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Press conference	Practice	In a press conference, one or more speakers may make a statement, which may be followed by questions from reporters. Sometimes only questioning occurs; sometimes there is a statement with no questions permitted.
Press conference	Practice	A media event at which no statements are made, and no questions allowed, is called a photo op. A government may wish to open their proceedings for the media to witness events, such as the passing of a piece of legislation from the government in parliament to the senate, via a media availability.American television stations and networks especially value press conferences: because today's TV news programs air for hours at a time, or even continuously, assignment editors have a steady appetite for ever-larger quantities of footage.
Press conference	Practice	News conferences are often held by politicians; by sports teams; by celebrities or film studios; by commercial organizations to promote products; by attorneys to promote lawsuits; and by almost anyone who finds benefit in the free publicity afforded by media coverage. Some people, including many police chiefs, hold press conferences reluctantly in order to avoid dealing with reporters individually. A press conference is often announced by sending an advisory or news release to assignment editors, preferably well in advance. Sometimes they are held spontaneously when several reporters gather around a newsmaker.
Press conference	Practice	News conferences can be held just about anywhere, in settings as formal as the White House room set aside for the purpose or as informal as the street in front of a crime scene. Hotel conference rooms and courthouses are often used for press conferences. Sometimes such gatherings are recorded for press use and later released on an interview disc.
Press conference	Media day	Media day is a special press conference event where rather than holding a conference after an event to field questions about the event that has recently transpired, a conference is held for the sole purpose of making newsmakers available to the media for general questions and photographs often before an event or series of events (such as an athletic season) occur. In athletics, teams and leagues host media days prior to the season and may host them prior to special events during the season like all-star games and championship games.
Hilbert's second problem	Hilbert's second problem	In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that the arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in Hilbert (1900), which include a second order completeness axiom. In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem. Some feel that Gödel's theorems give a negative solution to the problem, while others consider Gentzen's proof as a partial positive solution.
Hilbert's second problem	Hilbert's problem and its interpretation	In one English translation, Hilbert asks: "When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. ... But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results. In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. ... On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms." Hilbert's statement is sometimes misunderstood, because by the "arithmetical axioms" he did not mean a system equivalent to Peano arithmetic, but a stronger system with a second-order completeness axiom. The system Hilbert asked for a completeness proof of is more like second-order arithmetic than first-order Peano arithmetic.
Hilbert's second problem	Hilbert's problem and its interpretation	As a nowadays common interpretation, a positive solution to Hilbert's second question would in particular provide a proof that Peano arithmetic is consistent.
Hilbert's second problem	Hilbert's problem and its interpretation	There are many known proofs that Peano arithmetic is consistent that can be carried out in strong systems such as Zermelo–Fraenkel set theory. These do not provide a resolution to Hilbert's second question, however, because someone who doubts the consistency of Peano arithmetic is unlikely to accept the axioms of set theory (which is much stronger) to prove its consistency. Thus a satisfactory answer to Hilbert's problem must be carried out using principles that would be acceptable to someone who does not already believe PA is consistent. Such principles are often called finitistic because they are completely constructive and do not presuppose a completed infinity of natural numbers. Gödel's second incompleteness theorem (see Gödel's incompleteness theorems) places a severe limit on how weak a finitistic system can be while still proving the consistency of Peano arithmetic.
Hilbert's second problem	Gödel's incompleteness theorem	Gödel's second incompleteness theorem shows that it is not possible for any proof that Peano Arithmetic is consistent to be carried out within Peano arithmetic itself. This theorem shows that if the only acceptable proof procedures are those that can be formalized within arithmetic then Hilbert's call for a consistency proof cannot be answered. However, as Nagel & Newman (1958) explain, there is still room for a proof that cannot be formalized in arithmetic: "This imposing result of Godel's analysis should not be misunderstood: it does not exclude a meta-mathematical proof of the consistency of arithmetic. What it excludes is a proof of consistency that can be mirrored by the formal deductions of arithmetic. Meta-mathematical proofs of the consistency of arithmetic have, in fact, been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936, and by others since then. ... But these meta-mathematical proofs cannot be represented within the arithmetical calculus; and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program. ... The possibility of constructing a finitistic absolute proof of consistency for arithmetic is not excluded by Gödel’s results. Gödel showed that no such proof is possible that can be represented within arithmetic. His argument does not eliminate the possibility of strictly finitistic proofs that cannot be represented within arithmetic. But no one today appears to have a clear idea of what a finitistic proof would be like that is not capable of formulation within arithmetic."
Hilbert's second problem	Gentzen's consistency proof	In 1936, Gentzen published a proof that Peano Arithmetic is consistent. Gentzen's result shows that a consistency proof can be obtained in a system that is much weaker than set theory.
Hilbert's second problem	Gentzen's consistency proof	Gentzen's proof proceeds by assigning to each proof in Peano arithmetic an ordinal number, based on the structure of the proof, with each of these ordinals less than ε0. He then proves by transfinite induction on these ordinals that no proof can conclude in a contradiction. The method used in this proof can also be used to prove a cut elimination result for Peano arithmetic in a stronger logic than first-order logic, but the consistency proof itself can be carried out in ordinary first-order logic using the axioms of primitive recursive arithmetic and a transfinite induction principle. Tait (2005) gives a game-theoretic interpretation of Gentzen's method.
Hilbert's second problem	Gentzen's consistency proof	Gentzen's consistency proof initiated the program of ordinal analysis in proof theory. In this program, formal theories of arithmetic or set theory are assigned ordinal numbers that measure the consistency strength of the theories. A theory will be unable to prove the consistency of another theory with a higher proof theoretic ordinal.
Hilbert's second problem	Modern viewpoints on the status of the problem	While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether (or in what way) these theorems answer Hilbert's second problem. Simpson (1988) argues that Gödel's incompleteness theorem shows that it is not possible to produce finitistic consistency proofs of strong theories. Kreisel (1976) states that although Gödel's results imply that no finitistic syntactic consistency proof can be obtained, semantic (in particular, second-order) arguments can be used to give convincing consistency proofs. Detlefsen (1990) argues that Gödel's theorem does not prevent a consistency proof because its hypotheses might not apply to all the systems in which a consistency proof could be carried out. Dawson (2006) calls the belief that Gödel's theorem eliminates the possibility of a persuasive consistency proof "erroneous", citing the consistency proof given by Gentzen and a later one given by Gödel in 1958.
Kripke–Platek set theory with urelements	Kripke–Platek set theory with urelements	The Kripke–Platek set theory with urelements (KPU) is an axiom system for set theory with urelements, based on the traditional (urelement-free) Kripke–Platek set theory. It is considerably weaker than the (relatively) familiar system ZFU. The purpose of allowing urelements is to allow large or high-complexity objects (such as the set of all reals) to be included in the theory's transitive models without disrupting the usual well-ordering and recursion-theoretic properties of the constructible universe; KP is so weak that this is hard to do by traditional means.
Kripke–Platek set theory with urelements	Preliminaries	The usual way of stating the axioms presumes a two sorted first order language L∗ with a single binary relation symbol ∈ Letters of the sort p,q,r,... designate urelements, of which there may be none, whereas letters of the sort a,b,c,... designate sets. The letters x,y,z,... may denote both sets and urelements.
Kripke–Platek set theory with urelements	Preliminaries	The letters for sets may appear on both sides of ∈ , while those for urelements may only appear on the left, i.e. the following are examples of valid expressions: p∈a , b∈a The statement of the axioms also requires reference to a certain collection of formulae called Δ0 -formulae. The collection Δ0 consists of those formulae that can be built using the constants, ∈ , ¬ , ∧ , ∨ , and bounded quantification. That is quantification of the form ∀x∈a or ∃x∈a where a is given set.
Kripke–Platek set theory with urelements	Axioms	The axioms of KPU are the universal closures of the following formulae: Extensionality: ∀x(x∈a↔x∈b)→a=b Foundation: This is an axiom schema where for every formula ϕ(x) we have ∃a.ϕ(a)→∃a(ϕ(a)∧∀x∈a(¬ϕ(x))) Pairing: ∃a(x∈a∧y∈a) Union: ∃a∀c∈b.∀y∈c(y∈a) Δ0-Separation: This is again an axiom schema, where for every Δ0 -formula ϕ(x) we have the following ∃a∀x(x∈a↔x∈b∧ϕ(x)) Δ0-SCollection: This is also an axiom schema, for every Δ0 -formula ϕ(x,y) we have ∀x∈a.∃y.ϕ(x,y)→∃b∀x∈a.∃y∈b.ϕ(x,y) Set Existence: ∃a(a=a) Additional assumptions Technically these are axioms that describe the partition of objects into sets and urelements.
Kripke–Platek set theory with urelements	Applications	KPU can be applied to the model theory of infinitary languages. Models of KPU considered as sets inside a maximal universe that are transitive as such are called admissible sets.
Tritium Systems Test Assembly	Tritium Systems Test Assembly	The Tritium Systems Test Assembly (TSTA) was a facility at Los Alamos National Laboratory dedicated to the development and demonstration of technologies required for fusion-relevant deuterium-tritium processing. Facility design was launched in 1977. It was commissioned in 1982, and the first tritium was processed in 1984. The maximum tritium inventory was 140 grams.
Natural logarithm	Natural logarithm	The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
Natural logarithm	Natural logarithm	The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln 7.5 is 2.0149..., because e2.0149... = 7.5. The natural logarithm of e itself, ln e, is 1, because e1 = e, while the natural logarithm of 1 is 0, since e0 = 1.
Natural logarithm	Natural logarithm	The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a (with the area being negative when 0 < a < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see complex logarithm for more.
Natural logarithm	Natural logarithm	The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities: ln if ln if x∈R Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition: ln ln ln ⁡y.
Natural logarithm	Natural logarithm	Logarithms can be defined for any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter, log ln ln ln log b⁡e Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and scientific disciplines, and are used to solve problems involving compound interest.
Natural logarithm	History	The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649. Their work involved quadrature of the hyperbola with equation xy = 1, by determination of the area of hyperbolic sectors. Their solution generated the requisite "hyperbolic logarithm" function, which had the properties now associated with the natural logarithm.
Natural logarithm	History	An early mention of the natural logarithm was by Nicholas Mercator in his work Logarithmotechnia, published in 1668, although the mathematics teacher John Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619. It has been said that Speidell's logarithms were to the base e, but this is not entirely true due to complications with the values being expressed as integers.: 152
Natural logarithm	Notational conventions	The notations ln x and loge x both refer unambiguously to the natural logarithm of x, and log x without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many programming languages. In some other contexts such as chemistry, however, log x can be used to denote the common (base 10) logarithm. It may also refer to the binary (base 2) logarithm in the context of computer science, particularly in the context of time complexity.
Natural logarithm	Definitions	The natural logarithm can be defined in several equivalent ways.
Natural logarithm	Definitions	Inverse of exponential The most general definition is as the inverse function of ex , so that ln ⁡(x)=x . Because ex is positive and invertible for any real input x , this definition of ln ⁡(x) is well defined for any positive x. For the complex numbers, ez is not invertible, so ln ⁡(z) is a multivalued function. In order to make ln ⁡(z) a proper, single-output function, we therefore need to restrict it to a particular principal branch, often denoted by Ln ⁡(z) . As the inverse function of ez , ln ⁡(z) can be defined by inverting the usual definition of ez lim n→∞(1+zn)n Doing so yields: ln lim n→∞n⋅(zn−1) This definition therefore derives its own principal branch from the principal branch of nth roots.
Natural logarithm	Definitions	Integral definition The natural logarithm of a positive, real number a may be defined as the area under the graph of the hyperbola with equation y = 1/x between x = 1 and x = a. This is the integral ln ⁡a=∫1a1xdx. If a is in (0,1) then the region has negative area and the logarithm is negative. This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm: ln ln ln ⁡b. This can be demonstrated by splitting the integral that defines ln ab into two parts, and then making the variable substitution x = at (so dx = a dt) in the second part, as follows: ln ln ln ⁡b. In elementary terms, this is simply scaling by 1/a in the horizontal direction and by a in the vertical direction. Area does not change under this transformation, but the region between a and ab is reconfigured. Because the function a/(ax) is equal to the function 1/x, the resulting area is precisely ln b. The number e can then be defined to be the unique real number a such that ln a = 1. The natural logarithm also has an improper integral representation, which can be derived with Fubini's theorem as follows: ln ⁡(x)=∫1x1udu=∫1x∫0∞e−tudtdu=∫0∞∫1xe−tududt=∫0∞e−t−e−txtdt
Natural logarithm	Properties	The natural logarithm has the following mathematical properties: ln ⁡1=0 ln ⁡e=1 ln ln ln for and y>0 ln ln ln ⁡y ln ln for x>0 ln ln for 0<x<y lim ln ⁡(1+x)x=1 lim ln for x>0 ln for x>0 ln for and α≥1
Natural logarithm	Derivative	The derivative of the natural logarithm as a real-valued function on the positive reals is given by ln ⁡x=1x. How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral ln ⁡x=∫1x1tdt, then the derivative immediately follows from the first part of the fundamental theorem of calculus. On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for x > 0) can be found by using the properties of the logarithm and a definition of the exponential function. From the definition of the number lim u→0(1+u)1/u, the exponential function can be defined as lim lim h→0(1+hx)1/h , where u=hx,h=ux. The derivative can then be found from first principles. ln lim ln ln lim ln lim ln all above for logarithmic properties ln lim for continuity of the logarithm ln for the definition of lim for the definition of the ln as inverse function. Also, we have: ln ln ln ln ln ⁡x=1x. so, unlike its inverse function eax , a constant in the function doesn't alter the differential.
Natural logarithm	Series	Since the natural logarithm is undefined at 0, ln ⁡(x) itself does not have a Maclaurin series, unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if and x≠0, then ln ⁡x=∫1x1tdt=∫0x−111+udu=∫0x−1(1−u+u2−u3+⋯)du=(x−1)−(x−1)22+(x−1)33−(x−1)44+⋯=∑k=1∞(−1)k−1(x−1)kk. This is the Taylor series for ln x around 1. A change of variables yields the Mercator series: ln ⁡(1+x)=∑k=1∞(−1)k−1kxk=x−x22+x33−⋯, valid for \|x\| ≤ 1 and x ≠ −1.
Natural logarithm	Series	Leonhard Euler, disregarding x≠−1 , nevertheless applied this series to x = −1 to show that the harmonic series equals the natural logarithm of 1/(1 − 1), that is, the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated at N is close to the logarithm of N, when N is large, with the difference converging to the Euler–Mascheroni constant.
Natural logarithm	Series	The figure is a graph of ln(1 + x) and some of its Taylor polynomials around 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside this region, the higher-degree Taylor polynomials devolve to worse approximations for the function. A useful special case for positive integers n, taking x=1n , is: ln ⁡(n+1n)=∑k=1∞(−1)k−1knk=1n−12n2+13n3−14n4+⋯ If Re ⁡(x)≥1/2, then ln ln ⁡(1x)=−∑k=1∞(−1)k−1(1x−1)kk=∑k=1∞(x−1)kkxk=x−1x+(x−1)22x2+(x−1)33x3+(x−1)44x4+⋯ Now, taking x=n+1n for positive integers n, we get: ln ⁡(n+1n)=∑k=1∞1k(n+1)k=1n+1+12(n+1)2+13(n+1)3+14(n+1)4+⋯ If Re and x≠0, then ln ln ln ln ln ⁡(1−x−1x+1). Since ln ln ⁡(1−y)=∑i=1∞1i((−1)i−1yi−(−1)i−1(−y)i)=∑i=1∞yii((−1)i−1+1)=y∑i=1∞yi−1i((−1)i−1+1)=i−1→2k2y∑k=0∞y2k2k+1, we arrive at ln ⁡(x)=2(x−1)x+1∑k=0∞12k+1((x−1)2(x+1)2)k=2(x−1)x+1(11+13(x−1)2(x+1)2+15((x−1)2(x+1)2)2+⋯). Using the substitution x=n+1n again for positive integers n, we get: ln ⁡(n+1n)=22n+1∑k=0∞1(2k+1)((2n+1)2)k=2(12n+1+13(2n+1)3+15(2n+1)5+⋯). This is, by far, the fastest converging of the series described here. The natural logarithm can also be expressed as an infinite product: ln ⁡(x)=(x−1)∏k=1∞(21+x2k) Two examples might be: ln 16 )... 16 )... From this identity, we can easily get that: ln ⁡(x)=xx−1−∑k=1∞2−kx2−k1+x2−k For example: ln ⁡(2)=2−22+22−244+424−288+828⋯
Natural logarithm	The natural logarithm in integration	The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x): an antiderivative of g(x) is given by ln(\|f(x)\|). This is the case because of the chain rule and the following fact: ln ⁡\|x\|=1x,x≠0 In other words, when integrating over an interval of the real line that does not include x=0 then ln ⁡\|x\|+C where C is an arbitrary constant of integration.
Natural logarithm	The natural logarithm in integration	Likewise, when the integral is over an interval where f(x)≠0 , ln ⁡\|f(x)\|+C. For example, consider the integral of tan(x) over an interval that does not include points where tan(x) is infinite: tan sin cos cos cos ln cos ln sec ⁡x\|+C. The natural logarithm can be integrated using integration by parts: ln ln ⁡x−x+C. Let: ln ⁡x⇒du=dxx dv=dx⇒v=x then: ln ln ln ln ⁡x−x+C
Natural logarithm	Efficient computation	For ln(x) where x > 1, the closer the value of x is to 1, the faster the rate of convergence of its Taylor series centered at 1. The identities associated with the logarithm can be leveraged to exploit this: ln 123.456 ln 1.23456 10 ln 1.23456 ln 10 ln 1.23456 ln 10 ln 1.23456 2.3025851. Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above. Natural logarithm of 10 The natural logarithm of 10, which has the decimal expansion 2.30258509..., plays a role for example in the computation of natural logarithms of numbers represented in scientific notation, as a mantissa multiplied by a power of 10: ln 10 ln ln 10. This means that one can effectively calculate the logarithms of numbers with very large or very small magnitude using the logarithms of a relatively small set of decimals in the range [1, 10).
Natural logarithm	Efficient computation	High precision To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. Especially if x is near 1, a good alternative is to use Halley's method or Newton's method to invert the exponential function, because the series of the exponential function converges more quickly. For finding the value of y to give exp(y) − x = 0 using Halley's method, or equivalently to give exp(y/2) − x exp(−y/2) = 0 using Newton's method, the iteration simplifies to exp exp ⁡(yn) which has cubic convergence to ln(x).
Natural logarithm	Efficient computation	Another alternative for extremely high precision calculation is the formula ln ln ⁡2, where M denotes the arithmetic-geometric mean of 1 and 4/s, and s=x2m>2p/2, with m chosen so that p bits of precision is attained. (For most purposes, the value of 8 for m is sufficient.) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants ln 2 and π can be pre-computed to the desired precision using any of several known quickly converging series.) Or, the following formula can be used: ln ⁡x=πM(θ22(1/x),θ32(1/x)),x∈(1,∞) where θ2(x)=∑n∈Zx(n+1/2)2,θ3(x)=∑n∈Zxn2 are the Jacobi theta functions.Based on a proposal by William Kahan and first implemented in the Hewlett-Packard HP-41C calculator in 1979 (referred to under "LN1" in the display, only), some calculators, operating systems (for example Berkeley UNIX 4.3BSD), computer algebra systems and programming languages (for example C99) provide a special natural logarithm plus 1 function, alternatively named LNP1, or log1p to give more accurate results for logarithms close to zero by passing arguments x, also close to zero, to a function log1p(x), which returns the value ln(1+x), instead of passing a value y close to 1 to a function returning ln(y). The function log1p avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the ln. This keeps the argument, the result, and intermediate steps all close to zero where they can be most accurately represented as floating-point numbers.In addition to base e the IEEE 754-2008 standard defines similar logarithmic functions near 1 for binary and decimal logarithms: log2(1 + x) and log10(1 + x).
Natural logarithm	Efficient computation	Similar inverse functions named "expm1", "expm" or "exp1m" exist as well, all with the meaning of expm1(x) = exp(x) − 1.An identity in terms of the inverse hyperbolic tangent, log ⁡(1+x)=2artanh(x2+x), gives a high precision value for small values of x on systems that do not implement log1p(x). Computational complexity The computational complexity of computing the natural logarithm using the arithmetic-geometric mean (for both of the above methods) is O(M(n) ln n). Here n is the number of digits of precision at which the natural logarithm is to be evaluated and M(n) is the computational complexity of multiplying two n-digit numbers.
Natural logarithm	Continued fractions	While no simple continued fractions are available, several generalized continued fractions are, including: ln ⁡(1+x)=x11−x22+x33−x44+x55−⋯=x1−0x+12x2−1x+22x3−2x+32x4−3x+42x5−4x+⋱ ln ⁡(1+xy)=xy+1x2+1x3y+2x2+2x5y+3x2+⋱=2x2y+x−(1x)23(2y+x)−(2x)25(2y+x)−(3x)27(2y+x)−⋱ These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed, by repeatedly adding those of smaller numbers, with similarly rapid convergence. For example, since 2 = 1.253 × 1.024, the natural logarithm of 2 can be computed as: ln ln ln 125 27 45 63 253 759 1265 1771 −⋱. Furthermore, since 10 = 1.2510 × 1.0243, even the natural logarithm of 10 can be computed similarly as: ln 10 10 ln ln 125 20 27 45 63 18 253 759 1265 1771 −⋱. The reciprocal of the natural logarithm can be also written in this way: ln ⁡(x)=2xx2−112+x2+14x12+1212+x2+14x… For example: ln ⁡(2)=4312+5812+1212+58…
Natural logarithm	Complex logarithms	The exponential function can be extended to a function which gives a complex number as ez for any arbitrary complex number z; simply use the infinite series with x=z complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no x has ex = 0; and it turns out that e2iπ = 1 = e0. Since the multiplicative property still works for the complex exponential function, ez = ez+2kiπ, for all complex z and integers k.
Natural logarithm	Complex logarithms	So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued—any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2iπ at will. The complex logarithm can only be single-valued on the cut plane. For example, ln i = iπ/2 or 5iπ/2 or -3iπ/2, etc.; and although i4 = 1, 4 ln i can be defined as 2iπ, or 10iπ or −6iπ, and so on.
Natural logarithm	Complex logarithms	Plots of the natural logarithm function on the complex plane (principal branch)
Diffusion capacitance	Diffusion capacitance	Diffusion Capacitance is the capacitance that happens due to transport of charge carriers between two terminals of a device, for example, the diffusion of carriers from anode to cathode in a forward biased diode or from emitter to baseforward-biased junction of a transistor. In a semiconductor device with a current flowing through it (for example, an ongoing transport of charge by diffusion) at a particular moment there is necessarily some charge in the process of transit through the device. If the applied voltage changes to a different value and the current changes to a different value, a different amount of charge will be in transit in the new circumstances. The change in the amount of transiting charge divided by the change in the voltage causing it is the diffusion capacitance. The adjective "diffusion" is used because the original use of this term was for junction diodes, where the charge transport was via the diffusion mechanism. See Fick's laws of diffusion.
Diffusion capacitance	Diffusion capacitance	To implement this notion quantitatively, at a particular moment in time let the voltage across the device be V . Now assume that the voltage changes with time slowly enough that at each moment the current is the same as the DC current that would flow at that voltage, say I=I(V) (the quasistatic approximation). Suppose further that the time to cross the device is the forward transit time τF . In this case the amount of charge in transit through the device at this particular moment, denoted Q , is given by Q=I(V)τF .Consequently, the corresponding diffusion capacitance: Cdiff . is Cdiff=dQdV=dI(V)dVτF .In the event the quasi-static approximation does not hold, that is, for very fast voltage changes occurring in times shorter than the transit time τF , the equations governing time-dependent transport in the device must be solved to find the charge in transit, for example the Boltzmann equation. That problem is a subject of continuing research under the topic of non-quasistatic effects. See Liu , and Gildenblat et al.
DNA digital data storage	DNA digital data storage	DNA digital data storage is the process of encoding and decoding binary data to and from synthesized strands of DNA.While DNA as a storage medium has enormous potential because of its high storage density, its practical use is currently severely limited because of its high cost and very slow read and write times.In June 2019, scientists reported that all 16 GB of text from the English Wikipedia had been encoded into synthetic DNA. In 2021, scientists reported that a custom DNA data writer had been developed that was capable of writing data into DNA at 18 Mbps.
DNA digital data storage	Encoding methods	Countless methods for encoding data in DNA are possible. The optimal methods are those that make economical use of DNA and protect against errors. If the message DNA is intended to be stored for a long period of time, for example, 1,000 years, it is also helpful if the sequence is obviously artificial and the reading frame is easy to identify.
DNA digital data storage	Encoding methods	Encoding text Several simple methods for encoding text have been proposed. Most of these involve translating each letter into a corresponding "codon", consisting of a unique small sequence of nucleotides in a lookup table. Some examples of these encoding schemes include Huffman codes, comma codes, and alternating codes.
DNA digital data storage	Encoding methods	Encoding arbitrary data To encode arbitrary data in DNA, the data is typically first converted into ternary (base 3) data rather than binary (base 2) data. Each digit (or "trit") is then converted to a nucleotide using a lookup table. To prevent homopolymers (repeating nucleotides), which can cause problems with accurate sequencing, the result of the lookup also depends on the preceding nucleotide. Using the example lookup table below, if the previous nucleotide in the sequence is T (thymine), and the trit is 2, the next nucleotide will be G (guanine).
DNA digital data storage	Encoding methods	Various systems may be incorporated to partition and address the data, as well as to protect it from errors. One approach to error correction is to regularly intersperse synchronization nucleotides between the information-encoding nucleotides. These synchronization nucleotides can act as scaffolds when reconstructing the sequence from multiple overlapping strands.
DNA digital data storage	In vivo	The genetic code within living organisms can potentially be co-opted to store information. Furthermore synthetic biology can be used to engineer cells with "molecular recorders" to allow the storage and retrieval of information stored in the cell's genetic material. CRISPR gene editing can also be used to insert artificial DNA sequences into the genome of the cell. For encoding developmental lineage data (molecular flight recorder), roughly 30 trillion cell nuclei per mouse * 60 recording sites per nucleus * 7-15 bits per site yields about 2 TeraBytes per mouse written (but only very selectively read).
DNA digital data storage	In vivo	In-vivo light-based direct image and data recording A proof-of-concept in-vivo direct DNA data recording system was demonstrated through incorporation of optogenetically regulated recombinases as part of an engineered "molecular recorder" allows for direct encoding of light-based stimuli into engineered E.coli cells. This approach can also be parallelized to store and write text or data in 8-bit form through the use of physically separated individual cell cultures in cell-culture plates.
DNA digital data storage	In vivo	This approach leverages the editing of a "recorder plasmid" by the light-regulated recombinases, allowing for identification of cell populations exposed to different stimuli. This approach allows for the physical stimulus to be directly encoded into the "recorder plasmid" through recombinase action. Unlike other approaches, this approach does not require manual design, insertion and cloning of artificial sequences to record the data into the genetic code. In this recording process, each individual cell population in each cell-culture plate culture well can be treated as a digital "bit", functioning as a biological transistor capable of recording a single bit of data.
DNA digital data storage	History	The idea of DNA digital data storage dates back to 1959, when the physicist Richard P. Feynman, in "There's Plenty of Room at the Bottom: An Invitation to Enter a New Field of Physics" outlined the general prospects for the creation of artificial objects similar to objects of the microcosm (including biological) and having similar or even more extensive capabilities. In 1964–65, Mikhail Samoilovich Neiman, the Soviet physicist, published 3 articles about microminiaturization in electronics at the molecular-atomic level, which independently presented general considerations and some calculations regarding the possibility of recording, storage, and retrieval of information on synthesized DNA and RNA molecules. After the publication of the first M.S. Neiman's paper and after receiving by Editor the manuscript of his second paper (January, the 8th, 1964, as indicated in that paper) the interview with cybernetician Norbert Wiener was published. N. Wiener expressed ideas about miniaturization of computer memory, close to the ideas, proposed by M. S. Neiman independently. These Wiener's ideas M. S. Neiman mentioned in the third of his papers. This story is described in details.One of the earliest uses of DNA storage occurred in a 1988 collaboration between artist Joe Davis and researchers from Harvard University. The image, stored in a DNA sequence in E.coli, was organized in a 5 x 7 matrix that, once decoded, formed a picture of an ancient Germanic rune representing life and the female Earth. In the matrix, ones corresponded to dark pixels while zeros corresponded to light pixels.In 2007 a device was created at the University of Arizona using addressing molecules to encode mismatch sites within a DNA strand. These mismatches were then able to be read out by performing a restriction digest, thereby recovering the data.In 2011, George Church, Sri Kosuri, and Yuan Gao carried out an experiment that would encode a 659 kb book that was co-authored by Church. To do this, the research team did a two-to-one correspondence where a binary zero was represented by either an adenine or cytosine and a binary one was represented by a guanine or thymine. After examination, 22 errors were found in the DNA.In 2012, George Church and colleagues at Harvard University published an article in which DNA was encoded with digital information that included an HTML draft of a 53,400 word book written by the lead researcher, eleven JPEG images and one JavaScript program. Multiple copies for redundancy were added and 5.5 petabits can be stored in each cubic millimeter of DNA. The researchers used a simple code where bits were mapped one-to-one with bases, which had the shortcoming that it led to long runs of the same base, the sequencing of which is error-prone. This result showed that besides its other functions, DNA can also be another type of storage medium such as hard disk drives and magnetic tapes.In 2013, an article led by researchers from the European Bioinformatics Institute (EBI) and submitted at around the same time as the paper of Church and colleagues detailed the storage, retrieval, and reproduction of over five million bits of data. All the DNA files reproduced the information with an accuracy between 99.99% and 100%. The main innovations in this research were the use of an error-correcting encoding scheme to ensure the extremely low data-loss rate, as well as the idea of encoding the data in a series of overlapping short oligonucleotides identifiable through a sequence-based indexing scheme. Also, the sequences of the individual strands of DNA overlapped in such a way that each region of data was repeated four times to avoid errors. Two of these four strands were constructed backwards, also with the goal of eliminating errors. The costs per megabyte were estimated at $12,400 to encode data and $220 for retrieval. However, it was noted that the exponential decrease in DNA synthesis and sequencing costs, if it continues into the future, should make the technology cost-effective for long-term data storage by 2023.In 2013, a software called DNACloud was developed by Manish K. Gupta and co-workers to encode computer files to their DNA representation. It implements a memory efficiency version of the algorithm proposed by Goldman et al. to encode (and decode) data to DNA (.dnac files).The long-term stability of data encoded in DNA was reported in February 2015, in an article by researchers from ETH Zurich. The team added redundancy via Reed–Solomon error correction coding and by encapsulating the DNA within silica glass spheres via Sol-gel chemistry.In 2016 research by Church and Technicolor Research and Innovation was published in which, 22 MB of a MPEG compressed movie sequence were stored and recovered from DNA. The recovery of the sequence was found to have zero errors.In March 2017, Yaniv Erlich and Dina Zielinski of Columbia University and the New York Genome Center published a method known as DNA Fountain that stored data at a density of 215 petabytes per gram of DNA. The technique approaches the Shannon capacity of DNA storage, achieving 85% of the theoretical limit. The method was not ready for large-scale use, as it costs $7000 to synthesize 2 megabytes of data and another $2000 to read it.In March 2018, University of Washington and Microsoft published results demonstrating storage and retrieval of approximately 200MB of data. The research also proposed and evaluated a method for random access of data items stored in DNA. In March 2019, the same team announced they have demonstrated a fully automated system to encode and decode data in DNA.Research published by Eurecom and Imperial College in January 2019, demonstrated the ability to store structured data in synthetic DNA. The research showed how to encode structured or, more specifically, relational data in synthetic DNA and also demonstrated how to perform data processing operations (similar to SQL) directly on the DNA as chemical processes.In April 2019, due to a collaboration with TurboBeads Labs in Switzerland, Mezzanine by Massive Attack was encoded into synthetic DNA, making it the first album to be stored in this way.In June 2019, scientists reported that all 16 GB of Wikipedia have been encoded into synthetic DNA. In 2021, CATALOG reported that they had developed a custom DNA writer capable of writing data at 18 Mbps into DNA.The first article describing data storage on native DNA sequences via enzymatic nicking was published in April 2020. In the paper, scientists demonstrate a new method of recording information in DNA backbone which enables bit-wise random access and in-memory computing.
DNA digital data storage	Davos Bitcoin Challenge	On January 21, 2015, Nick Goldman from the European Bioinformatics Institute (EBI), one of the original authors of the 2013 Nature paper, announced the Davos Bitcoin Challenge at the World Economic Forum annual meeting in Davos. During his presentation, DNA tubes were handed out to the audience, with the message that each tube contained the private key of exactly one bitcoin, all coded in DNA. The first one to sequence and decode the DNA could claim the bitcoin and win the challenge. The challenge was set for three years and would close if nobody claimed the prize before January 21, 2018.Almost three years later on January 19, 2018, the EBI announced that a Belgian PhD student, Sander Wuyts, of the University of Antwerp and Vrije Universiteit Brussel, was the first one to complete the challenge. Next to the instructions on how to claim the bitcoin (stored as a plain text and PDF file), the logo of the EBI, the logo of the company that printed the DNA (CustomArray), and a sketch of James Joyce were retrieved from the DNA.
DNA digital data storage	The Lunar Library	The Lunar Library, launched on the Beresheet Lander by the Arch Mission Foundation, carries information encoded in DNA, which includes 20 famous books and 10,000 images. This was one of the optimal choices of storage, as DNA can last a long time. The Arch Mission Foundation suggests that it can still be read after billions of years.
DNA digital data storage	DNA of things	The concept of the DNA of Things (DoT) was introduced in 2019 by a team of researchers from Israel and Switzerland, including Yaniv Erlich and Robert Grass. DoT encodes digital data into DNA molecules, which are then embedded into objects. This gives the ability to create objects that carry their own blueprint, similar to biological organisms. In contrast to Internet of things, which is a system of interrelated computing devices, DoT creates objects which are independent storage objects, completely off-grid.
DNA digital data storage	DNA of things	As a proof of concept for DoT, the researcher 3D-printed a Stanford bunny which contains its blueprint in the plastic filament used for printing. By clipping off a tiny bit of the ear of the bunny, they were able to read out the blueprint, multiply it and produce a next generation of bunnies. In addition, the ability of DoT to serve for steganographic purposes was shown by producing non-distinguishable lenses which contain a YouTube video integrated into the material.
Ectrodactyly	Ectrodactyly	Ectrodactyly, split hand, or cleft hand (derived from Greek ektroma "miscarriage" and daktylos "finger") involves the deficiency or absence of one or more central digits of the hand or foot and is also known as split hand/split foot malformation (SHFM). The hands and feet of people with ectrodactyly (ectrodactyls) are often described as "claw-like" and may include only the thumb and one finger (usually either the little finger, ring finger, or a syndactyly of the two) with similar abnormalities of the feet.It is a substantial rare form of a congenital disorder in which the development of the hand is disturbed. It is a type I failure of formation – longitudinal arrest. The central ray of the hand is affected and usually appears without proximal deficiencies of nerves, vessels, tendons, muscles and bones in contrast to the radial and ulnar deficiencies. The cleft hand appears as a V-shaped cleft situated in the centre of the hand. The digits at the borders of the cleft might be syndactilyzed, and one or more digits can be absent. In most types, the thumb, ring finger and little finger are the less affected parts of the hand. The incidence of cleft hand varies from 1 in 90,000 to 1 in 10,000 births depending on the used classification. Cleft hand can appear unilateral or bilateral, and can appear isolated or associated with a syndrome.
Ectrodactyly	Ectrodactyly	Split hand/foot malformation (SHFM) is characterized by underdeveloped or absent central digital rays, clefts of hands and feet, and variable syndactyly of the remaining digits. SHFM is a heterogeneous condition caused by abnormalities at one of multiple loci, including SHFM1 (SHFM1 at 7q21-q22), SHFM2 (Xq26), SHFM3 (FBXW4/DACTYLIN at 10q24), SHFM4 (TP63 at 3q27), and SHFM5 (DLX1 and DLX 2 at 2q31). SHFM3 is unique in that it is caused by submicroscopic tandem chromosome duplications of FBXW4/DACTYLIN. SHFM3 is considered 'isolated' ectrodactyly and does not show a mutation of the tp63 gene.
Ectrodactyly	Presentation	Ectrodactyly can be caused by various changes to 7q. When 7q is altered by a deletion or a translocation, ectrodactyly can sometimes be associated with hearing loss. Ectrodactyly, or Split hand/split foot malformation (SHFM) type 1 is the only form of split hand/ malformation associated with sensorineural hearing loss.
Ectrodactyly	Genetics	A large number of human gene defects can cause ectrodactyly. The most common mode of inheritance is autosomal dominant with reduced penetrance, while autosomal recessive and X-linked forms occur more rarely. Ectrodactyly can also be caused by a duplication on 10q24. Detailed studies of a number of mouse models for ectrodactyly have also revealed that a failure to maintain median apical ectodermal ridge (AER) signalling can be the main pathogenic mechanism in triggering this abnormality.A number of factors make the identification of the genetic defects underlying human ectrodactyly a complicated process: the limited number of families linked to each split hand/foot malformation (SHFM) locus, the large number of morphogens involved in limb development, the complex interactions between these morphogens, the involvement of modifier genes, and the presumed involvement of multiple gene or long-range regulatory elements in some cases of ectrodactyly. In the clinical setting these genetic characteristics can become problematic and making predictions of carrier status and severity of the disease impossible to predict.In 2011, a novel mutation in DLX5 was found to be involved in SHFM.Ectrodactyly is frequently seen with other congenital anomalies. Syndromes in which ectrodactyly is associated with other abnormalities can occur when two or more genes are affected by a chromosomal rearrangement. Disorders associated with ectrodactyly include Ectrodactyly-Ectodermal Dysplasia-Clefting (EEC) syndrome, which is closely correlated to the ADULT syndrome and Limb-mammary (LMS) syndrome, Ectrodactyly-Cleft Palate (ECP) syndrome, Ectrodactyly-Ectodermal Dysplasia-Macular Dystrophy syndrome, Ectrodactyly-Fibular Aplasia/Hypoplasia (EFA) syndrome, and Ectrodactyly-Polydactyly. More than 50 syndromes and associations involving ectrodactyly are distinguished in the London Dysmorphology Database.
Ectrodactyly	Pathophysiology	The pathophysiology of cleft hand is thought to be a result of a wedge-shaped defect of the apical ectoderm of the limb bud (AER: apical ectodermal ridge). Polydactyly, syndactyly and cleft hand can occur within the same hand, therefore some investigators suggest that these entities occur from the same mechanism. This mechanism is not yet defined.
Ectrodactyly	Pathophysiology	Genetics The cause of cleft hand lies, for what is known, partly in genetics. The inheritance of cleft hand is autosomal dominant and has a variable penetrance of 70%. Cleft hand can be a spontaneous mutation during pregnancy (de novo mutation). The exact chromosomal defect in isolated cleft hand is not yet defined. However, the genetic causes of cleft hand related to syndromes have more clarity.
Ectrodactyly	Pathophysiology	The identified mutation for SHSF syndrome (split-hand/split-foot syndrome) a duplication on 10q24, and not a mutation of the tp63 gene as in families affected by EEC syndrome (ectrodactyly–ectodermal dysplasia–cleft syndrome). The p63 gene plays a critical role in the development of the apical ectodermal ridge (AER), this was found in mutant mice with dactylaplasia. Embryology Some studies have postulated that polydactyly, syndactyly and cleft hand have the same teratogenic mechanism. In vivo tests showed that limb anomalies were found alone or in combination with cleft hand when they were given Myleran. These anomalies take place in humans around day 41 of gestation.
Ectrodactyly	Diagnosis	Classification There are several classifications for cleft hand, but the most used classification is described by Manske and Halikis see table 3. This classification is based on the first web space. The first web space is the space between the thumb and the index finger. Table 3: Classification for cleft hand described by Manske and Halikis
Ectrodactyly	Treatment	The treatment of cleft hand is usually invasive and can differ each time because of the heterogeneity of the condition. The function of a cleft hand is mostly not restricted, yet improving the function is one of the goals when the thumb or first webspace is absent.The social and stigmatising aspects of a cleft hand require more attention. The hand is a part of the body which is usually shown during communication. When this hand is obviously different and deformed, stigmatisation or rejection can occur. Sometimes, in families with cleft hand with good function, operations for cosmetic aspects are considered marginal and the families choose not to have surgery.
Ectrodactyly	Treatment	Indications Surgical treatment of the cleft hand is based on several indications: Improving function Absent thumb Deforming syndactyly (mostly between digits of unequal length like index and thumb) Transverse bones (this will progress the deformity; growth of these bones will widen the cleft) Narrowed first webspace The feetAesthetical aspects Reducing deformity Timing of surgical interventions The timing of surgical interventions is debatable. Parents have to decide about their child in a very vulnerable time of their parenthood. Indications for early treatment are progressive deformities, such as syndactyly between index and thumb or transverse bones between the digital rays. Other surgical interventions are less urgent and can wait for 1 or 2 years.
Ectrodactyly	Treatment	Classification and treatment When surgery is indicated, the choice of treatment is based on the classification. Table 4 shows the treatment of cleft hand divided into the classification of Manske and Halikis.
Ectrodactyly	Treatment	Techniques described by Ueba, Miura and Komada and the procedure of Snow-Littler are guidelines; since clinical and anatomical presentation within the types differ, the actual treatment is based on the individual abnormality.Table 4: Treatment based on the classification of Manske and Halikis Snow-Littler The goal of this procedure is to create a wide first web space and to minimise the cleft in the hand. The index digit will be transferred to the ulnar side of the cleft. Simultaneously a correction of index malrotation and deviation is performed. To minimise the cleft, it is necessary to fix together the metacarpals which used to border the cleft. Through repositioning flaps, the wound can be closed.
Ectrodactyly	Treatment	Ueba Ueba described a less complicated surgery. Transverse flaps are used to resurface the palm, the dorsal side of the transposed digit and the ulnar part of the first web space. A tendon graft is used to connect the common extensor tendons of the border digits of the cleft to prevent digital separation during extension. The closure is simpler, but has cosmetic disadvantage because of the switch between palmar and dorsal skin.
Ectrodactyly	Treatment	Miura and Komada The release of the first webspace has the same principle as the Snow-Littler procedure. The difference is the closure of the first webspace; this is done by simple closure or closure with Z-plasties.
Ectrodactyly	History	Literature shows that cleft hand is described centuries ago. In City of God (426 A.D.), St. Augustine remarks: At Hippo-Diarrhytus there is a man whose hands are crescent-shaped, and have only two fingers each, and his feet similarly formed.
Ectrodactyly	History	The first modern reference to what might be considered a cleft hand was by Ambroise Paré in 1575. Hartsink (1770) wrote the first report of true cleft hand. In 1896, the first operation of the cleft hand was performed by Doctor Charles N. Dowed of New York City. However, the first certain description of what we know as a cleft hand as we know it today was described at the end of the 19th century.
Ectrodactyly	History	Symbrachydactyly Historically, a U-type cleft hand was also known as atypical cleft hand. The classification in which typical and atypical cleft hand are described was mostly used for clinical aspects and is shown in table 1. Nowadays, this "atypical cleft hand" is referred to as symbrachydactyly and is not a subtype of cleft hand.
Ectrodactyly	Notable cases	Bree Walker Once a popular television anchor woman in Los Angeles, she has appeared in the television drama Nip/Tuck as an inspirational character who battles her disease and counsels another family who have children with ectrodactyly Grady Stiles Sr. and Grady Stiles Jr.: known publicly as Lobster Boy and family, famous side show acts, featured on the AMC reality show, Freakshow.
Ectrodactyly	Notable cases	The Vadoma tribe in northern Zimbabwe Mikhail Tal, Soviet chess player, World Chess Champion 1960–61 Lee Hee-ah, a Korean pianist with only two fingers on each hand. Cédric Grégoire (better known as Lord Lokhraed) is the guitarist and lead vocalist of French black metal band Nocturnal Depression and has ectrodactyly on his fretting hand, which has only two fingers. Black Scorpion, freak show performer. Sam Schröder, 2020 US Open Quad Champion. Francesca Jones, British pro tennis player, former #149 in WTA rankings.
Ectrodactyly	Other animals	Ectrodactyly is not only a genetic characteristic in humans, but can also occur in frogs and toads, mice, salamanders, cows, chickens, rabbits, marmosets, cats and dogs, and even West Indian manatees. The following examples are studies showing the natural occurrence of ectrodactyly in animals, without the disease being reproduced and tested in a laboratory. In all three examples we see how rare the actual occurrence of ectrodactyly is.
Ectrodactyly	Other animals	Wood frog The Department of Biological Sciences at the University of Alberta in Edmonton, Alberta performed a study to estimate deformity levels in wood frogs in areas of relatively low disturbance. After roughly 22,733 individuals were examined during field studies, it was found that only 49 wood frogs had the ectrodactyly deformity.
Ectrodactyly	Other animals	Salamanders In a study performed by the Department of Forestry and Natural Resources at Purdue University, approximately 2000 salamanders (687 adults and 1259 larvae) were captured from a large wetland complex and evaluated for malformations. Among the 687 adults, 54 (7.9%) were malformed. Of these 54 adults, 46 (85%) had missing (ectrodactyly), extra (polyphalangy) or dwarfed digits (brachydactyly). Among the 1259 larvae, 102 were malformed, with 94 (92%) of the malformations involving ectrodactyly, polyphalangy, and brachydactyly. Results showed few differences in the frequency of malformations among life-history changes, suggesting that malformed larvae do not have substantially higher mortality than their adult conspecifics.
Ectrodactyly	Other animals	Cats and dogs Davis and Barry 1977 tested allele frequencies in domestic cats. Among the 265 cats observed, there were 101 males and 164 females. Only one cat was recorded to have the ectrodactyly abnormality, illustrating this rare disease. According to M.P. Ferreira, a case of ectrodactyly was found in a two-month-old male mixed Terrier dog. In another study, Carrig and co-workers also reported a series of 14 dogs with this abnormality proving that although ectrodactyly is an uncommon occurrence for dogs, it is not entirely unheard of.
Sequential auction	Sequential auction	A sequential auction is an auction in which several items are sold, one after the other, to the same group of potential buyers. In a sequential first-price auction (SAFP), each individual item is sold using a first price auction, while in a sequential second-price auction (SASP), each individual item is sold using a second price auction.
Sequential auction	Sequential auction	A sequential auction differs from a combinatorial auction, in which many items are auctioned simultaneously and the agents can bid on bundles of items. A sequential auction is much simpler to implement and more common in practice. However, the bidders in each auction know that there are going to be future auctions, and this may affect their strategic considerations. Here are some examples.
Sequential auction	Sequential auction	Example 1. There are two items for sale and two potential buyers: Alice and Bob, with the following valuations: Alice values each item as 5, and both items as 10 (i.e., her valuation is additive).
Sequential auction	Sequential auction	Bob values each item as 4, and both items as 4 (i.e., his valuation is unit demand).In a SASP, each item is put to a second-price-auction. Usually, such auction is a truthful mechanism, so if each item is sold in isolation, Alice wins both items and pays 4 for each item, her total payment is 4+4=8 and her net utility is 5 + 5 − 8 = 2. But, if Alice knows Bob's valuations, she has a better strategy: she can let Bob win the first item (e.g. by bidding 0). Then, Bob will not participate in the second auction at all, so Alice will win the second item and pay 0, and her net utility will be 5 − 0 = 5.
Sequential auction	Sequential auction	A similar outcome happens in a SAFP. If each item is sold in isolation, there is a Nash equilibrium in which Alice bids slightly above 4 and wins, and her net utility is slightly below 2. But, if Alice knows Bob's valuations, she can deviate to a strategy that lets Bob win in the first round so that in the second round she can win for a price slightly above 0.
Sequential auction	Sequential auction	Example 2. Multiple identical objects are auctioned, and the agents have budget constraints. It may be advantageous for a bidder to bid aggressively on one object with a view to raising the price paid by his rival and depleting his budget so that the second object may then be obtained at a lower price. In effect, a bidder may wish to “raise a rival’s costs” in one market in order to gain advantage in another. Such considerations seem to have played a significant role in the auctions for radio spectrum licenses conducted by the Federal Communications Commission. Assessment of rival bidders’ budget constraints was a primary component of the pre-bidding preparation of GTE’s bidding team.
Sequential auction	Nash equilibrium	A sequential auction is a special case of a sequential game. A natural question to ask for such a game is when there exists a subgame perfect equilibrium in pure strategies (SPEPS). When the players have full information (i.e., they know the sequence of auctions in advance), and a single item is sold in each round, a SAFP always has a SPEPS, regardless of the players' valuations. The proof is by backward induction:: 872–874 In the last round, we have a simple first price auction. It has a pure-strategy Nash equilibrium in which the highest-value agent wins by bidding slightly above the second-highest value.
Sequential auction	Nash equilibrium	In each previous round, the situation is a special case of a first-price auction with externalities. In such an auction, each agent may gain value, not only when he wins, but also when other agents win. In general, the valuation of agent i is represented by a vector vi[1],…,vi[n] , where vi[j] is the value of agent i when agent j wins. In a sequential auction, the externalities are determined by the equilibrium outcomes in the future rounds. In the introductory example, there are two possible outcomes: If Alice wins the first round, then the equilibrium outcome in the second round is that Alice buys an item worth $5 for $4, so her net gain is $1. Therefore, her total value for winning the first round is Alice Alice ]=5+1=6 If Bob wins the first round, then the equilibrium outcome in the second round is that Alice buys an item worth $5 for $0, so her net gain is $5. Therefore, her total value for letting Bob win is Alice Bob ]=0+5=5 Each first-price auction with externalities has a pure-strategy Nash equilibrium. In the above example, the equilibrium in the first round is that Bob wins and pays $1.
Sequential auction	Nash equilibrium	Therefore, by backward induction, each SAFP has a pure-strategy SPE.Notes: The existence result also holds for SASP. In fact, any equilibrium-outcome of a first-price auction with externalities is also an equilibrium-outcome of a second-price auction with the same externalities. The existence result holds regardless of the valuations of the bidders – they may have arbitrary utility functions on indivisible goods. In contrast, if all auctions are done simultaneously, a pure-strategy Nash equilibrium does not always exist, even if the bidders have subadditive utility functions.
Sequential auction	Social welfare	Once we know that a subgame perfect equilibrium exists, the next natural question is how efficient it is – does it obtain the maximum social welfare? This is quantified by the price of anarchy (PoA) – the ratio of the maximum attainable social welfare to the social welfare in the worst equilibrium. In the introductory Example 1, the maximum attainable social welfare is 10 (when Alice wins both items), but the welfare in equilibrium is 9 (Bob wins the first item and Alice wins the second), so the PoA is 10/9. In general, the PoA of sequential auctions depends on the utility functions of the bidders.
Sequential auction	Social welfare	The first five results apply to agents with complete information (all agents know the valuations of all other agents): Case 1: Identical items. There are several identical items. There are two bidders. At least one of them has a concave valuation function (diminishing returns). The PoA of SASP is at most 1.58 . Numerical results show that, when there are many bidders with concave valuation functions, the efficiency loss decreases as the number of users increases.
Sequential auction	Social welfare	Case 2: Additive bidders.: 885 The items are different, and all bidders regard all items as independent goods, so their valuations are additive set functions. The PoA of SASP is unbounded – the welfare in a SPEPS might be arbitrarily small. Case 3: Unit-demand bidders. All bidders regard all items as pure substitute goods, so their valuations are unit demand. The PoA of SAFP is at most 2 – the welfare in a SPEPS is at least half the maximum (if mixed strategies are allowed, the PoA is at most 4). In contrast, the PoA in SASP is again unbounded. These results are surprising and they emphasize the importance of the design decision of using a first-price auction (rather than a second-price auction) in each round.
Sequential auction	Social welfare	Case 4: submodular bidders. The bidders' valuations are arbitrary submodular set functions (note that additive and unit-demand are special cases of submodular). In this case, the PoA of both SAFP and SASP is unbounded, even when there are only four bidders. The intuition is that the high-value bidder might prefer to let a low-value bidder win, in order to decrease the competition that he might face in the future rounds.
Sequential auction	Social welfare	Case 5: additive+UD. Some bidders have additive valuations while others have unit-demand valuations. The PoA of SAFP might be at least min (n,m) , where m is the number of items and n is the number of bidders. Moreover, the inefficient equilibria persist even under iterated elimination of weakly dominated strategies. This implies linear inefficiency for many natural settings, including: Bidders with gross substitute valuations, capacitated valuations, budget-additive valuations, additive valuations with hard budget constraints on the payments.Case 6: unit-demand bidders with incomplete information. The agents do not know the valuations of the other agents, but only the probability-distribution from which their valuations are drawn. The sequential auction is then a Bayesian game, and its PoA might be higher. When all bidders have unit demand valuations, the PoA of a Bayesian Nash equilibrium in a SAFP is at most 3.
Sequential auction	Revenue maximization	An important practical question for sellers selling several items is how to design an auction that maximizes their revenue. There are several questions: 1. Is it better to use a sequential auction or a simultaneous auction? Sequential auctions with bids announced between sales seem preferable because the bids may convey information about the value of objects to be sold later. The auction literature shows that this information effect increases the seller's expected revenue since it reduces the winner's curse. However, there is also a deception effect which develops in the sequential sales. If a bidder knows that his current bid will reveal information about later objects then he has an incentive to underbid.
Sequential auction	Revenue maximization	2. If a sequential auction is used, in what order should the items be sold in order to maximize the seller's revenue?Suppose there are two items and there is a group of bidders who are subject to budget constraints. The objects have common values to all bidders but need not be identical, and may be either complement goods or substitute goods. In a game with complete information: 1. A sequential auction yields more revenue than a simultaneous ascending auction if: (a) the difference between the items' values is large, or (b) there are significant complementarities. A hybrid simultaneous-sequential form yields higher revenue than the sequential auction.

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