Narrow Results

In this paper, extensions between finite conformal modules over the Heisenberg–Virasoro conformal algebra are classified.

Let $(C,\underline{w})$ be a polarized nodal reducible curve. In this paper, we consider coherent systems of type $(r,d,k)$ on $C$ with $k<r$. We prove that the moduli spaces of $\underline{w}$-stable coherent systems stabilize for large $\alpha$ and we generalize several results known for the irreducible case when we choose a good polarization. Then, we study in detail the components of moduli spaces containing coherent systems arising from locally free sheaves.

It is well known that locally defined positive definite functions on Lie groups G generally do not extend to positive definite functions which are defined on the whole group. We introduce two stronger positivity concepts for locally defined functions, and show that they are equivalent to extendability. We then apply this to the case when G is the Heisenberg group. When an additional symmetry is imposed, we obtain a complete spectral analysis of the (locally defined) positive definite functions. The methods of proof are based on unitary dilation techniques (i.e., carefully chosen extensions of some underlying Hilbert space associated to the problem), and on spectral theory for noncommuting operators.

An open problem in the theory of Moufang loops is to classify those loops which are minimally nonassociative, that is, loops which are nonassociative but where all proper subloops are associative. A related question is to classify all integers n for which a minimally nonassociative loop exists. In [Possible orders of nonassociative Moufang loops, Comment. Math. Univ. Carolin.41(2) (2000) 237–244], O. Chein and the third author showed that a minimal nonassociative Moufang loop of order 2q³ can be constructed by using a non-abelian group of order q³. In [Moufang loops of odd order pq³, J. Algebra235 (2001) 66–93], the third author also proved that for odd primes p < q, a nonassociative Moufang loop of order pq³ exists if and only if q ≡ 1 (mod p). Here we complete the classification of minimally nonassociative Moufang loops of order pq³ for primes p < q.

Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.

A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second cohomology groups for specific topological quandles. A method of computing the cohomology groups of the inverse limit is applied to quandles.

Business process execution language (BPEL) does not provide a native support for defining data and metrics, required to perform business activity monitoring (BAM). Existing industry solutions are vendor-specific while platform-independent scientific approaches separate tightly coupled monitoring definitions from the core BPEL processes into external descriptors, thus increasing the complexity of the development, packaging and deployment processes. Furthermore, existing monitoring solutions reuse platform-specific audit trail events. This requires monitor model developers to be familiar with the details of the process implementation and the custom BPEL engine meta-model. To overcome these issues, this paper presents bpelx4bam — a new approach to defining monitoring data and metrics within BPEL processes, based on BPEL extensions. The bpelx4bam promotes these definitions into the first class citizens of the BPEL specification in order to unify and fully integrate BAM into the BPM lifecycle. It enables a cross-platform migration of BAM-enabled business processes, removes the need for separate models and specifications, eases development, packaging and deployment processes, aligns the separation of concerns with actual process roles and introduces BAM-specific and optimized events. Therefore, it presents a solid ground for a future specification or a standard in this area.

We study possible extensions of Hardy's inequalities induced by second, fourth and sixth-order linear differential operators with singular potentials.

In this paper, we translate the problem of extending an associative algebra by another associative algebra into the language of codifferentials. The authors have been constructing moduli spaces of algebras and studying their structure by constructing their versal deformations. The codifferential language is very useful for this purpose. The goal of this paper is to express the classical results about extensions in a form which leads to a simpler construction of moduli spaces of low-dimensional algebras.

The irreducible representations of the Witt algebra W are completely known. A classification of the irreducible U_χ(W)-modules was first established by Chang and later simplified by Strade. The aim of this paper is to give a classification of the extensions of the simple U_χ(W)-modules having p-character χ of height -1 ≤ r(χ) ≤ 1 and r(χ) = p - 1 where p denotes the characteristic of the ground field. As an application we give a stronger proof of the wildness of U_χ(W) for -1 ≤ r(χ) ≤ 1 such that (r(χ), p) ≠ (1, 5).

We characterize the abelian groups G for which Ext(G, -) commutes with direct sums.

In this paper we classify non-commutative quadrics and study their homological properties. In fact we find all non-commutative algebras of degree 2 up to isomorphism and we study these algebras via their homomorphic images onto the polynomial algebra k[x,y] as well as the Ext¹(k(p), k(q))-groups, where k(p) and k(q) are one-dimensional simple modules. Moreover some general results on simple finite-dimensional modules are obtained. Some of these results are applied to the special cases of non-commutative quadrics.

Let ${𝒜}$ and $𝔄$ be Banach algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with compatible actions. We define the product ${𝒜}⋊𝔄$, which is a strongly splitting Banach algebra extension of $𝔄$ by ${𝒜}$. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of ${𝒜}⋊𝔄$, we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of ${𝒜}⋊𝔄$ in relation to that of the algebras ${𝒜}$, $𝔄$ and the action of $𝔄$ on ${𝒜}$. We also compute the first cohomology group $H^1({𝒜}⋊𝔄,{({𝒜}⋊𝔄)}^{(n)})$ for all $n\in \mathbb{N}\cup \{0\}$ as well as the first-order cyclic cohomology group $H_{\lambda }^1({𝒜}⋊𝔄,{({𝒜}⋊𝔄)}^{(1)})$, where ${({𝒜}⋊𝔄)}^{(n)}$ is the $n$th dual space of ${𝒜}⋊𝔄$ when $n\in \mathbb{N}$ and ${𝒜}⋊𝔄$ itself when $n=0$. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and $n$-weak amenability of ${𝒜}⋊𝔄$. In this context, several open questions arise.

Let $\mathcal{𝒲}(b)$ be a class of free Lie conformal algebras of rank two with $ℂ[\partial ]$-basis $\{L,H\}$ and relations

A hom-Lie algebroid is a vector bundle together with a Lie algebroid like structure which is twisted by a homomorphism. In this paper, we use the idea of representations up to homotopy of Lie algebroids to construct a same structure for hom-Lie algebroids and we will explain how representations up to homotopy of length 1 are related to extensions of hom-Lie algebroids.

Hesitant fuzzy set has been able to receive an immense concern on the basis of several studies in both theoretical and practical viewpoints that have reported its adverse effects. During the last 12 years, there has been a growing interest in different extensions of hesitant fuzzy set theory which can generates and adopts to improve hesitant fuzzy capacity to address the increase in information complexity. Through this contribution, we intend to conduct a comprehensive literature review on “Hesitant Fuzzy” extensions those have been rarely reported previously. In view of this matter, this paper takes the literature on “Hesitant Fuzzy” extensions in the core collection database of Web of Science as the research object, and moreover, it uses the bibliometric software package of VOSviewer to visually analyze the themes of “Hesitant Fuzzy” researches from different aspects such as general statistics, top authors, the affiliation of top authors, the country of top authors, the document types and the subject areas. Furthermore, this contribution analyses the chronological development of “Hesitant Fuzzy” extensions which have been more frequently and thoroughly reviewed, including the general-based, the linguistic term-based, the q-rung orthopair-based and the probabilistic-based extensions of hesitant fuzzy sets. Last, a number of directions for future research are put forward.

We develop a local index theory for a class of operators associated with non-proper and non-isometric actions of Lie groupoids on smooth submersions. Such actions imply the existence of a short exact sequence of algebras, relating these operators to their non-commutative symbol. We then compute the connecting map induced by this extension on periodic cyclic cohomology. When cyclic cohomology is localized at appropriate isotropic submanifolds of the groupoid in question, we find that the connecting map is expressed in terms of an explicit Wodzicki-type residue formula, which involves the jets of non-commutative symbols at the fixed-point set of the action.

In this paper, some results on representations of Hom-Leibniz algebras are found. Specifically the adjoint representation and trivial representation of Hom-Leibniz algebras are studied in detail. Deformations and central extensions of Hom-Leibniz algebras are also studied as applications.