K. I. Appel :
“Horn sentences in identity theory J. Symb. Logic
24 : 4
(December 1959 ),
pp. 306–310 .
MR
132691 Zbl
0114.24502 article
Abstract
BibTeX
Horn [1951] obtained a sufficient condition for an elementary class to be closed under direct product. Chang and Morel [1958] showed that this is not a necessary condition. We will show that, if consideration is restricted to identity theory, that is, a first-order predicate calculus with equality but no other relation symbols or operation symbols, Horn’s condition is necessary and sufficient.
@article {key132691m,
AUTHOR = {Appel, K. I.},
TITLE = {Horn sentences in identity theory},
JOURNAL = {J. Symb. Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {24},
NUMBER = {4},
MONTH = {December},
YEAR = {1959},
PAGES = {306--310},
DOI = {10.2307/2963901},
NOTE = {MR:132691. Zbl:0114.24502.},
ISSN = {0022-4812},
}
K. I. Appel :
Two investigations on the borderline of logic and algebra Ph.D. thesis ,
University of Michigan ,
1959 .
Advised by R. C. Lyndon .
MR
2612854 phdthesis
People
BibTeX
@phdthesis {key2612854m,
AUTHOR = {Appel, Kenneth Ira},
TITLE = {Two investigations on the borderline
of logic and algebra},
SCHOOL = {University of Michigan},
YEAR = {1959},
PAGES = {66},
URL = {https://search.proquest.com/docview/301859460},
NOTE = {Advised by R. C. Lyndon.
MR:2612854.},
}
K. I. Appel :
“Partition rings of cyclic groups of odd prime power order Can. J. Math.
13
(1961 ),
pp. 373–391 .
MR
124419 Zbl
0100.03001 article
BibTeX
@article {key124419m,
AUTHOR = {Appel, K. I.},
TITLE = {Partition rings of cyclic groups of
odd prime power order},
JOURNAL = {Can. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics. Journal
Canadien de Math\'{e}matiques},
VOLUME = {13},
YEAR = {1961},
PAGES = {373--391},
DOI = {10.4153/CJM-1961-032-4},
NOTE = {MR:124419. Zbl:0100.03001.},
ISSN = {0008-414X},
}
K. I. Appel and F. M. Djorup :
“On the group generated by a free semigroup Proc. Am. Math. Soc.
15 : 5
(1964 ),
pp. 838–840 .
MR
168682 Zbl
0126.04701 article
Abstract
People
BibTeX
@article {key168682m,
AUTHOR = {Appel, K. I. and Djorup, F. M.},
TITLE = {On the group generated by a free semigroup},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {15},
NUMBER = {5},
YEAR = {1964},
PAGES = {838--840},
DOI = {10.2307/2034609},
NOTE = {MR:168682. Zbl:0126.04701.},
ISSN = {0002-9939},
}
K. I. Appel and T. G. McLaughlin :
“On properties of regressive sets Trans. Am. Math. Soc.
115
(1965 ),
pp. 83–93 .
MR
230616 Zbl
0192.05202 article
Abstract
People
BibTeX
It was recently demonstrated by R. Mansfield (unpublished) that complementary retraceable sets must be recursive. Our main result, proved in §3, is that at least one member of any complementary pair of regresssive sets is recursively enumerable. This is a generalization of Mansfield’s theorem, but the method of proof, in §3, is quite different. In §4, one of the two principal lemmas used by Mansfield is generalized, and some related material is developed, including an alternative derivation of the main theorem. In §5, we show that if the intersection of a pair of regressive sets is infinite then it has an infinite regressive subset. As a corollary to this last result, we prove that a coregressive hypersimple set is many-one incomparable with any hyperhypersimple set.
@article {key230616m,
AUTHOR = {Appel, K. I. and McLaughlin, T. G.},
TITLE = {On properties of regressive sets},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {115},
YEAR = {1965},
PAGES = {83--93},
DOI = {10.2307/1994258},
NOTE = {MR:230616. Zbl:0192.05202.},
ISSN = {0002-9947},
}
K. I. Appel :
“No recursively enumerable set is the union of finitely many immune retraceable sets Proc. Am. Math. Soc.
18 : 2
(1967 ),
pp. 279–281 .
MR
207548 Zbl
0183.01402 article
BibTeX
@article {key207548m,
AUTHOR = {Appel, K. I.},
TITLE = {No recursively enumerable set is the
union of finitely many immune retraceable
sets},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {18},
NUMBER = {2},
YEAR = {1967},
PAGES = {279--281},
DOI = {10.2307/2035279},
NOTE = {MR:207548. Zbl:0183.01402.},
ISSN = {0002-9939},
}
K. I. Appel :
“There exist two regressive sets whose intersection is not regressive J. Symb. Log.
32 : 3
(October 1967 ),
pp. 322–324 .
MR
215711 Zbl
0192.05203 article
Abstract
BibTeX
In [1962], as part of an analogy between the concepts of recursive emimerability and regressiveness, Dekker showed that the intersection of any two regressive sets which are recursively equivalent is a regressive set. In [1965], McLaughlin and the author showed that the intersection of two regressive sets, if infinite, has an infinite regressive subset. However, the following theorem shows that the analogy is not complete in this instance.
There exist two regressive sets whose intersection is not regressive.
@article {key215711m,
AUTHOR = {Appel, K. I.},
TITLE = {There exist two regressive sets whose
intersection is not regressive},
JOURNAL = {J. Symb. Log.},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {32},
NUMBER = {3},
MONTH = {October},
YEAR = {1967},
PAGES = {322--324},
DOI = {10.2307/2270773},
NOTE = {MR:215711. Zbl:0192.05203.},
ISSN = {0022-4812},
}
K. I. Appel and E. T. Parker :
“On unsolvable groups of degree \( p=4q+1 \) , \( p \) and \( q \) primes Can. J. Math.
19
(1967 ),
pp. 583–589 .
MR
217165 Zbl
0166.01903 article
Abstract
People
BibTeX
This paper presents two results. They are:
Let \( G \) be a doubly transitive permutation group of degree \( nq + 1 \) where \( q \) is a prime and \( n < q \) . If \( G \) is neither alternating nor symmetric, then \( G \) has Sylow \( q \) -subgroup of order only \( q \) .
There is no unsolvable transitive permutation group of degree \( p = 29 \) , 53, 149, 173, 269, 293, or 317 properly contained in the alternating group of degree \( p \) .
Result 2 was demonstrated by a computation on the Illiac II computer at the University of Illinois.
@article {key217165m,
AUTHOR = {Appel, K. I. and Parker, E. T.},
TITLE = {On unsolvable groups of degree \$p=4q+1\$,
\$p\$ and \$q\$ primes},
JOURNAL = {Can. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics. Journal
Canadien de Math\'ematiques},
VOLUME = {19},
YEAR = {1967},
PAGES = {583--589},
DOI = {10.4153/CJM-1967-051-2},
NOTE = {MR:217165. Zbl:0166.01903.},
ISSN = {0008-414X},
}
K. I. Appel and F. M. Djorup :
“On the equation \( z_1^n z_2^n\cdots z_k^n = y^n \) in a free semigroup Trans. Am. Math. Soc.
134 : 3
(December 1968 ),
pp. 461–470 .
MR
230827 Zbl
0169.02703 article
People
BibTeX
@article {key230827m,
AUTHOR = {Appel, K. I. and Djorup, F. M.},
TITLE = {On the equation \$z_1^n z_2^n\cdots z_k^n
= y^n\$ in a free semigroup},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {134},
NUMBER = {3},
MONTH = {December},
YEAR = {1968},
PAGES = {461--470},
DOI = {10.2307/1994869},
NOTE = {MR:230827. Zbl:0169.02703.},
ISSN = {0002-9947},
}
K. I. Appel :
“One-variable equations in free groups Proc. Am. Math. Soc.
19 : 4
(1968 ),
pp. 912–918 .
MR
232826 Zbl
0159.30502 article
Abstract
BibTeX
Let \( G \) be the free group on free generators \( a_1,\dots,a_r \) . Let \( w(x) \) be any freely reduced word in \( G^*\{x\} \) other than the empty word. Define \( U(w) \) to be the set of elements
\[ u = u(a_1,\dots,a_r) \]
of \( G \) which satisfy \( w(u) = 1 \) . Let \( \gamma_1, \gamma_2, \dots \) be a set of parameters (variables for integers). An expression
\[ W = w_1^{\alpha_1}w_2^{\alpha_2}\cdots w_m^{\alpha_m} \]
is called a parametric expression over \( G \) if the \( w_i \) , are words on \( a_1\cdots a_r \) , and the \( \alpha_j \) are polynomials in the parameters \( \gamma_j \) . A value of \( W \) is the word on \( a_1,\dots \) , \( a_r \) resulting from substituting integer values for the \( \gamma_i \) appearing in \( W \) . Lyndon [1960] effectively associates with \( w(x) \) a finite set of parametric expressions over \( G \) such that \( U(w) \) is precisely the set of all values of these parametric expressions. Lyndon’s argument does not bound the number of parameters required. Lorenc [1963] showed that at most two parameters were required in each parametric expression. We prove the following theorem.
\( U(w) \) is the set of values obtained by substitution of positive integers into a finite set of parametric expressions which are effectively obtainable from \( w(x) \) and contain at most one parameter in each expression.
@article {key232826m,
AUTHOR = {Appel, K. I.},
TITLE = {One-variable equations in free groups},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {19},
NUMBER = {4},
YEAR = {1968},
PAGES = {912--918},
DOI = {10.2307/2035339},
NOTE = {MR:232826. Zbl:0159.30502.},
ISSN = {0002-9939},
}
K. I. Appel :
“On two variable equations in free groups Proc. Am. Math. Soc.
21 : 1
(1969 ),
pp. 179–184 .
MR
257193 Zbl
0172.02701 article
Abstract
BibTeX
In [1960], Lyndon defined parametric words in groups as group theoretic expressions in the generators \( a_1,\dots \) , \( a_r \) that contain certain integer valued parameters \( \nu_1,\dots \) , \( \nu_d \) as exponents. The main theorem of his paper was that with any nontrivial equation in one variable in a free group can be associated a finite collection of parametric words such that any substitution of integers for parameters yields a solution, and all solutions are thus obtainable. (This result was refined in [1968] to show that each parametric word need have only one parameter occurrence.)
The main result of this paper is to show that no comparable statement can be made for equations in two variables. In particular, it will be shown that even with a somewhat more general type of parametric word the equation
\[ axbaya^{-1}x^{-1}b^{-1}a^{-1}y^{-1} = 1 \]
has a collection of solutions which cannot be described by any finite number of generalized parametric words.
@article {key257193m,
AUTHOR = {Appel, K. I.},
TITLE = {On two variable equations in free groups},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {21},
NUMBER = {1},
YEAR = {1969},
PAGES = {179--184},
DOI = {10.2307/2036887},
NOTE = {MR:257193. Zbl:0172.02701.},
ISSN = {0002-9939},
}
K. I. Appel and P. E. Schupp :
“The conjugacy problem for the group of any tame alternating knot is solvable Proc. Am. Math. Soc.
33 : 2
(1972 ),
pp. 329–336 .
MR
294460 Zbl
0243.20036 article
Abstract
People
BibTeX
@article {key294460m,
AUTHOR = {Appel, K. I. and Schupp, P. E.},
TITLE = {The conjugacy problem for the group
of any tame alternating knot is solvable},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {33},
NUMBER = {2},
YEAR = {1972},
PAGES = {329--336},
DOI = {10.2307/2038056},
NOTE = {MR:294460. Zbl:0243.20036.},
ISSN = {0002-9939},
}
K. I. Appel :
“On the conjugacy problem for knot groups ,”
pp. 18
in
Conference in group theory
(Racine, WI, 28–30 June 1972 ).
Edited by R. W. Gatterdam and K. W. Weston .
Lecture Notes in Mathematics 319 .
Springer (Berlin ),
1973 .
Abstract only.
Abstract for an article eventually published in Math. Z. 138 :3 (1974)
Zbl
0256.20045 incollection
People
BibTeX
@incollection {key0256.20045z,
AUTHOR = {Appel, K. I.},
TITLE = {On the conjugacy problem for knot groups},
BOOKTITLE = {Conference in group theory},
EDITOR = {Gatterdam, R. W. and Weston, K. W.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {319},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1973},
PAGES = {18},
NOTE = {(Racine, WI, 28--30 June 1972). Abstract
only. Abstract for an article eventually
published in \textit{Math. Z.} \textbf{138}:3
(1974). Zbl:0256.20045.},
ISSN = {0075-8434},
}
K. I. Appel :
“On the conjugacy problem for knot groups Math. Z.
138 : 3
(1974 ),
pp. 273–294 .
An abstract was published in Conference in group theory (1972)
MR
357622 Zbl
0276.20033 article
Abstract
BibTeX
In this paper a technique is developed for solving the conjugacy problem for a large class of knot groups. In particular it provides the first known proof of the solvability of the conjugacy problem for the group of a non-alternating knot. The technique is applied to solve the conjugacy problem for the infinite class of (non-alternating) cable knots of type \( (2,1) \) on knotted tori. It is also applied to solve the conjugacy problem for two more complicated knots, and to provide a different proof for the theorem of Weinbaum [1971] on prime alternating knots.
The approach is based on the small cancellation arguments of Lyndon and Schupp. We employ the Wirtinger presentation of the knot group (rather than the Dehn presentation employed in [Weinbaum 1971] and [Appel and Schupp 1972]). Since this presentation satisfies almost none of the small cancellation conditions on relator regions, we are forced to use somewhat differently defined faces in the graph theoretic part of the approach. The principal tool employed is the dual of a (small cancellation type) conjugacy diagram, and the resulting natural faces which we call sections. In this paper we consider only prime knots. With considerable additional machinery the approach was extended to all alternating knots in the work announced in [Appel 1971], but it seems reasonable that the method in [Appel and Schupp 1972] will eventually show that the conjugacy problem for a composite knot is solvable if those for its prime parts are solvable.
While the basic techniques are developed in a uniform manner, their application makes use of specific properties of the knots examined. There is as yet no uniform procedure for applying the technique to all knots, but there are no known cases where it does not apply.
@article {key357622m,
AUTHOR = {Appel, Kenneth I.},
TITLE = {On the conjugacy problem for knot groups},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {138},
NUMBER = {3},
YEAR = {1974},
PAGES = {273--294},
DOI = {10.1007/BF01237125},
NOTE = {An abstract was published in \textit{Conference
in group theory} (1972). MR:357622.
Zbl:0276.20033.},
ISSN = {0025-5874},
}
K. I. Appel :
“Book review: J. N. Crossley, et al., ‘What is mathematical logic?’ Math. Mag.
47 : 4
(September 1974 ),
pp. 236–237 .
MR
1572110 article
People
BibTeX
@article {key1572110m,
AUTHOR = {Appel, K. I.},
TITLE = {Book review: {J}.~{N}. {C}rossley, et
al., ``{W}hat is mathematical logic?''},
JOURNAL = {Math. Mag.},
FJOURNAL = {Mathematics Magazine},
VOLUME = {47},
NUMBER = {4},
MONTH = {September},
YEAR = {1974},
PAGES = {236--237},
URL = {http://www.jstor.org/stable/2689225},
NOTE = {MR:1572110.},
ISSN = {0025-570X},
}
K. Appel and W. Haken :
“The existence of unavoidable sets of geographically good configurations Ill. J. Math.
20 : 2
(1976 ),
pp. 218–297 .
MR
392641 Zbl
0322.05141 article
Abstract
People
BibTeX
A set of configurations is unavoidable if every planar map contains at least one element of the set. A configuration \( \mathscr{C} \) is called geographically good if whenever a member country \( M \) of \( \mathscr{C} \) has any three neighbors \( N_1 \) , \( N_2 \) , \( N_3 \) which are not members of \( \mathscr{C} \) then \( N_1 \) , \( N_2 \) , \( N_3 \) are consecutive (in some order) about \( M \) .
The main result is a constructive proof that there exist finite unavoidable sets of geographically good configurations. This result is the first step in an investigation of an approach towards the Four Color Conjecture.
@article {key392641m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {The existence of unavoidable sets of
geographically good configurations},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {20},
NUMBER = {2},
YEAR = {1976},
PAGES = {218--297},
URL = {http://projecteuclid.org/euclid.ijm/1256049898},
NOTE = {MR:392641. Zbl:0322.05141.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“Every planar map is four colorable Bull. Am. Math. Soc.
82 : 5
(September 1976 ),
pp. 711–712 .
MR
424602 Zbl
0331.05106 article
Abstract
People
BibTeX
@article {key424602m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {Every planar map is four colorable},
JOURNAL = {Bull. Am. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical
Society},
VOLUME = {82},
NUMBER = {5},
MONTH = {September},
YEAR = {1976},
PAGES = {711--712},
DOI = {10.1090/S0002-9904-1976-14122-5},
NOTE = {MR:424602. Zbl:0331.05106.},
ISSN = {0002-9904},
}
K. Appel and W. Haken :
“A proof of the four color theorem Discrete Math.
16 : 2
(October 1976 ),
pp. 179–180 .
MR
543791 Zbl
0339.05109 article
Abstract
People
BibTeX
@article {key543791m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {A proof of the four color theorem},
JOURNAL = {Discrete Math.},
FJOURNAL = {Discrete Mathematics},
VOLUME = {16},
NUMBER = {2},
MONTH = {October},
YEAR = {1976},
PAGES = {179--180},
DOI = {10.1016/0012-365X(76)90147-3},
NOTE = {MR:543791. Zbl:0339.05109.},
ISSN = {0012-365X},
}
K. Appel and W. Haken :
“Every planar map is four colorable ,”
J. Recreat. Math.
9 : 3
(1976–1977 ),
pp. 161–169 .
MR
543797 Zbl
0357.05043 article
People
BibTeX
@article {key543797m,
AUTHOR = {Appel, Kenneth and Haken, Wolfgang},
TITLE = {Every planar map is four colorable},
JOURNAL = {J. Recreat. Math.},
FJOURNAL = {Journal of Recreational Mathematics},
VOLUME = {9},
NUMBER = {3},
YEAR = {1976--1977},
PAGES = {161--169},
NOTE = {MR:543797. Zbl:0357.05043.},
ISSN = {0022-412x},
}
K. Appel and W. Haken :
“Every planar map is four colorable, I: Discharging Ill. J. Math.
21 : 3
(1977 ),
pp. 429–490 .
A microfiche supplement to both parts was published in Ill. J. Math. 21 :3 (1977)
MR
543792 Zbl
0387.05009 article
Abstract
People
BibTeX
We begin by describing, in chronological order, the earlier results which led to the work of this paper. The proof of the Four Color Theorem requires the results of Sections 2 and 3 of this paper and the reducibility results of Part II. Sections 4 and 5 will be devoted to an attempt to explain the difficulties of the Four Color Problem and the unusual nature of the proof.
@article {key543792m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {Every planar map is four colorable,
{I}: {D}ischarging},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {429--490},
URL = {http://projecteuclid.org/euclid.ijm/1256049011},
NOTE = {A microfiche supplement to both parts
was published in \textit{Ill. J. Math.}
\textbf{21}:3 (1977). MR:543792. Zbl:0387.05009.},
ISSN = {0019-2082},
}
K. Appel, W. Haken, and J. Koch :
“Every planar map is four colorable, II: Reducibility Ill. J. Math.
21 : 3
(1977 ),
pp. 491–567 .
A microfiche supplement to both parts was published in Ill. J. Math. 21 :3 (1977)
MR
543793 Zbl
0387.05010 article
Abstract
People
BibTeX
In Part I of this paper, a discharging procedure is defined which yields the unavoidability (in planar triangulations) of a set \( \mathscr{U} \) of configurations of ring size fourteen or less. In this part, \( \mathscr{U} \) is presented (as Table \( \mathscr{U} \) consisting of Figures 1–63) together with a discussion of the reducibility proofs of its members.
@article {key543793m,
AUTHOR = {Appel, K. and Haken, W. and Koch, J.},
TITLE = {Every planar map is four colorable,
{II}: {R}educibility},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {491--567},
URL = {http://projecteuclid.org/euclid.ijm/1256049012},
NOTE = {A microfiche supplement to both parts
was published in \textit{Ill. J. Math.}
\textbf{21}:3 (1977). MR:543793. Zbl:0387.05010.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“The class check lists corresponding to the supplement to ‘Every planar map is four colorable. Part I and Part II’ ,”
Ill. J. Math.
21 : 3
(1977 ),
pp. C1–C210 .
Microfiche supplement.
Extra material to accompany the supplement published in Ill. J. Math. 21 :3 (1977)
MR
543794 article
People
BibTeX
@article {key543794m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {The class check lists corresponding
to the supplement to ``{E}very planar
map is four colorable. {P}art {I} and
{P}art {II}''},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {C1--C210},
NOTE = {Microfiche supplement. Extra material
to accompany the supplement published
in \textit{Ill. J. Math.} \textbf{21}:3
(1977). MR:543794.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“Microfiche supplement to ‘Every planar map is four colorable. Part I and Part II’ Ill. J. Math.
21 : 3
(1977 ),
pp. 1–251 .
Microfiche supplement.
Supplement to the two part article published as Ill. J. Math. 21 :3 (1977)Ill. J. Math. 21 :3 (1977)Ill. J. Math. 21 :3 (1977)
MR
543795 article
People
BibTeX
@article {key543795m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {Microfiche supplement to ``{E}very planar
map is four colorable. {P}art {I} and
{P}art {II}''},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {1--251},
URL = {https://projecteuclid.org/euclid.ijm/1256049023},
NOTE = {Microfiche supplement. Supplement to
the two part article published as \textit{Ill.
J. Math.} \textbf{21}:3 (1977) and \textit{Ill.
J. Math.} \textbf{21}:3 (1977). A class
check list was also published as \textit{Ill.
J. Math.} \textbf{21}:3 (1977). MR:543795.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“The solution of the four-color-map problem Sci. Amer.
237 : 4
(October 1977 ),
pp. 108–121 .
MR
543796 article
Abstract
People
BibTeX
@article {key543796m,
AUTHOR = {Appel, Kenneth and Haken, Wolfgang},
TITLE = {The solution of the four-color-map problem},
JOURNAL = {Sci. Amer.},
FJOURNAL = {Scientific American},
VOLUME = {237},
NUMBER = {4},
MONTH = {October},
YEAR = {1977},
PAGES = {108--121},
DOI = {10.1038/scientificamerican1077-108},
URL = {https://www.jstor.org/stable/24953967},
NOTE = {MR:543796.},
ISSN = {0036-8733},
}
J. S. Graber, K. Appel, K. Fleming, D. F. Bailey, and M. V. Subbarao :
“News & letters Math. Mag.
50 : 3
(May 1977 ),
pp. 173–175 .
MR
1572218 article
People
BibTeX
@article {key1572218m,
AUTHOR = {Graber, James S. and Appel, Kenneth
and Fleming, Kirk and Bailey, Donald
F. and Subbarao, M. V.},
TITLE = {News \& letters},
JOURNAL = {Math. Mag.},
FJOURNAL = {Mathematics Magazine},
VOLUME = {50},
NUMBER = {3},
MONTH = {May},
YEAR = {1977},
PAGES = {173--175},
DOI = {10.1080/0025570X.1977.11976642},
NOTE = {MR:1572218.},
ISSN = {0025-570X},
}
K. Appel and W. Haken :
“The four-color problem ,”
pp. 153–180
in
Mathematics today: Twelve informal essays .
Edited by L. A. Steen .
Springer (Berlin ),
1978 .
incollection
People
BibTeX
@incollection {key80470812,
AUTHOR = {K. Appel and W. Haken},
TITLE = {The four-color problem},
BOOKTITLE = {Mathematics today: Twelve informal essays},
EDITOR = {L. A. Steen},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {153--180},
}
K. Appel and W. Haken :
“An unavoidable set of configurations in planar triangulations J. Comb. Theory, Ser. B
26 : 1
(February 1979 ),
pp. 1–21 .
MR
525813 Zbl
0407.05035 article
Abstract
People
BibTeX
@article {key525813m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {An unavoidable set of configurations
in planar triangulations},
JOURNAL = {J. Comb. Theory, Ser. B},
FJOURNAL = {Journal of Combinatorial Theory. Series
B},
VOLUME = {26},
NUMBER = {1},
MONTH = {February},
YEAR = {1979},
PAGES = {1--21},
DOI = {10.1016/0095-8956(79)90038-8},
NOTE = {MR:525813. Zbl:0407.05035.},
ISSN = {0095-8956},
}
K. Appel, W. Haken, and J. Mayer :
“Triangulation à \( v_5 \) séparés dans le problème des quatre couleurs Separated triangulation of \( v_5 \) in the four-color problem ],
J. Comb. Theory, Ser. B
27 : 2
(October 1979 ),
pp. 130–150 .
MR
546856 Zbl
0344.05113 article
Abstract
People
BibTeX
Considérant la notion classique minimal planaire 5-chromatique, les auteurs étudient les triangulations du plan dont tons les sommets sont de degré \( \geq 5 \) et dont les sommets de degré 5 sont séparés (aucune arete ne retie deux sommets de degré 5); its prouvent qu’un graphe minimal comporte nécessairement une arête 5-5. L’article présente: 1) une démonstration fondée sur un ensemble minimum de 14 configurations réductibles, 2) une demonstration fondée sur un algorithme applicable au cas général.
@article {key546856m,
AUTHOR = {Appel, Kenneth and Haken, Wolfgang and
Mayer, Jean},
TITLE = {Triangulation \`a \$v_5\$ s\'epar\'es
dans le probl\`eme des quatre couleurs
[Separated triangulation of \$v_5\$ in
the four-color problem]},
JOURNAL = {J. Comb. Theory, Ser. B},
FJOURNAL = {Journal of Combinatorial Theory. Series
B},
VOLUME = {27},
NUMBER = {2},
MONTH = {October},
YEAR = {1979},
PAGES = {130--150},
DOI = {10.1016/0095-8956(79)90075-3},
NOTE = {MR:546856. Zbl:0344.05113.},
ISSN = {0095-8956},
}
K. I. Appel :
“Un nouveau type de preuve mathématique. Le théorème des quatre couleurs, II A new type of mathematical proof: The four-color theorem, II ],
Publ. Dép. Math., Lyon
16 : 3–4
(1979 ),
pp. 81–88 .
In collaboration with W. Haken.
MR
602656 Zbl
0455.05031 article
People
BibTeX
@article {key602656m,
AUTHOR = {Appel, K. I.},
TITLE = {Un nouveau type de preuve math\'ematique.
{L}e th\'eor\`eme des quatre couleurs,
{II} [A new type of mathematical proof:
{T}he four-color theorem, {II}]},
JOURNAL = {Publ. D\'ep. Math., Lyon},
FJOURNAL = {Publications du D\'epartement de Math\'ematiques.
Facult\'e des Sciences de Lyon},
VOLUME = {16},
NUMBER = {3--4},
YEAR = {1979},
PAGES = {81--88},
URL = {http://www.numdam.org/item/PDML_1979__16_3-4_81_0/},
NOTE = {In collaboration with W. Haken. MR:602656.
Zbl:0455.05031.},
ISSN = {0076-1056},
}
W. Abikoff, K. Appel, and P. Schupp :
“Lifting surface groups to \( \mathrm{SL}(2,\mathbb{C}) \) 1–5
in
Kleinian groups and related topics
(Oaxtepec, Mexico, 10–14 August 1981 ).
Edited by D. M. Gallo and R. M. Porter .
Lecture Notes in Mathematics 971 .
Springer (Berlin ),
1983 .
MR
690272 Zbl
0531.30037 incollection
People
BibTeX
@incollection {key690272m,
AUTHOR = {Abikoff, W. and Appel, K. and Schupp,
P.},
TITLE = {Lifting surface groups to \$\mathrm{SL}(2,\mathbb{C})\$},
BOOKTITLE = {Kleinian groups and related topics},
EDITOR = {Gallo, D. M. and Porter, R. M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {971},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1983},
PAGES = {1--5},
DOI = {10.1007/BFb0067067},
NOTE = {(Oaxtepec, Mexico, 10--14 August 1981).
MR:690272. Zbl:0531.30037.},
ISSN = {0075-8434},
ISBN = {9783540394266},
}
K. I. Appel and P. E. Schupp :
“Artin groups and infinite Coxeter groups Invent. Math.
72 : 2
(June 1983 ),
pp. 201–220 .
MR
700768 Zbl
0536.20019 article
People
BibTeX
@article {key700768m,
AUTHOR = {Appel, K. I. and Schupp, P. E.},
TITLE = {Artin groups and infinite {C}oxeter
groups},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {72},
NUMBER = {2},
MONTH = {June},
YEAR = {1983},
PAGES = {201--220},
DOI = {10.1007/BF01389320},
NOTE = {MR:700768. Zbl:0536.20019.},
ISSN = {0020-9910},
}
Contributions to group theory Contemp. Math.
33 .
Issue edited by K. I. Appel, J. G. Ratcliffe, and P. E. Schupp .
American Mathematical Society (Providence, RI ),
1984 .
Papers dedicated to Roger C. Lyndon on the occasion of his sixty-fifth birthday.
MR
767092 Zbl
0539.00007 book
People
BibTeX
@book {key767092m,
TITLE = {Contributions to group theory},
EDITOR = {Appel, Kenneth I. and Ratcliffe, John
G. and Schupp, Paul E.},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1984},
DOI = {10.1090/conm/033/767093},
NOTE = {Published as \textit{Contemp. Math.}
\textbf{33}. Papers dedicated to {R}oger
{C}. {L}yndon on the occasion of his
sixty-fifth birthday. MR:767092. Zbl:0539.00007.},
ISSN = {0271-4132},
ISBN = {9780821850350},
}
K. I. Appel :
“Roger C. Lyndon: A biographical and personal note 1–10
in
Contributions to group theory .
Edited by K. I. Appel, J. G. Ratcliffe, and P. E. Schupp .
Contemporary Mathematics 33 .
American Mathematical Society (Providence, RI ),
1984 .
MR
767093 Zbl
0546.01008 incollection
People
BibTeX
@incollection {key767093m,
AUTHOR = {Appel, Kenneth I.},
TITLE = {Roger {C}. {L}yndon: {A} biographical
and personal note},
BOOKTITLE = {Contributions to group theory},
EDITOR = {Appel, Kenneth I. and Ratcliffe, John
G. and Schupp, Paul E.},
SERIES = {Contemporary Mathematics},
NUMBER = {33},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1984},
PAGES = {1--10},
DOI = {10.1090/conm/033/767093},
NOTE = {MR:767093. Zbl:0546.01008.},
ISSN = {0271-4132},
ISBN = {9780821850350},
}
K. I. Appel :
“On Artin groups and Coxeter groups of large type 50–78
in
Contributions to group theory .
Edited by K. I. Appel, J. G. Ratcliffe, and P. E. Schupp .
Contemporary Mathematics 33 .
American Mathematical Society (Providence, RI ),
1984 .
MR
767099 Zbl
0576.20021 incollection
Abstract
People
BibTeX
@incollection {key767099m,
AUTHOR = {Appel, Kenneth I.},
TITLE = {On {A}rtin groups and {C}oxeter groups
of large type},
BOOKTITLE = {Contributions to group theory},
EDITOR = {Appel, Kenneth I. and Ratcliffe, John
G. and Schupp, Paul E.},
SERIES = {Contemporary Mathematics},
NUMBER = {33},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1984},
PAGES = {50--78},
DOI = {10.1090/conm/033/767099},
NOTE = {MR:767099. Zbl:0576.20021.},
ISSN = {0271-4132},
ISBN = {9780821850350},
}
K. Appel and W. Haken :
“The four color proof suffices Math. Intell.
8 : 1
(1986 ),
pp. 10–20 .
MR
823216 Zbl
0578.05022 article
People
BibTeX
@article {key823216m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {The four color proof suffices},
JOURNAL = {Math. Intell.},
FJOURNAL = {The Mathematical Intelligencer},
VOLUME = {8},
NUMBER = {1},
YEAR = {1986},
PAGES = {10--20},
DOI = {10.1007/BF03023914},
NOTE = {MR:823216. Zbl:0578.05022.},
ISSN = {0343-6993},
}
K. Appel and W. Haken :
Every planar map is four colorable Contemporary Mathematics 98 .
American Mathematical Society (Providence, RI ),
1989 .
With the collaboration of J. Koch.
MR
1025335 Zbl
0681.05027 book
People
BibTeX
@book {key1025335m,
AUTHOR = {Appel, Kenneth and Haken, Wolfgang},
TITLE = {Every planar map is four colorable},
SERIES = {Contemporary Mathematics},
NUMBER = {98},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1989},
PAGES = {xvi+741},
DOI = {10.1090/conm/098},
NOTE = {With the collaboration of J. Koch. MR:1025335.
Zbl:0681.05027.},
ISSN = {0271-4132},
ISBN = {9780821851036},
}
