The local finiteness of some PI-algebras

1978 ◽  
Vol 18 (4) ◽  
pp. 663-666 ◽  
Author(s):  
K. I. Beidar ◽  
V. D. T�n
Keyword(s):  
Local Finiteness

scholarly journals The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

scholarly journals On the laws of certain finite groups

1970 ◽  
Vol 11 (4) ◽  
pp. 441-489 ◽  
Author(s):  
John Cossey ◽  
Sheila Oates MacDonald ◽  
Anne Penfold Street
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classification Problems ◽  
Sylow Subgroups ◽  
Local Finiteness ◽  
Special Cases

In recent years a great deal of attention has been devoted to the study of finite simple groups, but one aspect which seems to have been little considered is that of the laws they satisfy. In a recent paper [3], the first two of the present authors gave a basis for laws of PSL(2, 5). The techniques of [3] can be used to show that (modulo certain classification problems) a basis for the laws of PSL(2, pn) can be made up from laws of the following types:(1) an exponent law,(2) laws which determine the Sylow subgroups,(3) laws which determine the normalisers of the Sylow subgroups,(4) in certain special cases, laws which determine subvarieties of smaller exponent, e.g. the subvariety of exponent 12 for those PSL(2, pn) which contain S4,(5) a law implying local finiteness.

Some groups of exponent 72

2014 ◽  
Vol 17 (6) ◽  
Author(s):  
Enrico Jabara ◽  
Daria V. Lytkina ◽  
Victor D. Mazurov
Keyword(s):  
Local Finiteness

AbstractLocal finiteness is proved for groups of exponent dividing 72 with no elements of order 6.

Matric generators of coalgebras and bialgebras

2019 ◽  
Vol 18 (08) ◽  
pp. 1950144
Author(s):  
Hiroshi Kihara
Keyword(s):  
Commutative Ring ◽  
Finite Subset ◽  
Finiteness Theorem ◽  
Finitely Generated ◽  
Semihereditary Ring ◽  
Local Finiteness

Takeuchi asserted that if a bialgebra [Formula: see text] over a field [Formula: see text] is finitely generated as a [Formula: see text]-algebra, then [Formula: see text] is a matric bialgebra. We introduce the notion of a matric coalgebra over a commutative ring [Formula: see text]. We show that if [Formula: see text] is faithfully projective as a [Formula: see text]-module, then [Formula: see text] is a matric coalgebra. Using this, we also show that if a bialgebra [Formula: see text] over a semihereditary ring [Formula: see text] is projective as a [Formula: see text]-module, then any finite subset of [Formula: see text] is contained in some matric subbialgebra. This result is a generalization of Takeuchi’s assertion and can be regarded as a local finiteness theorem on bialgebras.

scholarly journals Local Finiteness, Distinguishing Numbers, and Tucker's Conjecture

10.37236/4873 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Florian Lehner ◽  
Rögnvaldur G. Möller
Keyword(s):  
Finite Number ◽  
Finite Graph ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Local Finiteness ◽  
Vertex Set

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a non-locally finite graph for which no finite number of colours suffices.

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