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{{for|the railroad executive|Daniel Willard}}
'''Dan Edward Willard''' is an American computer scientist and logician, and is a professor of computer science at the [[University at Albany]].

==Education and career==
Willard did his undergraduate studies in mathematics at [[Stony Brook University]], graduating in 1970. He went on to graduate studies in mathematics at [[Harvard University]], earning a masters degree in 1972 and a doctorate in 1978. After leaving Harvard, he worked at [[Bell Labs]] for four years before joining the Albany faculty in 1983.<ref name="cv">[http://www.cs.albany.edu/~dew/outside.pdf Curriculum vitae], accessed 2013-06-04.</ref>

==Contributions==
Although trained as a mathematician and employed as a computer scientist, Willard's most highly cited publication is in [[evolutionary biology]]. In 1973, with biologist [[Robert Trivers]], Willard published the [[Trivers–Willard hypothesis]], that female mammals could control the [[sex ratio]] of their offspring, and that it would be evolutionally advantageous for healthier or higher-status females to have more male offspring and for less healthy or lower-status females to have more female offspring.<ref group="paper">{{citation
| last1 = Trivers | first1 = R. L. | author1-link = Robert Trivers
| last2 = Willard | first2 = D. E.
| bibcode = 1973Sci...179...90T
| doi = 10.1126/science.179.4068.90
| issue = 4068
| journal = [[Science (journal)|Science]]
| jstor = 1734960
| pages = 90–2
| pmid = 4682135
| title = Natural selection of parental ability to vary the sex ratio of offspring
| volume = 179
| year = 1973}}.</ref> Controversial at the time, especially because it proposed no mechanism for this control, this theory was later validated through observation,<ref>{{citation
| last1 = Simpson | first1 = M. J. A.
| last2 = Simpson | first2 = A. E.
| doi = 10.1038/300440a0
| journal = [[Nature (journal)|Nature]]
| pages = 440–441
| title = Birth sex ratios and social rank in rhesus monkey mothers
| volume = 300
| year = 1982|bibcode = 1982Natur.300..440S }}.</ref> and it has been called "one of
the most influential and highly cited papers of 20th century evolutionary biology".<ref>{{citation
| last = Mathews | first = Paul
| issue = 2
| journal = Society, Biology, & Human Affairs
| pages = 11–23
| title = Is there a psychological proximate mechanism for inducing a Trivers–Willard effect in humans? Results of an internet experiment looking at the desired sex composition of children after mortality priming
| url = http://www.biosocsoc.org/sbha/resources/76_2/SBHA_76_2_Matthews.pdf
| volume = 76
| year = 2011}}.</ref>

Willard's 1978 thesis work on [[range searching]] [[data structure]]s<ref group="paper">{{citation
| last = Willard | first = D. E.
| publisher = Harvard University
| series = Ph.D. thesis
| title = Predicate-Oriented Database Search Algorithms
| year = 1978}}.</ref> was one of the predecessors to the technique of [[fractional cascading]],<ref>{{citation
| last1 = de Berg | first1 = M.
| last2 = van Kreveld | first2 = M.
| last3 = Overmars | first3 = M. H. | author3-link = Mark Overmars
| last4 = Schwarzkopf | first4 = O.
| edition = 3rd
| isbn = 9783540779735
| page = 116
| publisher = Springer-Verlag
| title = Computational Geometry: Algorithms and Applications
| url = http://books.google.com/books?id=tkyG8W2163YC&pg=PA116
| year = 2008}}.</ref> and throughout the 1980s Willard continued to work on related data structure problems. As well as continuing to work on range searching, he did important early work on the [[order-maintenance problem]],<ref group="paper">{{citation
| last = Willard | first = Dan E.
| contribution = Maintaining dense sequential files in a dynamic environment
| doi = 10.1145/800070.802183
| pages = 114–121
| title = [[Symposium on Theory of Computing|Proc. 14th ACM Symposium on Theory of Computing (STOC '82)]]
| year = 1982}}.</ref> and invented the [[x-fast trie]] and [[y-fast trie]], data structures for storing and searching sets of small integers with low memory requirements.<ref group="paper">{{citation
| last = Willard | first = Dan E.
| doi = 10.1016/0020-0190(83)90075-3
| issue = 2
| journal = [[Information Processing Letters]]
| mr = 731126
| pages = 81–84
| title = Log-logarithmic worst-case range queries are possible in space Θ(''N'')
| volume = 17
| year = 1983}}.</ref>

In computer science, Willard is best known for his work with [[Michael Fredman]] in the early 1990s on [[integer sorting]] and related data structures. Before their research, it had long been known that [[comparison sort]]ing required time <math>\Theta(n\log n)</math> to sort a set of <math>n</math> items, but that faster algorithms were possible if the keys by which the items were sorted could be assumed to be integers of moderate size. For instance, sorting keys in the range from <math>1</math> to <math>N</math> could be accomplished in time <math>O(n(1+\tfrac{\log N}{\log n}))</math> by [[radix sorting]]. However, it was assumed that integer sorting algorithms would necessarily have a time bound depending on <math>N</math>, and would necessarily be slower than comparison sorting for sufficiently large values of <math>N</math>. In research originally announced in 1990, Fredman and Willard changed these assumptions by introducing the [[transdichotomous model]] of computation. In this model, they showed that integer sorting could be done in time <math>O(n\tfrac{\log n}{\log\log n})</math> by an algorithm using their [[fusion tree]] data structure as a [[priority queue]].<ref group="paper">{{citation
| last1 = Fredman | first1 = Michael L. | author1-link = Michael Fredman
| last2 = Willard | first2 = Dan E.
| doi = 10.1016/0022-0000(93)90040-4
| mr = 1248864
| issue = 3
| journal = [[Journal of Computer and System Sciences]]
| pages = 424–436
| title = Surpassing the information-theoretic bound with fusion trees
| volume = 47
| year = 1993}}.</ref><ref>{{citation|title=Computing 'fusion trees' to explode barriers: an algorithm that speeds up how fast computers can sort information|first=Ivars|last=Peterson|authorlink=Ivars Peterson|journal=[[Science News]]|date=June 29, 1991|url=http://www.thefreelibrary.com/Computing+'fusion+trees'+to+explode+barriers.-a010980507}}.</ref> In a follow-up to this work, Fredman and Willard also showed that similar speedups could be applied to other standard algorithmic problems including [[minimum spanning tree]]s and [[shortest path problem|shortest paths]].<ref group="paper">{{citation
| last1 = Fredman | first1 = Michael L. | author1-link = Michael Fredman
| last2 = Willard | first2 = Dan E.
| issue = 3
| journal = [[Journal of Computer and System Sciences]]
| pages = 533–551
| title = Trans-dichotomous algorithms for minimum spanning trees and shortest paths
| volume = 48
| year = 1994 | doi=10.1016/S0022-0000(05)80064-9}}.</ref>

Since 2000, Willard's publications have primarily concerned [[self-verifying theories]]: systems of logic that have been weakened sufficiently, compared to more commonly studied systems, to prevent [[Gödel's incompleteness theorems]] from applying to them. Within these systems, it is possible to prove that the systems themselves are logically consistent, without this deduction leading to the self-contradiction that Gödel's theorem implies for stronger systems.<ref group="paper">{{citation
| last = Willard | first = Dan E.
| doi = 10.2307/2695030
| issue = 2
| journal = [[Journal of Symbolic Logic]]
| mr = 1833464
| pages = 536–596
| title = Self-verifying axiom systems, the incompleteness theorem and related reflection principles
| volume = 66
| year = 2001}}.</ref>

==Selected publications==
{{reflist|group="paper"}}

==References==
{{reflist}}

{{DEFAULTSORT:Willard, Dan Edward}}
[[Category:Year of birth missing (living people)]]
[[Category:Living people]]
[[Category:American computer scientists]]
[[Category:American logicians]]
[[Category:Mathematical logicians]]
[[Category:Theoretical computer scientists]]
[[Category:Stony Brook University alumni]]
[[Category:University at Albany, SUNY faculty]]

Taraxerol synthase - Revision history

en>Dcirovic: ←Created page with '{{Infobox enzyme | Name = Taraxerol synthase | EC_number = 5.4.99.35 | CAS_number = | IUBMB_EC_number = 5/4/99/35 | GO_code = | image = | width = | caption =...'