Convergence of the Nelder-Mead method

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

Purely Iterative Algorithms for Newton’s Maps and General Convergence

The aim of this paper is to study the local dynamical behaviour of a broad class of purely iterative algorithms for Newton’s maps. In particular, we describe the nature and stability of fixed points and provide a type of scaling theorem. Based on those results, we apply a rigidity theorem in order to study the parameter space of cubic polynomials, for a large class of new root finding algorithms. Finally, we study the relations between critical points and the parameter space.

On Convergence Rates of Some Limits

In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each y > 0 , x L ( x + y ) L ( x ) − 1 → 0 as x → ∞ . Very recently, we have extended this result by considering a wider class of functions U related to the following more general condition. For each y > 0 , r ( x ) U ( x + y g ( x ) ) U ( x ) − 1 → 0 as x → ∞ , for some functions r and g. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.

Convergence theorems for the Nelder-Mead method

We develop a matrix form of the Nelder-Mead method and after discussing the concept of convergence we prove a general convergence theorem. The new theorem is demonstrated in low dimensional spaces.

Convergence rates for estimators of geodesic distances and Frèchet expectations

Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

Human nature, science of, in the 18th century

Eighteenth-century speculation on human nature is distinguishable by its approach and underlying assumptions. Taking their cue from Francis Bacon and Isaac Newton, many philosophers of the Enlightenment endeavoured to extend the methods of natural science to the moral sciences. Perhaps the most explicit of such endeavours was David Hume’s ambition for a ‘science of man’, but he was not alone. There was a general convergence on the idea that human nature is constant and uniform in its operating principles – that is, its determining motives (passions), its source of knowledge (sense experience) and its mode of operation (association of ideas). By virtue of this constancy human nature was predictable, so that once it was scientifically understood, then social institutions could be designed to effect desired outcomes.