A Note on Relative Efficiency of Axiom Systems

We introduce a notion of relative efficiency for axiom systems. Given an axiom system A_β for a theory T consistent with S¹₂, we show that the problem of deciding whether an axiom system A_α for the same theory is more efficient than A_β is II₂-hard. Several possibilities of speed-up of proofs are examined in relation to pairs of axiom systems A_α, A_β, with A_α ? A_β, both in the case of A_α, A_β having the same language, and in the case of the language of A_α extending that of A_β: in the latter case, letting Pr_α, Pr_β denote the theories axiomatized by A_α, A_β, respectively, and assuming Pr_α to be a conservative extension of Pr_β, we show that if A_α — A_β contains no nonlogical axioms, then A_α can only be a linear speed-up of A_β; otherwise, given any recursive function g and any A_β, there exists a finite extension A_α of A_β such that A_α is a speed-up of A_β with respect to g. Mathematics Subject Classification: 03F20, 03F30.     

Catalytic Asymmetric Reactions of 4‐Substituted Indoles with Nitroethene: A Direct Entry to Ergot Alkaloid Structures

A domino Friedel–Crafts/nitro‐Michael reaction between 4‐substituted indoles and nitroethene is presented. The reaction is catalyzed by BINOL‐derived phosphoric acid catalysts, and delivers the corresponding 3,4‐ring‐fused indoles with very good results in terms of yields and diastereo‐ and enantioselectivities. The tricyclic benzo[cd]indole products bear a nitro group at the right position to serve as precursors of ergot alkaloids, as demonstrated by the formal synthesis of 6,7‐secoagroclavine from one of the adducts. DFT calculations suggest that the outcome of the reaction stems from the preferential evolution of a key nitronic acid intermediate through a nucleophilic addition pathway, rather than to the expected “quenching” through protonation.     

Data filtering for cluster analysis by $$\ell _0$$-norm regularization

A data filtering method for cluster analysis is proposed, based on minimizing a least squares function with a weighted \(\ell _0\)-norm penalty. To overcome the discontinuity of the objective function, smooth non-convex functions are employed to approximate the \(\ell _0\)-norm. The convergence of the global minimum points of the approximating problems towards global minimum points of the original problem is stated. The proposed method also exploits a suitable technique to choose the penalty parameter. Numerical results on synthetic and real data sets are finally provided, showing how some existing clustering methods can take advantages from the proposed filtering strategy.     

Exploiting damped techniques for nonlinear conjugate gradient methods

In this paper we propose the use of damped techniques within Nonlinear Conjugate Gradient (NCG) methods. Damped techniques were introduced by Powell and recently reproposed by Al-Baali and till now, only applied in the framework of quasi-Newton methods. We extend their use to NCG methods in large scale unconstrained optimization, aiming at possibly improving the efficiency and the robustness of the latter methods, especially when solving difficult problems. We consider both unpreconditioned and Preconditioned NCG. In the latter case, we embed damped techniques within a class of preconditioners based on quasi-Newton updates. Our purpose is to possibly provide efficient preconditioners which approximate, in some sense, the inverse of the Hessian matrix, while still preserving information provided by the secant equation or some of its modifications. The results of an extensive numerical experience highlights that the proposed approach is quite promising.     

Mathematical modelling and the learning trajectory: tools to support the teaching of linear algebra

In this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and span. The objective is to evaluate this didactic proposal by determining the level of match between the hypothetical learning trajectory (HLT) designed in this study with the actual learning trajectory in the second experimental cycle of an investigation design-based research more extensive. The results show a high level of match between the trajectories in more than half of the conjectures, which gives evidence that the HLT has supported, in many cases, the achievement of the learning objective, and that additionally mathematical modelling contributes to the construction of these linear algebra concepts.     

An upper bound on the Chebotarev invariant of a finite group

A subset {g ₁,..., g _d } of a finite group G invariably generates \(\left\{ {g_1^{{x_1}}, \ldots ,g_d^{{x_d}}} \right\}\) generates G for every choice of x _i ∈ G. The Chebotarev invariant C(G) of G is the expected value of the random variable n that is minimal subject to the requirement that n randomly chosen elements of G invariably generate G. The first author recently showed that \(C\left( G \right) \leqslant \beta \sqrt {\left| G \right|} \) for some absolute constant β. In this paper we show that, when G is soluble, then β is at most 5/3. We also show that this is best possible. Furthermore, we show that, in general, for each ε > 0 there exists a constant c _ε such that \(C\left( G \right) \leqslant \left( {1 + \in } \right)\sqrt {\left| G \right|} + {c_ \in }\).