates obtained by Ahlfors in [5, p. 402]. Let us mention that Ahlfors used such estimates to solve a particular case of Nevanlinna's inverse problem. See [START_REF] Papadopoulos | Value distribution theory and Teichmüller's paper Einfache Beispiele zur Wertverteilungslehre[END_REF] for further details.

The Main Lemma

As mentionned earlier, §6 of Teichmüller's paper is devoted to the proof of the Main Lemma. Teichmüller starts this section by defining a quasiconformal mapping as a one-to-one continuous mapping w = w(z) between two domains of the complex plane which, except at isolated exceptional points, is continuously differentiable with a nonzero Jacobian (no assumption about orientation-preservation is made). He then recalls that for such a mapping w = w(z) one can define the dilatation quotient as w (z + re iα ) -w(z) re iα at a point z in the direction α, and where the maximum and the minimum are taken over all directions α. For a conformal mapping, there is a well-defined notion of derivative which does not depend on the direction. Thus, the directional derivative at each point is constant and the dilatation quotient is equal to 1. The converse is also true: at a regular point, if the dilatation quotient is equal to 1, then the derivative does not depend on the direction, which implies that the map is conformal. As recalled by Teichmüller, the value D(z) is also equal to the ratio of the major to the minor axis of an ellipse into which an infinitesimal circle is mapped.

Unlike in Teichmüller's previous paper [START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF], the dilatation quotient is not assumed to be bounded. He declares that with such an assumption "one does not necessarily get to the intended function-theoretic applications." Indeed, it is now known that this unboundedness condition on the dilatation quotient leads to applications in uniformisation (or the type problem) or in value distribution theory. See for instance [START_REF] Goldberg | Value Distribution of Meromorphic Functions[END_REF] and also the papers [START_REF] Alberge | Teichmüller's work on the type problem[END_REF] and [START_REF] Papadopoulos | Value distribution theory and Teichmüller's paper Einfache Beispiele zur Wertverteilungslehre[END_REF] in the present volume. Let us point out that in the modern definition of quasiconformality, the mapping is assumed to be sense-preserving, the dilatation quotient is assumed to be bounded and the differentiablilty assumption made by Teichmüller (with isolated singularities) is replaced by a weaker assumption of absolute continuity on lines.

Before Teichmüller, Lavrentieff, in his papers [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF][START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF], already considered such mappings under the term "almost analytic functions" (cf. the commentary [START_REF] Alberge | A commentary on Lavrentieff's paper Sur une classe de représentations continues[END_REF] in the present volume).

One can also add that the study of mappings with unbounded dilatation quotient began to attract significant attention after the publication of the work of David [START_REF] David | Solutions de l'equation de Beltrami avec µ ∞ = 1[END_REF]. See also the exposition [START_REF] Otal | Quasiconformal and BMO-quasiconformal homeomorphisms[END_REF] by Otal in Volume III of the present Handbook. One of the results in David's paper are sufficient conditions on the dilatation that allow one to solve a so-called degenerate Beltrami equation. Another approach using estimates of modules of ring domains appears in [START_REF] Lehto | Homeomorphisms with a given dilatation[END_REF][START_REF] Brakalova | On solutions of the Beltrami equation[END_REF][START_REF] Gutlyanskiȋ | On the degenerate Beltrami equation[END_REF][START_REF] Brakalova | On solutions of the Beltrami equation[END_REF]. The investigation of the solutions of the degenerate Beltrami equation led to the development of the theory of mappings of exponentially integrable distortion which are not necessarily quasiconformal in the modern sense. Many of the known properties of quasiconformal mappings were extended to this larger class [START_REF] Astala | Elliptic partial differential equations and quasiconformal mappings in the plane[END_REF][START_REF] Iwaniec | Geometric function theory and nonlinear analysis[END_REF].

After introducing this notion of quasiconformality and before proving the Main Lemma, Teichmüller provides an estimate on the distortion of a ring domain under a quasiconformal mapping. More precisely, he proves by means of the length-area method the following result. Then (16)

r 2 r 1 1 C(r) dr r ≤ log ρ 2 -log ρ 1 ≤ r 2 r 1 C(r) dr r .
It is worth noting that this result implies a well-known distortion inequality for quasiconformal (in the modern sense) mappings which says that the module of the image of a ring domain by a quasiconformal mapping does not exceed a certain upper bound.

As consequences of such a lemma, Teichmüller first gives a necessary condition on the dilatation quotient for having a quasiconformal mapping that maps either the punctured plane onto the unit disc or the unit disc onto the punctured plane. More specifically, he applies [START_REF] Belinskiȋ | Behavior of a quasiconformal mapping at an isolated point (in Russian)[END_REF] to show that if there exists a quasiconformal mapping from the punctured plane onto the unit disc such that D ≤ C (|z|), then [START_REF] Besicovitch | On two problems of Löwner[END_REF] ∞ dr r • C(r) < ∞.

On the other hand, if it is the unit disc that is mapped quasiconformally onto the punctured plane such that D ≤ C (|z|), then by the same techniques he gets (18)

1 C(r) • dr r = ∞.
The condition given in [START_REF] Besicovitch | On two problems of Löwner[END_REF], as pointed out by Teichmüller, was already obtained by Lavrentieff in [42, Théorème 1] (see also [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF]Théorème 7]) and as the latter, Teichmüller uses it in his §7 to obtain a result on the type problem. See [START_REF] Alberge | Teichmüller's work on the type problem[END_REF] for further details. Another consequence of Lemma 5.1 is the Main Lemma that we state again for the convenience of the reader. In order to prove this lemma, Teichmüller considers the family of curves C λ which are the images of |z| = e λ by the given quasiconformal mapping. For sufficiently large λ, the family so defined satisfies conditions (i). and (ii). stated at the end of §5 of this commentary. He uses Let us recall that a proof of this lemma that uses Ahlfors' distortion theorem was obtained by Wittich in [START_REF] Wittich | Zum Beweis eines Satzes über quasikonforme Abbildungen[END_REF]. Another proof can be found in [30, p. 345ff].

Later on, at the end of his most quoted paper [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF], Teichmüller presents a conjecture which says that under the same hypothesis than Theorem 5.2, one has [START_REF] Brakalova | On the local behavior of certain homeomorphisms[END_REF] w(z) ∼ const • z as z → ∞.

In order to justify such a conjecture, Teichmüller says that it "is supported by the 'spiraling value distribution' for certain functions and the induced order of growth. In particular, this holds for the mapping w = ze iη(|z|) (η(r) real)." Teichmüller continues by giving an idea of the proof. Later, a full proof was obtained by Belinskiȋ in [START_REF] Belinskiȋ | Behavior of a quasiconformal mapping at an isolated point (in Russian)[END_REF] and Lehto in [START_REF] Lehto | On the differentiability of quasiconformal mappings with prescribed complex dilatation[END_REF]. See also [48, V.6.6]. This result is now referred to as the Teichmüller-Wittich-Belinskiȋ theorem. A survey of this result with historical comments was written by Drasin in [START_REF] Drasin | On the Teichmüller-Wittich-Belinskii theorem[END_REF]. For further applications of this theorem one refers to [48, V.7] and [START_REF] Bojarski | Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane[END_REF]Ch. 11]. One can also find improvements of this theorem in [START_REF] Reich | On the Behavior of Quasiconformal Mappings at a Point[END_REF][START_REF] Brakalova | On the local behavior of certain homeomorphisms[END_REF][START_REF] Gutlyanskiȋ | Conformality of a quasiconformal mapping at a point[END_REF][START_REF] Brakalova | Sufficient and necessary conditions for conformality. Part I. Geometric viewpoint[END_REF][START_REF] Brakalova | Sufficient and Necessary Conditions for Conformality. Part II. Analytic Viewpoint[END_REF][START_REF] Shishikura | Conformality of quasiconformal mappings at a point, revisited[END_REF]. The last section of Teichmüller's paper concerns the type problem and is discussed in our paper [START_REF] Alberge | Teichmüller's work on the type problem[END_REF] in the present volume.

Index

1 2 -

 2 1 ." We refer to §5 below for the definition of a quasiconformal mapping and the definition of the associated dilatation quotient D(z).

Cµ |z| and r 2

 2 (µ) = max Cµ |z|. The conditions that Teichmüller seeks are given by the following result: (14) lim µ→∞ ω(µ) = 0 ⇐⇒ lim χ→∞ χ<λ<µ {M(χ, µ) -M(χ, λ) -M(λ, µ)} .

Lemma 4 . 4 .

 44 Let |M(λ, µ) -(µ -λ)| ≤ ϕ(λ), for λ < κ, M(λ, µ) ≤ K where lim λ→∞ ϕ(λ) = 0. Then for some constant α lim λ→∞ (log r 1 (λ) -λ) = lim λ→∞ (log r 2 (λ) -λ) = α.

Lemma 5 . 1 .

 51 Let r 1 < |z| < r 2 be a circular ring mapped quasiconformally onto a circular ring ρ 1 < |w| < ρ 2 by a map whose dilatation quotient satisfies D(z) ≤ C(|z|).

Theorem 5 . 2 (

 52 Main Lemma). Let the punctured z-plane be mapped one-to-one and quasiconformally onto the punctured w-plane. Let the dilatation quotient satisfy D(z) ≤ C(|z|), where C(r) → 1, as r → ∞, so fast that ∞ (C(r) -1) dr r converges. Then, by approaching infinity |w| ∼ const • |z|.

Lemma 5 . 1 ,

 51 and Lemma 4.4 by setting ϕ (λ) = ∞ e λ (C(r) -1) dr r to arrive at the conclusion of the statement.

Cf. the historical notes in Ahlfors' book Conformal invariants [8, p. 81].

Cf. L. V. Ahlfors', Collected works[START_REF] Ahlfors | Collected works, in 2 volumes, Series: Contemporary Mathematicians Ser[END_REF], Vol. 1, p. 1

A different and shorter proof of a variation of Theorem 4.1, together with an estimate equivalent to[START_REF] Alberge | Teichmüller's work on the type problem[END_REF], is due to Pommerenke[START_REF] Ch | Boundary behavior of conformal mappings[END_REF], p. 201-202. The result proved by Pommerenke uses the additional assumption that G ′ ∪ G ′′ = G, and thus the estimate[START_REF] Alberge | On five papers by Herbert Grötzsch[END_REF] concerns the points lying on the joint boundary components ∂G ′ ∩ ∂G ′′ . Pommerenke's proof uses the method of extremal length, properties of univalent conformal mappings, the area theorem, and coefficient estimates. A proof of the Modulsatz, following Pommerenke's method, and further applications can be found in the monograph byGarnett and Marshall [29,.

Acknowledgements. The first and third authors acknowledge support from the U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 "RNMS: GEometric structures And Representation varieties" (the GEAR Network).