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We prove that the category of F-coalgebras admits exact sequences, under condition F preserves pullbacks; the underlying category 풞 is assumed to be cocomplete with epi-(regular mono) factorizations, finite products and exact sequences. [ABSTRACT FROM AUTHOR]

Finite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to $\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space $\textbf{H}^s(\text{curl};\,\Omega)$ , and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for $\textbf{H}(\text{curl})$ -elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the $H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments. [ABSTRACT FROM AUTHOR]

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 define the concepts of a "strict morphism" as a suitable map between geometric spaces (as a generalization of hyperstructures), and its kernel. Then we introduce the concepts of a "quotient geometric space", an "exact sequence", an "exact hypersequence", a "homology" and a "hyperhomology" of geometric spaces. Finally we generalize some famous theorems from groups and modules in this direction, such as the "isomorphism theorem", the "five short lemma", the "snake lemma", the "exact triangle", etc., by some conditions that are naturally valid for groups. [ABSTRACT FROM AUTHOR]

The main objective of the research is to study the first natural triangular representation of the symmetric groups over a field K of characteristic p≠ 2 which deals with the partition λ = (n − 4,3,1) of the positive integer n. Furthermore, this work has proven that the S(n − 4,3,1) is a submodule of F1 . The F1 = KSn(x2x4x6x7² −x2x3x6x7² − x2x4x5x7² + x²x³x5x7² ) can be only split whenp|(n − 5). الهدف من هذا البحث هو دراسة [ABSTRACT FROM AUTHOR]

An extension of X by Y is a short exact sequence of quasi Banach modules and homomorphisms 0 -→ Y -→ Z -→ X -→ 0. When properly organized all these extensions constitute a linear space denoted by ExtB(X, Y ), where B is the underlying (Banach) algebra. In this paper we "compute" the spaces of extensions for the Schatten classes when they are regarded in its natural (left) module structure over B = B(H), the algebra of all operators on the ground Hilbert space. Our main results can be summarized as follows: ... In the first case, every extension 0 -→ Sq -→ Z -→ Sp -→ 0 splits and so X = Sq -Sp. In the second case, every self-extension of Sp arises (and gives rise) to a minimal extension of S1 in the quasi Banach category, that is, a short exact sequence 0 -→ C -→ M -→ S1 -→ 0. In the third case, each extension corresponds to a "twisted Hilbert space", that is, a short exact sequence 0 -→ H -→ T -→ H -→ 0. Thus, the subject of the paper is closely connected to the early "three-space" problems studied (and solved) in the seventies by Enflo, Lindenstrauss, Pisier, Kalton, Peck, Ribe, Roberts, and others. [ABSTRACT FROM AUTHOR]

In this work, we prove by employing mapping Cone that the sequence and the subsequence of the characteristic-zero are exact and subcomplex respectively in the case of partition (6,6,4) . [ABSTRACT FROM AUTHOR]

Let X be a Hopf manifold with non-Abelian fundamental group and E be a holomorphic vector bundle over X, with trivial pull-back to ℂn − {0}. The authors show that there exists a line bundle L over X such that E ⊗ L has a nowhere vanishing section. It is proved that in case dim(X) ≥ 3, π*(E) is trivial if and only if E is filtrable by vector bundles. With the structure theorem, the authors get the cohomology dimension of holomorphic bundle E over X with trivial pull-back and the vanishing of Chern class of E. [ABSTRACT FROM AUTHOR]

It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper, we establish such a generalization of the "salamander lemma" due to G. M. Bergman, in a self-dual axiomatic context (developed originally by Z. Janelidze), which applies to all usual non-abelian group-like structures and also covers axiomatic contexts such as semi-abelian categories in the sense of G. Janelidze, L. Márki and W. Tholen and exact categories in the sense of M. Grandis. [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]

In this paper, the notion of (short) quasi-exact sequence of S-acts is introduced. We study the behaviour of quasi-exact sequences in regard to some algebraic properties of S-acts including principal weak injectivity, principal weak flatness, regularity and torsion freeness. Moreover, some results concerning commutative diagrams of modules are generalized to acts over monoids. [ABSTRACT FROM AUTHOR]

A straightforward way is offered to convert a system of high order linear functional equations of various type into a system of first order equations. [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]

Quasi module is an axiomatisation of the hyperspace structure based on a module. We proposed this structure in our paper [2]. It can be thought of as a generalisation of the module structure in the sense that every module can be embedded into a quasi module and every quasi module contains a module. The structure of quasi module is designed on a commutative semigroup with an external ring multiplication and a compatible partial order which has a very intrinsic effect in the entire structure and plays a key role in any development of the theory of quasi module. In this paper we have introduced the concept of exact sequence on quasi modules and proved some Theorems related to exact sequences. [ABSTRACT FROM AUTHOR]

We prove that if E and F are large ideals of B(G) for which the associated coaction functors are exact, then the same is true for E ∩ F. We also give an example of a coaction functor whose restriction to the maximal coactions does not come from any large ideal. [ABSTRACT FROM AUTHOR]

A straightforward way requiring no calculations is offered to convert a system of high order constant coefficient linear PDEs into a system of first order PDEs. [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]

We study the Bredon-Illman cohomology with local coefficients for a G-space X in the case of G being a totally disconnected, locally compact group. We prove that any short exact sequence of equivariant local coefficients systems on X gives a long exact sequence of the associated Bredon-Illman cohomology groups with local coefficients. [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]

In this note, a construction of state representations of singular linear time-invariant differential systems is described directly in terms of trajectories. [ABSTRACT FROM AUTHOR]