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We examine Lie (super)algebroids equipped with a homological section, i.e., an odd section that ‘self-commutes’, we refer to such Lie algebroids as inner Q-algebroids: these provide natural examples of suitably “superised” Q-algebroids in the sense of Mehta. Such Lie algebroids are a natural generalisation of Q-manifolds and Lie superalgebras equipped with a homological element. Amongst other results, we show that, via the derived bracket formalism, the space of sections of an inner Q-algebroid comes equipped with an odd Loday–Leibniz bracket.  相似文献   
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Space, as we typically represent it in mathematics and physics, is composed of dimensionless, indivisible points. On an alternative, region-based approach to space, extended regions together with the relations of ‘parthood’ and ‘contact’ are taken as primitive; points are represented as mathematical abstractions from regions.Region-based theories of space have been traditionally modeled in regular closed (or regular open) algebras, in work that goes back to [5] and [21]. Recently, logics for region-based theories of space were developed in [3] and [19]. It was shown that these logics have both a nice topological and relational semantics, and that the minimal logic for contact algebras, Lmincont (defined below), is complete for both.The present paper explores the question of completeness of Lmincont and its extensions for individual topological spaces of interest: the real line, Cantor space, the rationals, and the infinite binary tree. A second aim is to study a different, algebraic model of logics for region-based theories of space, based on the Lebesgue measure algebra (or algebra of Borel subsets of the real line modulo sets of Lebesgue measure zero). As a model for point-free space, the algebra was first discussed in [2]. The main results of the paper are that Lmincont is weakly complete for any zero-dimensional, dense-in-itself metric space (including, e.g., Cantor space and the rationals); the extension Lmincont+(Con) is weakly complete for the real line and the Lebesgue measure contact algebra. We also prove that the logic Lmincont+(Univ) is weakly complete for the infinite binary tree.  相似文献   
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Let A and B be two factor von Neumann algebras. For A, B ∈ A, define by [A, B]_*= AB-BA~*the skew Lie product of A and B. In this article, it is proved that a bijective map Φ : A → B satisfies Φ([[A, B]_*, C]_*) = [[Φ(A), Φ(B)]_*, Φ(C)]_*for all A, B, C ∈ A if and only if Φ is a linear *-isomorphism, or a conjugate linear *-isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.  相似文献   
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Novel main-chain liquid crystalline Diels—Alder dynamic networks (LCDANs) were prepared that exhibit unprecedented ease for actuator programming and reprocessing compared to existing liquid crystalline network (LCN) systems. Following cooling from 125 °C, LCDANs are deformed with aligned mesogens self-locked at room temperature by slowly formed Diels–Alder (DA) bonds, which allows for the formation of solid 3D actuators capable of reversible shape change, and strip walker and wheel-capable light-driven locomotion upon either thermally or optically induced order–disorder phase transition. Any actuator can readily be erased at 125 °C and reprogrammed into a new one under ambient conditions. Moreover, LCDANs can be processed directly from melt (for example, fiber drawing) and from solution (for example, casting tubular actuators), which cannot be achieved with LCNs using exchangeable covalent bonds. The combined attributes of LCDANs offer significant progress toward developing easily programmable/processable LCN actuators.  相似文献   
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Let F be an infinite field and let Mn(F) be the algebra of n×n matrices over F endowed with an elementary grading whose neutral component coincides with the main diagonal. In this paper, we find a basis for the graded polynomial identities of Mn(F) with the transpose involution. Our results generalize for infinite fields of arbitrary characteristic previous results in the literature, which were obtained for the field of complex numbers and for a particular class of elementary G-gradings.  相似文献   
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We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acting on a regular rooted tree acts absolutely non-freely on the boundary of the tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II1-factor representations, and infinitely many continuous ergodic invariant random subgroups.  相似文献   
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