Maximal Type Elements for Families

In this paper, we present a variety of existence theorems for maximal type elements in a general setting. We consider multivalued maps with continuous selections and multivalued maps which are admissible with respect to Gorniewicz and our existence theory is based on the author’s old and new coincidence theory. Particularly, for the second section we present presents a collectively coincidence coercive type result for different classes of maps. In the third section we consider considers majorized maps and presents a variety of new maximal element type results. Coincidence theory is motivated from real-world physical models where symmetry and asymmetry play a major role.

Generalized Bernstein Kantorovich operators: Voronovskaya type results, convergence in variation

This paper includes Voronovskaya type results and convergence in variation for the exponential Bernstein Kantorovich operators. The Voronovskaya type result is accompanied by a relation between the mentioned operators and suitable auxiliary discrete operators. Convergence of the operators with respect to the variation seminorm is obtained in the space of functions with bounded variation. We propose a general framework covering the results provided by previous literature.

Guaranteed Upper Bounds For The Velocity Error Of Pressure-Robust Stokes Discretisations

This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one.

Some New Results on Bicomplex Bernstein Polynomials

The aim of this work is to consider bicomplex Bernstein polynomials attached to analytic functions on a compact C2-disk and to present some approximation properties extending known approximation results for the complex Bernstein polynomials. Furthermore, we obtain and present quantitative estimate inequalities and the Voronovskaja-type result for analytic functions by bicomplex Bernstein polynomials.

Yet Another New Variant of Szász–Mirakyan Operator

In this paper, we construct a new variant of the classical Szász–Mirakyan operators, Mn, which fixes the functions 1 and eax,x≥0,a∈R. For these operators, we provide a quantitative Voronovskaya-type result. The uniform weighted convergence of Mn and a direct quantitative estimate are obtained. The symmetry of the properties of the classical Szász–Mirakyan operator and of the properties of the new sequence is investigated. Our results improve and extend similar ones on this topic, established in the last decade by many authors.

Fourier Optimization and Quadratic Forms

We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\ge 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun–Titchmarsh-type results in short intervals and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.

Vocal production learning in mammals revisited

Vocal production learning, the ability to modify the structure of vocalizations as a result of hearing those of others, has been studied extensively in birds but less attention has been given to its occurrence in mammals. We summarize the available evidence for vocal learning in mammals from the last 25 years, updating earlier reviews on the subject. The clearest evidence comes from cetaceans, pinnipeds, elephants and bats where species have been found to copy artificial or human language sounds, or match acoustic models of different sound types. Vocal convergence, in which parameter adjustments within one sound type result in similarities between individuals, occurs in a wider range of mammalian orders with additional evidence from primates, mole-rats, goats and mice. Currently, the underlying mechanisms for convergence are unclear with vocal production learning but also usage learning or matching physiological states being possible explanations. For experimental studies, we highlight the importance of quantitative comparisons of seemingly learned sounds with vocal repertoires before learning started or with species repertoires to confirm novelty. Further studies on the mammalian orders presented here as well as others are needed to explore learning skills and limitations in greater detail. This article is part of the theme issue ‘Vocal learning in animals and humans’.

Relational Quantum Mechanics and the PBR Theorem: A Peaceful Coexistence

According to Relational Quantum Mechanics (RQM) the wave function $$\psi$$ ψ is considered neither a concrete physical item evolving in spacetime, nor an object representing the absolute state of a certain quantum system. In this interpretative framework, $$\psi$$ ψ is defined as a computational device encoding observers’ information; hence, RQM offers a somewhat epistemic view of the wave function. This perspective seems to be at odds with the PBR theorem, a formal result excluding that wave functions represent knowledge of an underlying reality described by some ontic state. In this paper we argue that RQM is not affected by the conclusions of PBR’s argument; consequently, the alleged inconsistency can be dissolved. To do that, we will thoroughly discuss the very foundations of the PBR theorem, i.e. Harrigan and Spekkens’ categorization of ontological models, showing that their implicit assumptions about the nature of the ontic state are incompatible with the main tenets of RQM. Then, we will ask whether it is possible to derive a relational PBR-type result, answering in the negative. This conclusion shows some limitations of this theorem not yet discussed in the literature.

Markovian random iterations of homeomorphisms of the circle

In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle $S^{1}$ , providing a new result on stochastic circle dynamics even for $C^1$ -diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains.