Orthogonality relations on Abelian groups

Notes on p-blocks of characters of finite groups

The rings of this paper are assumed to have relations of orthogonality defined on them. Such relations are uniquely determined by complete boolean algebras of ideals. Using the Stone space of these boolean algebras, and following J. Dauns and K.H. Hofmann, a sheaf-theoretic representation is obtained for rings with orthogonality relations, and the rings of global sections of these sheaves are characterized. Baer rings, f–rings and commutative semi-prime rings have natural orthogonality relations and among these the Baer rings are isomorphic to their associated rings of global sections. A special type of ideal is singled out in commutative semi-prime rings and following G. Spirason and E. Strzelecki, in an unpublished note, a characterization of a class of such rings is obtained.

We develop an abstract framework for the investigation of quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representations. To illustrate the usefulness of our abstract method, we show that it behaves well with respect to infinite tensor products. This construction subsumes examples from the study of magnetic Weyl calculus, magnetic pseudo-differential Weyl calculus, metaplectic representation on locally compact abelian groups, irreducible representations associated with finite-dimensional coadjoint orbits of some special infinite-dimensional Lie groups, and square-integrability properties shared by arbitrary irreducible representations of nilpotent Lie groups.

In this paper, we discuss sections of higher homotopy groupoid bundles those yield by group actions. We study some basic properties related to the left-invariant (G-invariant) of such sections and also present some conditions for such sections to be left-invariant (G-invariant). We concentrate a kind of group action on the Core higher homotopy groupoid bundle of certain topological spaces which undergo a regular topological group action. Interestingly, each element of orbit space yields continuous sections of the Core higher homotopy groupoid bundle having beautiful algebraic properties. Moreover, each of those continuous sections behaves nicely as a left-invariant section with respect to topological group action by G. Since higher homotopy groups are abelian, the set of such sections forms always an abelian group. Moreover, each element of the group is a left-invariant section, therefore, it is convenient to call them commutative group of left-invariant sections of the Core higher homotopy groupoid bundle. Further, we show this group is a topological invariant and study some basic properties of it. At the end of the paper, we give interrelations between the retraction map and this group.

Comfort, Hernandez and Trigos-Arrieta [2] introduced the class ACCS(#) of abelian groups H such that the natural map φ: G → G/H, where G is the divisible hull of H, has a cross section Γ: G/H → G that is continuous in the Bohr topology of G and G/H. They showed that ACCS(#) is closed under finite products and contains all finitely generated groups (and, of course, all divisible groups). They also gave an example of a group that does not belong to ACCS(#). We give further examples of groups from ACCS(#) (e.g., the groups of P-adic integers) and we find some new restraints for the groups from ACCS(#). This entails that large powers of nondivisible abelian groups never belong to ACCS(#) and gives an upper bound for the size of the reduced groups in ACCS(#) (roughly speaking, most of the abelian groups do not belong to ACCS(#)).

The conditional Cauchy functional equation for a mappingF: (X, +, ⊥) → (Y, +), i.e.,F(x + y) = F(x) + F(y) for allx, y ∈ X withx ⊥ y, (*) on a real vector space equipped with an abstract relation ⊥ (calledorthogonality), was first studied by Gudder and Strawther in 1975. They defined ⊥ by a system consisting of five axioms and described the general hemi-continuous real valued solution of (*) showing that the existence of non-trivial even ones characterize inner product orthogonality. Using the more restrictive axioms of Ratz (introduced in 1980 to obtain the general solution without regularity conditions: odd solutions are additive, while the even ones are quadratic), recently we have proved the same assuming arbitrary mappingsF with values in an abelian group but for dimX ⩾ 3. In 1989, Ratz and the author modified the system of axioms so that it should include the orthogonality induced by an isotropic symmetric bilinear form ϕ and still ensure the additive/quadratic representation. In this context, the main purpose of this note is to characterize on a real vector space the symmetric bilinear orthogonality ⊥ϕ as the essentially unique extension of an orthogonality relation satisfying certain weak axioms and admitting non-trivial even hemi-continuous solutions of (*) with values in a Hausdorff topological abelian group.

Let O,#X = OD' I o 0' define a semigroup on the Banach space F(M, E) of continuous sections of E over M. It is known that (O', S') is hyperbolic iff V# has spectrum off the unit circle for t # 0. We prove that a third equivalent condition is that the (unbounded!) infinitesimal generator L of {f}) have its spectrum disjoint from the imaginary axis. In two dimensions this property coincides with the Poincare stability criterion for a periodic orbit of a planar dynamical system.

Conditions under which a function generates a multiresolution analysis are investigated. The definition of the spectral function of a shift invariant space is generalized from ℝn to a locally compact abelian group and the union density and intersection triviality properties of a multiresolution analysis are characterized in terms of the spectral functions. Finally, all multiresolution analysis conditions are characterized in terms of the scaling and the spectral functions.

Published and unpublished data on bio-, chemo- and magnetostratigraphic events spanning the late Paleocene-early Eocene are reviewed and calibrated to a revised Geomagnetic Polarity Time Scale (GPTS), itself already in need of revision. This timescale serves as a template for placing the upper Paleocene-lower Eocene stratigraphic succession of NW Europe (Anglo-Paris-Belgian Basins) in a sequence stratigraphic framework. It is clear that an approach that integrates sequence- and magnetobioisotope stratigraphy provides a unifying correlation framework within which to delineate the disjunct depositional histories of marginal and deep marine basinal stratigraphies and allows the stratigrapher to move from rock to time. Significant conclusions of this integration include the following: 1. The major climatic warming, weakening of atmospheric circulation and faunal extinction events, which are seen in the deep sea stratigraphic record, are seen to be closely associated with the mid-part of the (redefined) planktonic foraminiferal Zone P5 and calcareous nannoplankton Zone NP9. 2. A relatively rapid ( c. 1000 years) decrease of c. 3 to 4 per mil in the δ 13 C of marine carbonates has now been recognized in high southern and northern latitude sites and at several scattered, intermediate locations. A similar excursion has been observed in mammalian teeth and soil carbonates near the base of the Wasatchian North American Land Mammal Age in the Big Horn Basin a few metres above the Argile plastique bariolée (‘type’ Sparnacian) in the Paris Basin. This δ 13 C excursion is seen to be a truly global event which occurs in both marine ( c. mid-Zone NP9) and terrestrial stratigraphies at a level estimated at c. 55.5 Ma. 3. An evaluation of the calcareous plankton biostratigraphy and stable isotope records at several deep sea sites suggests the presence of multiple unconformities in the Paleocene/Eocene boundary interval. There would appear to be no unequivocally demonstrated continuous stratigraphic section through the c. 2.55 million year interval of Chron C24r in the deep sea record or from outcrops on land, and a composite record is required to construct the sequence of events that occurred during this interval. 4. The Paleocene/Eocene boundary (which awaits the determination of a Global Stratigraphic Section and Point [GSSP]) is bracketed by the base of the Ieper Clay Formation (= Ypresian Stage), estimated here at > 54.6 Ma, and the P5/P6a zonal boundary (LAD Morozovella velascoensis ) at c. 54.7 Ma (above), and by the δ 13 C spike (and associated events) in mid-Zone NP9 at c. 55.5 Ma (below). The ‘boundary interval’ encompasses the NP9/10 zonal boundary at 55 Ma and the base of the Harwich Formation (base of the London Clay Formation Oldhaven Beds of previous authors) at c. 54.8 Ma. 5. Calcareous nannoplankton studies of recently recollected Thanetian localities in the eastern part of the Paris Basin have revealed that the Thanetian Sables de Châlons-sur-Vesles are older (Zones NP6 and NP7) than the Sables de Bracheux s.l. (= Sables du Tillet; Zone NP8). This leads to a revised framework of correlation between the upper Paleocene formations of northwest Europe. 6. The base of the Sparnacian base, taken as the Paleocene/Eocene boundary by most vertebrate palaeontologists, is correlative with a level within Zone NP9, c. 0.5 million year older than the NP9/10 zonal boundary (used by some marine micropalaeontologists to delineate the Paleocene/Eocene boundary) and > one million years older than the base of the Ieper Clay (base Ypresian) in Belgium.

The Silurian–Devonian Boundary of China: Review and perspectives

Abstract. The minus partial order linear algebraic methods have proven to be useful in the study of complex matrices. This paper extends study of minus partial orders to general rings. It is shown that the condition a < b, where < is the minus partial order, defines two triples of orthogonal idempotents, and thus, a decomposition of a ring into a direct sum of abelian groups. Hence, several well-known results concerning minus partial order on real and complex matrices are generalized. Analogous decompositions for rectangular matrices over a ring and for Banach space operators are obtained. The equivalent conditions for the invariance of ab(1)a under the choices of b(1) are also obtained. An original inspiration for this work came from the study of minus partial order on complex matrices and from linear algebra methods.

In this chapter we deal with semigroups S in which, for any a, b ∈ S, the assumption ab = ba implies axb = bxa for all x ∈ S. These semigroups are called conditionally commutative semigroups. In the beginning of the chapter we present equivalent conditions for a conditionally commutative semigroup to be a semilattice of archimedean semigroups, a rectangular band of t-archimedean semigroups, or a semilattice of t-archimedean semigroups, respectively. We prove the followig results. A conditionally commutative semigroup is a semilattice of archimedean semigroups if and only if it is a band of t-archimedean semigroups. A conditionally commutative semigroup is a rectangular band of t-archimedean semigroups if and only if it is archimedean. A conditionally commutative semigroup S is a semilattice of t-archimedean semigroups if and only if, for every a, b ∈ S, there is a positive integer k such that ( a b)k (b a ) We also present results about weakly separative conditionally commutative semigroups. It is shown that every weakly separative conditionally commutative semigroup is a disjoint union of commutative cancellative pover joined semigroups. It is also proved that a conditionally commutative semigroup is weakly separative and regular if and only if it is a normal band of abelian groups.