Plus and minus logarithms and Amice transform

We give a new description of Pollack's plus and minus $p$-adic logarithms $\log_p^\pm$ in terms of distributions. In particular, if $\mu_\pm$ denote the pre-images of $\log_p^\pm$ under the Amice transform, we give explicit formulae for the values $\mu_\pm(a+p^n\mathbb{Z}_p)$ for all $a\in \mathbb{Z}_p$ and all integers $n\ge1$. Our formulae imply that the distribution $\mu_-$ agrees with a distribution studied by Koblitz in 1977. Furthermore, we show that a similar description exists for Loeffler's two-variable analogues of these plus and minus logarithms.Comment: 9 page

A review of the value of innovation in inhalers for COPD and asthma

Background: Appropriate use of inhaled therapies for asthma and chronic obstructive pulmonary disease (COPD) is critical to ensuring good patient outcomes, efficient use of healthcare resources and limiting the effects of high-morbidity. The appropriate choice of inhaler and active therapy, incorporating patient preferences, can help improve treatment adherence and long-term outcomes. Despite this, many current inhalers are non-intuitive to use, and require extensive training. Methods: In this review, an expert panel considers the evidence for the use of inhaler devices in management of COPD and asthma. The panel also evaluates the value of innovation in inhaler technologies, which optimise the use of existing molecules from a clinical, economic and societal perspective. Conclusions: The panel conclusion is that there remains a substantial unmet need in inhaler technology and that innovation in inhaler devices can provide real-world health benefits to patients. Furthermore, we recommend that these innovations should be supported by healthcare systems through appropriate pricing and reimbursement mechanisms. Keywords: Asthma; Chronic obstructive pulmonary disease; innovation; patient preference; adherence; patient compliance; metered dose inhaler; dry powder inhaler; medication errors; valu