Narrow Results

We classify certain extensions of A𝕋-algebras using the six-term exact sequence in K-theory together with the Elliott invariants of the ideal and quotient. We also give certain necessary and sufficient conditions for such extension algebras being A𝕋-algebras.

In this paper, we provide some integral conditions to extend the Ricci flow coupled with harmonic map flow. Our results generalize the corresponding results for Ricci flow obtained by Wang [On the conditions to extend Ricci flow, Int. Math. Res. Not.8 (2008), Articles: rnn012, 30 pp].

Chinese Herbal Medicines (CHM) are the most common interventions of traditional Chinese medicine (TCM), typically administered as either single herbs or formulas. Systematic reviews (SRs) are essential references for evaluating the efficacy and safety of CHM treatments accurately and reliably. Unfortunately, the reporting quality of SRs with CHM is not optimal, especially the reporting of CHM interventions and the rationale of why these interventions were selected. To address this problem, a group of TCM clinical experts, methodologists, epidemiologists, and editors has developed a PRISMA extension for CHM interventions (PRISMA-CHM) through a comprehensive process, including registration, literature review, consensus meeting, three-round Delphi survey, and finalization. The PRISMA checklist was extended by introducing the concept of TCM Pattern and the characteristics of CHM interventions. A total of twenty-four items (including sub-items) are included in the checklist, relating to title (1), structured summary (2), rationale (3), objectives (4), eligibility criteria (6), data items (11), synthesis of results (14, 21), additional analyses (16, 23), study characteristics (18), summary of evidence (24), and conclusions (26). Illustrative examples and explanations are also provided. The group hopes that PRISMA-CHM 2020 will improve the reporting quality of SRs of CHM.

We study the geometry of the central paths of linear programming theory. These paths are the solution curves of the Newton vector field of the logarithmic barrier function. This vector field extends to the boundary of the polytope and we study the main properties of this extension: continuity, analyticity, singularities.

We introduce the concept of an extension of a semilattice of groups $A$ by a group $G$ and describe all the extensions of this type which are equivalent to the crossed products $A{\ast }_{\mathrm{\Theta }}G$ by twisted partial actions $\mathrm{\Theta }$ of $G$ on $A$. As a consequence, we establish a one-to-one correspondence, up to an isomorphism, between twisted partial actions of groups on semilattices of groups and so-called Sieben twisted modules over $E$-unitary inverse semigroups.

We review Gröbner–Shirshov bases for Lie algebras and survey some new results on Gröbner–Shirshov bases for $\mathrm{\Omega }$-Lie algebras, Gelfand–Dorfman–Novikov algebras, Leibniz algebras, etc. Some applications are given, in particular, some characterizations of extensions of groups, associative algebras and Lie algebras are given.

Group object quandles are sets with compatible group and quandle operations. Their endomorphisms will themselves be group objects and function composition becomes a third compatible operation, giving the structure of a nearring, a ring-like structure lacking one distributive property. We give an orbit decomposition of group object quandles in terms of cosets of a certain subgroup of the additive group, and describe them as central extensions of this subgroup. We then describe and count the elements of the endomorphism nearring and study its ideal structure.

In this paper we propose an operational interpretation of general fuzzy measures. On the basis of this interpretation, we define the concept of coherence with respect to a partial information, and propose a rule of inference similar to the natural extension ⁷. We also suggest a definition of independence for fuzzy measures.

We consider the relations between some properties of the certainty equivalent and the form of a utility function under Cumulative Prospect Theory.

We apply Hilbert module methods to show that normal completely positive maps admit weak tensor dilations. Appealing to a duality between weak tensor dilations and extensions of CP-maps, we get an existence proof for certain extensions. We point out that this duality is part of a far reaching duality between a von Neumann bimodule and its commutant in which other dualities, known and new, also find their natural common place.

We show that the Owen value for TU games with a cooperation structure extends the Shapley value in a consistent way. In particular, the Shapley value is the expected Owen value for all symmetric distributions on the partitions of the player set. Similar extensions of the Banzhaf value do not show this property.

In this paper, we give a purely cohomological interpretation of the extension problem for (super) Lie algebras, that is, the problem of extending a Lie algebra by another Lie algebra. We then give a similar interpretation of infinitesimal deformations of extensions. In particular, we consider infinitesimal deformations of representations of a Lie algebra.

In this paper, we study the extension and restriction theorems of the anisotropic Besov spaces in a Lipschitz hyper-surface in space-time domain. We hope that such theorems will be useful in solving a parabolic type equations in a time-varying domain.

We obtain all Dirichlet spaces ℱ_q, q ∈ ℝ, of holomorphic functions on the unit ball of ℂ^N as weighted symmetric Fock spaces over ℂ^N. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the ℱ_q. We pick out the analytic Hilbert modules from among the ℱ_q. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each ℱ_q. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C*-algebras generated by the shift operators on the ℱ_q fit in exact sequences that are in the same Ext class. We identify the groups K₀ and K₁ of the Toeplitz algebras on the ℱ_q arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces ℱ_b that depend on a weight sequence b.

In this paper, we study the class (Ω, ∞) of groups whose every infinite subset contains two distinct elements generating an Ω-group where Ω is either the class of Černikov groups, or the class of Černikov-by-nilpotent groups and we deduce some characterizations of finite-by-nilpotent groups.

We classify the abelian groups G, for which the following property holds: for every subgroup H, every ϕ ∈ Aut (H) has an extension ψ ∈ Aut(G). We also classify the infinite polycyclic-by-finite groups and the finite nilpotent 2′-groups having this property. Fuchs, Bertholf, Walls and Tomkinson did similar work for groups which have the property that homomorphisms of its subgroups extend to the whole group.

A submodule $W$ of $V$ is summand absorbing, if $x+y\in W$ implies $x\in W,y\in W$ for any $x,y\in V$. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more generally of upper bound modules.

We adapt and generalize results of Loganathan on the cohomology of inverse semigroups to the cohomology of ordered groupoids. We then derive a five-term exact sequence in cohomology from an extension of ordered groupoids, and show that this sequence leads to a classification of extensions by a second cohomology group. Our methods use structural ideas in cohomology as far as possible, rather than computation with cocycles.

In this paper, we describe the split exact sequences of skew left braces. We define a free action of the second cohomology group of a skew left brace $H$ by $\mathrm{\text{Ann}}(I)$ on ${\mathrm{\text{Ext}}}_{\alpha }(H,I)$ and show that this action becomes transitive if $I$ is a trivial skew brace. We also generalize the Well’s type exact sequence for extensions by the trivial skew brace.

Let $G$ be the direct product of a generalized extraspecial $\mathbb{Q}$-group and finitely many copies of ${\mathbb{Q}}_{\pi }$, where $\pi$ is a set of primes. It is proved that every polynomial automorphism of $G$ is an inner automorphism. As an application of this result, the structure of the group generated by all polynomial automorphisms of an extension of $\mathbb{Q}$ by a direct sum of finitely many copies of $\mathbb{Q}$ is determined.