Conveniently Hölder Homomorphisms are Smooth in the Convenient Sense

We show that every ‘conveniently Hölder’ homomorphism between Lie groups in the sense of convenient differential calculus is smooth (in the convenient sense). In particular, every ℓip⁰-homomorphism is smooth.     

Digital Cofibration and Digital Lusternik-Schnirelmann Category in the Sense of Subdivision

In this paper, using the notion of subdivision, the authors generalize the definition of cofibration in digital topology and show that this kind of cofibration is injective in the sense of subdivision. Meanwhile, they give the necessary condition under which a digital map is a cofibration. Furthermore, they consider the Lusternik-Schnirelmann category of digital maps in the sense of subdivision and give several fundamental homotopy properties about it.     

Generalized Gerstewitz's Functions and Vector Variational Principle for-Efficient Solutions in the Sense of Németh

In this paper, we first generalize Gerstewitz's functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order principle [Qiu, J. H.: A pre-order principle and set-valued Ekeland variational principle.J. Math. Anal. Appl., 419, 904–937(2014)]. By using the generalized Gerstewitz's functions and the partial order principle, we obtain a vector EVP for-efficient solutions in the sense of N′emeth, which essentially improves the earlier results by completely removing a usual assumption for boundedness of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.     

How Fast and in what Sense(s) Does the Calderón Reproducing Formula Converge?

We prove that a very general form of the Calderón reproducing formula converges in L ^p (w), for all 1<p<∞, whenever w belongs to the Muckenhoupt class A _p . We show that the formula converges whether we interpret its defining integral, in very natural senses, as a limit of L ^p (w)-valued Riemann or Lebesgue integrals. We give quantitative estimates on their rates of convergence (or, equivalently, on the speed at which the errors go to 0) in L ^p (w).     

Moral Sensitivity and Desire Attachment: In What Sense are they Constituents of One’s Rational Profile?

A quantitative interpretation is given of the (in)coherence that moral agents experience as a tension between their ordered moral judgments over n physically incompatible actions, and the competitive ordering of motivating intensities (or, desires). Then a model describing one’s tendency to reduce the experienced in-coherence is constructed. In this model, moral sensitivity (S) and desire attachment (e) function as primitives that motivate from opposing perspectives the reduction of incoherence. Two distinct sub-processes of this reduction are therefore initiated by (S) and (e) co-ordinated (more or less efficiently) by the agent’s degree of rationality (R) characteristic of her capacity to handle such internal tensions. This process ends when a new equilibrium between what motivates and what resists (further) reduction has been reached. A macro-equilibrium is described involving (R) constrained by weakness-of-will (W _w ). A reinterpretation of the Aristotelian characters (enkratês, akratês, etc.) and an exegesis of Hume’s ‘Calm Passions’ follow as applications.

Making Sense of Abstract Algebra: Exploring Secondary Teachers’ Understandings of Inverse Functions in Relation to Its Group Structure

This article draws on semi-structured, task-based interviews to explore secondary teachers’ (N = 7) understandings of inverse functions in relation to abstract algebra. In particular, a concept map task is used to understand the degree to which participants, having recently taken an abstract algebra course, situated inverse functions within its group structure (i.e., the set of invertible functions under composition). In addition, their particular conceptions of functions and function composition throughout the interviews were then also considered as a means to explore further their responses during the interviews. Findings indicate that only two participants showed evidence of the desirable mathematically powerful understandings from abstract algebra in relation to inverse functions, and further analysis suggests a variety of challenges in terms of developing meaningful connections, which were more related to conceptions about secondary content than to the abstract algebra content. Implications for the mathematical preparation of secondary teachers are discussed.