Asymptotic and structural properties of special cases of the Wright function arising in probability theory

This analysis paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory and the theory of stochastic processes. Specifically, we establish new asymptotic properties of the particular Wright function 1Ψ1(ρ, k; ρ, 0; x) = X∞ n=0 Γ(k + ρn) Γ(ρn) x n n! (|x| < ∞) when the parameter ρ ∈ (−1, 0)∪(0, ∞) and the argument x is real. In the probability theory applications, which are focused on studies of the Poisson-Tweedie mixtures, the parameter k is a non-negative integer. Several representations involving well-known special functions are given for certain particular values of ρ. The asymptotics of 1Ψ1(ρ, k; ρ, 0; x) are obtained under numerous assumptions on the behavior of the arguments k and x when the parameter ρ is both positive and negative. We also provide some integral representations and structural properties involving the ‘reduced’ Wright function 0Ψ1(−−; ρ, 0; x) with ρ ∈ (−1, 0) ∪ (0, ∞), which might be useful for the derivation of new properties of members of the power-variance family of distributions. Some of these imply a reflection principle that connects the functions 0Ψ1(−−;±ρ, 0; ·) and certain Bessel functions. Several asymptotic relationships for both particular cases of this function are also given. A few of these follow under additional constraints from probability theory results which, although previously available, were unknown to analysts.

The function of the self-attention network

This commentary links Humphrey and Sui’s proposed Self-attention Network (SAN) to the memory advantage associated with self-relevant information (i.e., the self-reference effect). Articulating this link elucidates the functional quality of the SAN in ensuring that information of potential importance to self is not lost. This adaptive system for self-processing mirrors the cognitive response to threat stimuli, which also elicit attentional biases and produce characteristically enhanced, episodic representations in memory. Understanding the link between the SAN and memory is key to comprehending more broadly the operation of the self in cognition.

The effects of ageing and exercise on skeletal muscle structure and function

Musculoskeletal ageing is associated with profound morphological and functional changes that increase fall risk and disease incidence and is characterised by age-related reductions in motor unit number and atrophy of muscle fibres, particularly type II fibres. Decrements in functional strength and power are relatively modest until the 6th decade, after which the rate of loss exponentially accelerates, particularly beyond the 8th decade of life. Physical activity is a therapeutic modality that can significantly attenuate age-related decline. The underlying signature of ageing, as manifested by perturbed redox homeostasis, leads to a blunting of acute and chronic redox regulated exercise adaptations. Impaired redox regulated exercise adaptations are mechanistically related to altered exercise-induced reactive oxygen and nitrogen species generation and a resultant failure to properly activate redox regulated signaling cascades. Despite the aforementioned specific impairment in redox signaling, exercise induces a plethora of beneficial effects, irrespective of age. There is, therefore, strong evidence for promoting regular physical exercise, especially progressive resistance training as a lifelong habitual practice.