Hans Lausch - Academia.edu (original) (raw)
Papers by Hans Lausch
Semigroups, 1980
Abstract (An) Let be a variety, A an algebra in , and n > 0 an integer; A ( A n ) is an algebr... more Abstract (An) Let be a variety, A an algebra in , and n > 0 an integer; A ( A n ) is an algebra in . Let P n ( A ) be the subalgebra of A ( A n ) generated by the n projections from A n to A and the constants; then P n ( A ) is called the algebra of n -place polynomial functions on A. A map ϕ from A n to A is said to be a k -local polynomial map if, for any k elements a 1 , …, a k e A n , there exists a polynomial function p such that p ( a i ) = ϕ( a i ), i = 1, …, k . The set of k -local polynomial maps from A n to A is denoted by L k P n ( A ). The behaviour of the chain L 1 P n ( A ) ⊇ L 2 P n ( A )⊇ … ⊇ L k P n ( A )⊇ … has been investigated by various authors for a number of varieties, e.g. for any abelian group A and any n , L 4 P n ( A ) = L k P n ( A ), for all k ⩾ > 4, and L 3 P 1 ( A ) = L k P 1 ( A ) for all k ⩾ 3 (Hule and Nobauer [1977]). It will be shown that, for any semilattice S, L n +2 P n ( S ) = L n P n ( S ), for k ⩾ n + 2.
The Concise Handbook of Algebra, 2002
In an earlier article [1], where the activities of the Problems Committee of the Australian Mathe... more In an earlier article [1], where the activities of the Problems Committee of the Australian Mathematical Olympiad Committee (AMOC) were summarised, readers were invited to donate problems for use in national, regional and international mathematics competitions. Perhaps another call for olympiad-type problems will serve as a reminder to all who have been aware of those contests and at the same time inform those who have joined our mathematical community more recently of a noble way of enhancing school mathematics for budding mathematicians. Participants in those competitions are usually senior secondary-school students, although brilliant younger students have been identified through the mathematical olympiads almost every year. The problems they have to solve are from “pre-calculus ” areas: number theory, geometry (with a strong preference for “Euclidean ” geometry), algebra, discrete mathemat-ics, inequalities, functional equations. In [1], a sample of five competition problems was...
About once in three years the Senior Problems Committee of the Australian Math-ematical Olympiad ... more About once in three years the Senior Problems Committee of the Australian Math-ematical Olympiad Committee (AMOC) turns to our mathematical community with an appeal for problem donations that can be used in national, regional and international senior-secondary-school mathematics competitions. The latest appeal [2] provided examples of competition problems that had been set for various con-tests in Australia and in the Asia-Pacific region between 2008 and early 2010. The present article is to repeat this exercise with problems from competitions held between 2010 and early 2013. Problems chosen for these competitions are from ‘pre-calculus ’ areas such as geometry (with a strong preference for ‘Euclidean’ geometry), number theory, algebra and combinatorics. Before exhibiting various sample problems, it may be appropriate to put the vari-ous competitions serviced by the AMOC Senior Problems Committee into context. 1. The AMOC Senior Contest is held in August of each year. About 75 stu-...
The American Mathematical Monthly, 1998
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Mathematische Nachrichten, 1988
Selecta Mathematica, 2003
When Menger’s book on curve theory [23], written in co-operation with Georg Nobeling, appeared in... more When Menger’s book on curve theory [23], written in co-operation with Georg Nobeling, appeared in 1932 he already had over 60 publications to his credit. Among them were his group theoretical investigations, an offshoot of his seminal paper “Untersuchungen uber allgemeine Metrik” [18]. The Mathematisches Kolloquium at the University of Vienna discussed the ideas developed in this paper and in this way other young mathematicians, especially Abraham Wald (cf. [36]), Olga Taussky, Franz Alt and Gustav Beer, became actively interested in Menger’s distance geometry. At the 13th meeting of the Kolloquium held on 14 March 1930, Menger presented new results under the heading “u ber eine metrische Geometrie in Gruppen” [19]. They were published in the Mathematische Zeitschrift as “Beitrage zur Gruppentheorie. I. uber eine Gruppenmetrik”, [21]1. The second of Menger’s three investigations in [18], “Die euklidische Metrik”, opens with a proof of the following theorem discovered by M. M. Biedermann: a connected metric space M is homeomorphic with a subspace of l i if, for each triple (a, b, c) of distinct points of M, one of the points lies between the other two. 2 However, Menger observed that Biedermann’s condition does not characterise metric spaces which are isometric (“abstandsgleich” in [18], “kongruent” in [20]) to some subspace of ℝ1. Thus he posed the problem of finding necessary and sufficient conditions for a semimetric space3 to be isometrically embeddable (“abstandstreu einbettbar”) into — or, more specifically, to be isometric to —the euclidean n-space ℝ n .
Lecture Notes in Mathematics, 1977
I ° Introduction This talk is to present a few results on cohomology of groups arising from an in... more I ° Introduction This talk is to present a few results on cohomology of groups arising from an investigation into the cohomological behaviour of inverse semigroups ([6], [7], [8]). In this context, relative cohomology in the sense of Auslander ([i]) is just a special case of a more general situation, and various long exact sequences, some of which are well-known ([9], [ii]) and have been established by various authors individually, can be obtained by one and the same method. The starting point for our consideration is the concept of a semilattice of groups, that is a functor S from a semilattice E regarded as a category to the category Gv of groups. Thus for e, f ( E , e ~_ f , there is exactly one group homomorphism ~e,f : S(e) ~ S(f) such that ~e,e is the identity on S(e) ,
Proceedings of the American Mathematical Society, 1969
Monatshefte f�r Mathematik, 1965
Monatshefte f�r Mathematik, 1966
Diese Arbeit stellt eine Fortsetzung und Erg~nzung zu [1] dar, es werden in ihr n/iralieh im Ansc... more Diese Arbeit stellt eine Fortsetzung und Erg~nzung zu [1] dar, es werden in ihr n/iralieh im Anschlul] an die Untersuehungen yon [1] verschiedene weitere Aussagen fiber die Gruppe u(| aller umkehrbar eindeutigen Polynomabbfldungen einer endlichen Gruppe (~ auf sich gewonnen. Im ersten Teil der Arbeit setzen wir (~ als abelsch voraus, im zweiten Tefl als nilpotent der Klasse 2, der dritte Tell enthalt einige Ergebnisse fiber u((~) als Permutationsgruppe auf der Menge q6. In der Bezeichnungsweise schlieSen wir uns an [1 ] an. Nochmals sei bemerkt, dab | stets als endlich vorausgesetzt wird.
Monatshefte f�r Mathematik, 1965
Mathematische Zeitschrift, 1977
Mathematische Zeitschrift, 1973
Mathematische Zeitschrift, 1966
Gegeben sei eine endliche Gruppe G. Es wird die Menge aller Transformationen x -+ al xa2 x ... a ... more Gegeben sei eine endliche Gruppe G. Es wird die Menge aller Transformationen x -+ al xa2 x ... a r x G + 1 (a~e G fest, x durchl/iuft G) betrachtet. Unter diesen bilden diejenigen Transformationen, die Permutationen auf der Menge G sind, eine Gruppe u (G). Diese Gruppe wurde in [1], [2] studiert. Unter anderem wurde folgender Satz bewiesen: Genau dann ist G abelsch, wenn alle (peu(G) dutch Abbildungen der Gestalt ~o : x ~ a x ~, a e G, (r, exp G) = 1 geliefert werden. Im folgenden soil nun gezeigt werden:
The Mathematical Gazette, 1998
Journal of the Australian Mathematical Society, 1979
Let A be a universal algebra. A function q>: A*-+A is called a /-local polynomial function, if <p... more Let A be a universal algebra. A function q>: A*-+A is called a /-local polynomial function, if <p can be interpolated on any / places of A k by a polynomial function-for the definition of a polynomial function on A, see Lausch and Nobauer (1973). Let Pt.{A) be the set of all polynomial functions, L,P k (A) the set of all /-local polynomial functions on A and LP k (A) the intersection of al\L t P t (A), then
Semigroups, 1980
Abstract (An) Let be a variety, A an algebra in , and n > 0 an integer; A ( A n ) is an algebr... more Abstract (An) Let be a variety, A an algebra in , and n > 0 an integer; A ( A n ) is an algebra in . Let P n ( A ) be the subalgebra of A ( A n ) generated by the n projections from A n to A and the constants; then P n ( A ) is called the algebra of n -place polynomial functions on A. A map ϕ from A n to A is said to be a k -local polynomial map if, for any k elements a 1 , …, a k e A n , there exists a polynomial function p such that p ( a i ) = ϕ( a i ), i = 1, …, k . The set of k -local polynomial maps from A n to A is denoted by L k P n ( A ). The behaviour of the chain L 1 P n ( A ) ⊇ L 2 P n ( A )⊇ … ⊇ L k P n ( A )⊇ … has been investigated by various authors for a number of varieties, e.g. for any abelian group A and any n , L 4 P n ( A ) = L k P n ( A ), for all k ⩾ > 4, and L 3 P 1 ( A ) = L k P 1 ( A ) for all k ⩾ 3 (Hule and Nobauer [1977]). It will be shown that, for any semilattice S, L n +2 P n ( S ) = L n P n ( S ), for k ⩾ n + 2.
The Concise Handbook of Algebra, 2002
In an earlier article [1], where the activities of the Problems Committee of the Australian Mathe... more In an earlier article [1], where the activities of the Problems Committee of the Australian Mathematical Olympiad Committee (AMOC) were summarised, readers were invited to donate problems for use in national, regional and international mathematics competitions. Perhaps another call for olympiad-type problems will serve as a reminder to all who have been aware of those contests and at the same time inform those who have joined our mathematical community more recently of a noble way of enhancing school mathematics for budding mathematicians. Participants in those competitions are usually senior secondary-school students, although brilliant younger students have been identified through the mathematical olympiads almost every year. The problems they have to solve are from “pre-calculus ” areas: number theory, geometry (with a strong preference for “Euclidean ” geometry), algebra, discrete mathemat-ics, inequalities, functional equations. In [1], a sample of five competition problems was...
About once in three years the Senior Problems Committee of the Australian Math-ematical Olympiad ... more About once in three years the Senior Problems Committee of the Australian Math-ematical Olympiad Committee (AMOC) turns to our mathematical community with an appeal for problem donations that can be used in national, regional and international senior-secondary-school mathematics competitions. The latest appeal [2] provided examples of competition problems that had been set for various con-tests in Australia and in the Asia-Pacific region between 2008 and early 2010. The present article is to repeat this exercise with problems from competitions held between 2010 and early 2013. Problems chosen for these competitions are from ‘pre-calculus ’ areas such as geometry (with a strong preference for ‘Euclidean’ geometry), number theory, algebra and combinatorics. Before exhibiting various sample problems, it may be appropriate to put the vari-ous competitions serviced by the AMOC Senior Problems Committee into context. 1. The AMOC Senior Contest is held in August of each year. About 75 stu-...
The American Mathematical Monthly, 1998
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Mathematische Nachrichten, 1988
Selecta Mathematica, 2003
When Menger’s book on curve theory [23], written in co-operation with Georg Nobeling, appeared in... more When Menger’s book on curve theory [23], written in co-operation with Georg Nobeling, appeared in 1932 he already had over 60 publications to his credit. Among them were his group theoretical investigations, an offshoot of his seminal paper “Untersuchungen uber allgemeine Metrik” [18]. The Mathematisches Kolloquium at the University of Vienna discussed the ideas developed in this paper and in this way other young mathematicians, especially Abraham Wald (cf. [36]), Olga Taussky, Franz Alt and Gustav Beer, became actively interested in Menger’s distance geometry. At the 13th meeting of the Kolloquium held on 14 March 1930, Menger presented new results under the heading “u ber eine metrische Geometrie in Gruppen” [19]. They were published in the Mathematische Zeitschrift as “Beitrage zur Gruppentheorie. I. uber eine Gruppenmetrik”, [21]1. The second of Menger’s three investigations in [18], “Die euklidische Metrik”, opens with a proof of the following theorem discovered by M. M. Biedermann: a connected metric space M is homeomorphic with a subspace of l i if, for each triple (a, b, c) of distinct points of M, one of the points lies between the other two. 2 However, Menger observed that Biedermann’s condition does not characterise metric spaces which are isometric (“abstandsgleich” in [18], “kongruent” in [20]) to some subspace of ℝ1. Thus he posed the problem of finding necessary and sufficient conditions for a semimetric space3 to be isometrically embeddable (“abstandstreu einbettbar”) into — or, more specifically, to be isometric to —the euclidean n-space ℝ n .
Lecture Notes in Mathematics, 1977
I ° Introduction This talk is to present a few results on cohomology of groups arising from an in... more I ° Introduction This talk is to present a few results on cohomology of groups arising from an investigation into the cohomological behaviour of inverse semigroups ([6], [7], [8]). In this context, relative cohomology in the sense of Auslander ([i]) is just a special case of a more general situation, and various long exact sequences, some of which are well-known ([9], [ii]) and have been established by various authors individually, can be obtained by one and the same method. The starting point for our consideration is the concept of a semilattice of groups, that is a functor S from a semilattice E regarded as a category to the category Gv of groups. Thus for e, f ( E , e ~_ f , there is exactly one group homomorphism ~e,f : S(e) ~ S(f) such that ~e,e is the identity on S(e) ,
Proceedings of the American Mathematical Society, 1969
Monatshefte f�r Mathematik, 1965
Monatshefte f�r Mathematik, 1966
Diese Arbeit stellt eine Fortsetzung und Erg~nzung zu [1] dar, es werden in ihr n/iralieh im Ansc... more Diese Arbeit stellt eine Fortsetzung und Erg~nzung zu [1] dar, es werden in ihr n/iralieh im Anschlul] an die Untersuehungen yon [1] verschiedene weitere Aussagen fiber die Gruppe u(| aller umkehrbar eindeutigen Polynomabbfldungen einer endlichen Gruppe (~ auf sich gewonnen. Im ersten Teil der Arbeit setzen wir (~ als abelsch voraus, im zweiten Tefl als nilpotent der Klasse 2, der dritte Tell enthalt einige Ergebnisse fiber u((~) als Permutationsgruppe auf der Menge q6. In der Bezeichnungsweise schlieSen wir uns an [1 ] an. Nochmals sei bemerkt, dab | stets als endlich vorausgesetzt wird.
Monatshefte f�r Mathematik, 1965
Mathematische Zeitschrift, 1977
Mathematische Zeitschrift, 1973
Mathematische Zeitschrift, 1966
Gegeben sei eine endliche Gruppe G. Es wird die Menge aller Transformationen x -+ al xa2 x ... a ... more Gegeben sei eine endliche Gruppe G. Es wird die Menge aller Transformationen x -+ al xa2 x ... a r x G + 1 (a~e G fest, x durchl/iuft G) betrachtet. Unter diesen bilden diejenigen Transformationen, die Permutationen auf der Menge G sind, eine Gruppe u (G). Diese Gruppe wurde in [1], [2] studiert. Unter anderem wurde folgender Satz bewiesen: Genau dann ist G abelsch, wenn alle (peu(G) dutch Abbildungen der Gestalt ~o : x ~ a x ~, a e G, (r, exp G) = 1 geliefert werden. Im folgenden soil nun gezeigt werden:
The Mathematical Gazette, 1998
Journal of the Australian Mathematical Society, 1979
Let A be a universal algebra. A function q>: A*-+A is called a /-local polynomial function, if <p... more Let A be a universal algebra. A function q>: A*-+A is called a /-local polynomial function, if <p can be interpolated on any / places of A k by a polynomial function-for the definition of a polynomial function on A, see Lausch and Nobauer (1973). Let Pt.{A) be the set of all polynomial functions, L,P k (A) the set of all /-local polynomial functions on A and LP k (A) the intersection of al\L t P t (A), then