The local theorem for tilings

In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

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Centrally symmetric convex bodies and sections having maximal quermassintegrals

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.49.2012.2.1197 ◽

2012 ◽

Vol 49 (2) ◽

pp. 189-199

Author(s):

E. Makai ◽

H. Martini

Keyword(s):

Convex Body ◽

Convex Bodies ◽

The Body ◽

Symmetric Convex ◽

Local Theorem ◽

Centrally Symmetric ◽

Euclidean Unit Ball ◽

Analogous Theorem ◽

Lower Dimensional ◽

Dimensional Surface

Let d ≧ 2, and let K ⊂ ℝd be a convex body containing the origin 0 in its interior. In a previous paper we have proved the following. The body K is 0-symmetric if and only if the following holds. For each ω ∈ Sd−1, we have that the (d − 1)-volume of the intersection of K and an arbitrary hyperplane, with normal ω, attains its maximum if the hyperplane contains 0. An analogous theorem, for 1-dimensional sections and 1-volumes, has been proved long ago by Hammer (see [2]). In this paper we deal with the ((d − 2)-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small C2-perturbations, or C3-perturbations of the Euclidean unit ball, respectively.

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Local theorems and group extensions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410004740x ◽

1973 ◽

Vol 73 (1) ◽

pp. 7-20 ◽

Cited By ~ 5

Author(s):

Kenneth K. Hickin ◽

Richard E. Phillips

Keyword(s):

Local System ◽

Local Theorem ◽

Group Extensions ◽

Finite Set

The concept of a local system of a set W is defined in ((8), p. 166) and ((12), p. 126). Recall that a set ℒ of subsets of W is a local system if Uℒ = W and ℒ is directed in the following sense: for every finite set H1, …, Hn of elements of ℒ, there is an M∈ℒ such that Hi⊂M for 1≤i≤n. If Σ is a class of groups, L(Σ) is the class of all groups G that possess a local system of Σ subgroups. Σ satisfies the local theorem, or is L-closed, if L(Σ)⊂Σ. Many classes of groups which satisfy the local theorem are discussed in ((12), pp. 126–144).

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On the lower estimation of the rate of convergence in the local theorem

Lithuanian Mathematical Journal ◽

10.15388/lmj.1967.19981 ◽

1967 ◽

Vol 7 (3) ◽

pp. 405-408

Author(s):

N. Gamkrelidze

Keyword(s):

Rate Of Convergence ◽

Local Theorem ◽

Lower Estimation

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Н. Г. Гамкрелидзе. Об одной нижней оценке для скорости сходимости в локальной теореме N. Gamkrelidzė. Apie vieną konvergavimo greičio apatinį įvertinimą lokalinėje teoremoje

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