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We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding \mathbb {N}\trianglelefteq \mathbb {E} of locally compact groups and a twisted action (\alpha,\tau) thereof on a (post)liminal C^*-algebra A the twisted crossed product A\rtimes _{\alpha,\tau }\mathbb {E} is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup \mathbb {N}\trianglelefteq \mathbb {E} is type-I as soon as \mathbb {E} is. This happens for instance if \mathbb {N} is discrete and \mathbb {E} is Lie, or if \mathbb {N} is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group \mathbb {G} type-I-preserving if all semidirect products \mathbb {N}\rtimes \mathbb {G} are type-I as soon as \mathbb {N} is, and linearly type-I-preserving if the same conclusion holds for semidirect products V\rtimes \mathbb {G} arising from finite-dimensional \mathbb {G}-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie. [ABSTRACT FROM AUTHOR]

In this paper, we study an excision theorem of the dihedral and reflexive (co)homology theory of associative algebras. That is, for such an extension, we obtain a six-term exact sequence in the dihedral cohomology. Also, we present and prove the relation between cyclic and dihedral cohomology of algebras and some examples. [ABSTRACT FROM AUTHOR]

Different first order representations of high order linear PDEs, depending on which notion of the degree of a polynomial is chosen, are possible. Two notions of degree, namely, the total degree and multi-degree, are standard and the most natural. In our recent article, a straightforward way was provided to convert a system of high order constant coefficient linear PDEs into a system of first order PDEs based on the notion of total degree. In the present article, an analogous development is carried out in the multi-degree setting. [ABSTRACT FROM AUTHOR]

In this paper we find a generalization of the snake lemma and Schanuel's lemma in Hv-modules. We define the isomorph sequences and determine the conditions to split the exact sequences in Hv-modules. Some interesting results on these concepts are given. [ABSTRACT FROM AUTHOR]

In this paper we provide an axiomatic analysis of the classical diagram chasing method of Mac Lane, that relies on duality and chasing elements of pointed sets, which allows one to generalize this method from abelian categories to non-abelian ones. Among the examples where the generalized method can be used are all modular semiexact categories in the sense of Grandis (which include all Puppe-Mitchell exact categories) and sequentiable categories in the sense of Bourn (which include all semi-abelian categories, and in particular, the categories of group-like structures). The method turns out to be closely related to the essential features of these categories. At the same time, in some sense, it simplifies the usual proofs of some of the standard diagram lemmas in them. [ABSTRACT FROM AUTHOR]

The largest class of multivalued systems satisfying the module-like axioms are the Hv -modules. The main tools concerning the class of Hv -modules with the ordinary modules are the fundamental relations. Based on the relation ε* , exact sequences in Hv -modules are defined. In this paper, we introduce the Hv -module M[A] and determine its heart and the connection between equivalence relations ε* M[A] and ε* A . Moreover, we define the M[-] and -[M] functors and investigate the exactness and some concepts related to them. Finally, we prove the five short lemma in Hv -modules. [ABSTRACT FROM AUTHOR]

Let V be a vertex operator algebra. The fusion products in this paper are defined by logarithmic intertwining operators. Under this setting, we prove the pure exactness of any extension of the V -module V . Namely, if 0 → Q → ϵ P → ρ V → 0 is a short exact sequence of V -modules and W is a V -module, then 0 → Q ⊠ W → ϵ ⊠ 1 W P ⊠ W → ρ ⊠ 1 W V ⊠ W → 0 is also exact, where we view it as a sequence of g ( V ) -modules. [ABSTRACT FROM AUTHOR]

Abstract: By means of a mixture of linear and nonlinear techniques, the paper treats quasi-Banach function spaces as -modules and studies the basic algebraical constructions and their interactions, placing the emphasis on tensor products. Sample result: if is an exact sequence of -modules and homomorphisms, where Y and Z are function spaces, and V is another function space, then the tensorized sequence is exact too. [Copyright &y& Elsevier]

The paper establishes a relation between exact sequences, parametric finite elements, and perfectly matched layer (PML) techniques. We illuminate the analogy between the Piola-like maps used to define parametric H-, H(curl)-, H(div)-, and L-conforming elements, and the corresponding PML complex coordinates stretching for the same energy spaces. We deliver a method for obtaining PML-stretched bilinear forms (constituting the new weak formulation for the original problem with PML absorbing boundary layers) directly from their classical counterparts. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that diagram lemmas for abelian groups (and more generally in abelian categories) used in algebraic topology, can be suitably extended to “non-abelian” structures such as groups, rings, loops, etc. Moreover, they are equivalent to properties which arise in the axiomatic study of these structures. For the five lemma this is well known, and in the present paper we establish this for the snake lemma and the lemma, which, when suitably formulated, turn out to be equivalent to each other for all (pointed) algebraic structures, and also in general categories of a special type. In particular, we show that among varieties of universal algebras, they are satisfied precisely in the so-called (pointed) ideal determined varieties. [Copyright &y& Elsevier]

We construct, for any double complex in an abelian category, certain "short-distance" maps, and an exact sequence involving these, instances of which can be pieced together to give the "long-distance" maps and exact sequences of results such as the Snake Lemma. Further applications are given. We also note what the building blocks of an analogous study of triple complexes would be [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell–Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups of the projective plane were originally studied by Van Buskirk during the 1960s, and are of particular interest due to the fact that they have torsion. The group (resp. ) is isomorphic to the cyclic group of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If , we first prove that the lower central series of is constant from the commutator subgroup onwards. We observe that is isomorphic to , where denotes the free group of rank k, and denotes the quaternion group of order 8, and that is an extension of an index 2 subgroup K of by . As for the derived series of , we show that for all , it is constant from the derived subgroup onwards. The group being finite and soluble for , the critical cases are . We are able to determine completely the derived series of . The subgroups , and are isomorphic respectively to , and , and we compute the derived series quotients of these groups. From onwards, the derived series of , as well as its successive derived series quotients, coincide with those of . We analyse the derived series of and its quotients up to , and we show that is a semi-direct product of by . Finally, we give a presentation of . [Copyright &y& Elsevier]

Abstract: Fuhrmann’s state-space construction (in its generalized form) is used to obtain a general theory of first order representations of Fliess models defined over an arbitrary noetherian commutative ring. The case of arbitrary linear delay differential equations is involved. [Copyright &y& Elsevier]

Abstract: A duality of two categories is introduced. It generalizes the Malcev description of torsion free finite rank abelian groups. An equivalence of two categories is also introduced. It generalizes the Kurosh description of p-primitive groups. The composition of the duality and the equivalence is the duality earlier introduced by W. Wickless and the author. It is shown that the last duality preserves exactness for short exact sequences of homomorphisms. [Copyright &y& Elsevier]