Fundamental Investigations on the Isomorphism of Commutative Group Algebras in Bulgaria

The isomorphism problem of arbitrary algebraic structures plays always a central role in the study of a given algebraic object. In this paper we give the first investigations and also some basic results on the isomorphism problem of commutative group algebras in Bulgaria.

Isomorphism of Commutative Group Algebras of Finite Abelian Groups

Let a commutative ring R be a direct product of indecomposable rings with identity and let G be a finite abelian p-group. In the present paper we give a complete system of invariants of the group algebra RG of G over R when p is an invertible element in R. These investigations extend some classical results of Berman (1953 and 1958), Sehgal (1970) and Karpilovsky (1984) as well as a result of Mollov (1986).

Some classes of semisimple group (and loop) algebras over finite fields

We determine the structure of the semisimple group algebra of certain groups over the rationals and over those finite fields where the Wedderburn decompositions have the least number of simple components We apply our work to obtain similar information about the loop algebras of mdecomposable RA loops and to produce negative answers to the isomorphism problem over various fields (C) 2010 Elsevier Inc All rights reserved

Isomorphism of Commutative Modular Group Algebras

∗ The work was supported by the National Fund “Scientific researches” and by the Ministry of Education and Science in Bulgaria under contract MM 70/91.

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Group algebras of torsion groups and Lie nilpotence

Let * be an involution of a group algebra FG induced by an involution of the group G. For char F not equal 2, we classify the torsion groups G with no elements of order 2 whose Lie algebra of *-skew elements is nilpotent.

Unitary SK1 of unitary and graded division algebras

The reduced unitary Whitehead group $\SK$ of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary $\SK$ of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary $\SK$ of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary $\SK$ for much wider classes of division algebras over henselian fields.

Computing Generators of Free Modules over Orders in Group Algebras

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] \cong \oplus_xM_x is explicitly computable and each M_x is in fact a matrix ring over a field, this leads to an algorithm that either gives elements \alpha_1,...,\alpha_d \in X such that X = A\alpha_1 \oplus ... \oplusA\alpha_d or determines that no such elements exist. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A(E[G];O_L) \subseteq E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = \IQ.

L1 acquisition across Portuguese dialects: modular and interdisciplinary interfaces as sources of explanation

This study investigates the child (L1) acquisition of properties at the interfaces of morpho-syntax, syntax-semantics and syntax-pragmatics, by focusing on inflected infinitives in European Portuguese (EP). Three child groups were tested, 6–7-year-olds, 9–10-year-olds and 11–12-year-olds, as well as an adult control group. The data demonstrate that children as young as 6 have knowledge of the morpho-syntactic properties of inflected infinitives, although they seem at first glance to show partially insufficient knowledge of their syntax–semantic interface properties (i.e. non-obligatory control properties), differently from children aged 9 and older, who show clearer evidence of knowledge of both types of properties. However, in general, both morpho-syntactic and syntax–semantics interface properties are also accessible to 6–7-year-old children, although these children give preference to a range of interpretations partially different from the adults; in certain cases, they may not appeal to certain pragmatic inferences that permit additional interpretations to adults and older children. Crucially, our data demonstrate that EP children master the two types of properties of inflected infinitives years before Brazilian Portuguese children do (Pires and Rothman, 2009a and Pires and Rothman, 2009b), reasons for and implications of which we discuss in detail.

L1 Acquisition of interface properties across Portuguese dialects: modular and interdisciplinary interfaces as sources of explanation

This study investigates the child (L1) acquisition of properties at the interfaces of morphosyntax, syntax-semantics and syntax-pragmatics, by focusing on inflected infinitives in European Portuguese (EP). Three child groups were tested, 6–7-year-olds, 9–10-year-olds and 11–12-year-olds, as well as an adult control group. The data demonstrate that children as young as 6 have knowledge of the morpho-syntactic properties of inflected infinitives, although they seem at first glance to show partially insufficient knowledge of their syntax–semantic interface properties (i.e. non-obligatory control properties), differently from children aged 9 and older, who show clearer evidence of knowledge of both types of properties. However, in general, both morpho-syntactic and syntax–semantics interface properties are also accessible to 6–7-year-old children, although these children give preference to a range of interpretations partially different from the adults; in certain cases, they may not appeal to certain pragmatic inferences that permit additional interpretations to adults and older children. Crucially, our data demonstrate that EP children master the two types of properties of inflected infinitives years before Brazilian Portuguese children do (Pires and Rothman, 2009a,b), reasons for and implications of which we discuss in detail.

Horizontal projections of area 17 in Cebus monkeys: metric features, and modular and laminar distribution

Metric features and modular and laminar distributions of intrinsic projections of area 17 were studied in Cebus apella. Anterogradely and retrogradely labeled cell appendages were obtained using both saturated pellets and iontophoretic injections of biocytin into the operculum. Laminar and modular distributions of the labeled processes were analyzed using Nissl counterstaining, and/or cytochrome oxidase and/or NADPH-diaphorase histochemistry. We distinguished three labeled cell types: pyramidal, star pyramidal and stellate cells located in supragranular cortical layers (principally in layers IIIa, IIIb α, IIIb ß and IIIc). Three distinct axon terminal morphologies were found, i.e., Ia, Ib and II located in granular and supragranular layers. Both complete and partial segregation of group I axon terminals relative to the limits of the blobs of V1 were found. The results are compatible with recent evidence of incomplete segregation of visual information flow in V1 of Old and New World primates.

FREE GROUP ALGEBRAS IN DIVISION RINGS

Let k be an algebraically closed field of characteristic zero and let L be an algebraic function field over k. Let sigma : L -> L be a k-automorphism of infinite order, and let D be the skew field of fractions of the skew polynomial ring L[t; sigma]. We show that D contains the group algebra kF of the free group F of rank 2.

Fundamental Investigations on the Unit Groups of Commutative Group Algebras in Bulgaria

In this paper we give the first investigations and also some basic results on the unit groups of commutative group algebras in Bulgaria. These investigations continue some classical results. Namely, it is supposed that the cardinality of the starting group is arbitrary.

Localization of quantum biequivariant D-modules and q-W algebras

Peer reviewed

Effect of poly(acrylic acid) molecular mass and end-group functionality on calcium oxalate crystal morphology and growth

A number of series of poly(acrylic acids) (PAA) of differing end-groups and molecular mass were used to study the inhibition of calcium oxalate crystallization. The effects of the end-group on crystal speciation and morphology were significant and dramatic, with hexyl-isobutyrate end groups giving preferential formation of calcium oxalate dihydrate (COD) rather than the more stable calcium oxalate monohydrate (COM), while both more hydrophobic end-groups and less-hydrophobic end groups led predominantly to formation of the least thermodynamically stable form of calcium oxalate, calcium oxalate trihydrate. Conversely, molecular mass had little impact on calcium oxalate speciation or crystal morphology. It is probable that the observed effects are related to the rate of desorption of the PAA moiety from the crystal (lite) surfaces and that the results point to a major role for end-group as well as molecular mass in controlling desorption rate.