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In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant π and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind.

In this paper, we present new inequalities which reveal further relationship for both the inverse tangent function arctan(x) and the inverse hyperbolic function arctanh(x). At the same time, we give the analogue for inverse hyperbolic tangent and other corresponding functions.

In this paper, we obtain some new inequalities which reveal the further relationship between the inverse tangent function arctanx and the inverse hyperbolic sine function sinh−1x. At the same time, we give the analogue for inverse hyperbolic tangent and inverse sine.

Meta-analysis is a method to obtain a weighted average of results from various studies. In addition to pooling effect sizes, meta-analysis can also be used to estimate disease frequencies, such as incidence and prevalence. In this article we present methods for the meta-analysis of prevalence. We discuss the logit and double arcsine transformations to stabilise the variance. We note the special situation of multiple category prevalence, and propose solutions to the problems that arise. We describe the implementation of these methods in the MetaXL software, and present a simulation study and the example of multiple sclerosis from the Global Burden of Disease 2010 project. We conclude that the double arcsine transformation is preferred over the logit, and that the MetaXL implementation of multiple category prevalence is an improvement in the methodology of the meta-analysis of prevalence.

The arcsine square root transformation has long been standard procedure when analyzing proportional data in ecology, with applications in data sets containing binomial and non-binomial response variables. Here, we argue that the arcsine transform should not be used in either circumstance. For binomial data, logistic regression has greater interpretability and higher power than analyses of transformed data. However, it is important to check the data for additional unexplained variation, i.e., overdispersion, and to account for it via the inclusion of random effects in the model if found. For non-binomial data, the arcsine transform is undesirable on the grounds of interpretability, and because it can produce nonsensical predictions. The logit transformation is proposed as an alternative approach to address these issues. Examples are presented in both cases to illustrate these advantages, comparing various methods of analyzing proportions including untransformed, arcsine- and logit-transformed linear models and logistic regression (with or without random effects). Simulations demonstrate that logistic regression usually provides a gain in power over other methods.

In the paper, the authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers, apply a newly established series expansion to derive a closed-form formula for specific partial Bell polynomials and to derive a series representation of generalized logsine function, and deduce combinatorial identities involving the first kind Stirling numbers.

Interspecific killing is a key determinant of the abundances and distributions of carnivores, their prey, and nonprey community members. Similarity of body size has been proposed to lead competitors to seek similar prey, which increases the likelihood of interference encounters, including lethal ones. We explored the influence of body size, diet, predatory habits, and taxonomic relatedness on interspecific killing. The frequency of attacks depends on differences in body size: at small and large differences, attacks are less likely to occur; at intermediate differences, killing interactions are frequent and related to diet overlap. Further, the importance of interspecific killing as a mortality factor in the victim population increases with an increase in body size differences between killers and victims. Carnivores highly adapted to kill vertebrate prey are more prone to killing interactions, usually with animals of similar predatory habits. Family‐level taxonomy influences killing interactions; carnivores tend to interact more with species in the same family than with species in different families. We conclude that although resource exploitation (diet), predatory habits, and taxonomy are influential in predisposing carnivores to attack each other, relative body size of the participants is overwhelmingly important. We discuss the implications of interspecific killing for body size and the dynamics of geographic ranges.

In this paper, we provide a new simple proof of a double inequality of Masjed-Jamei type proved by L. Zhu [1 ] .

This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions and their normalized remainders, explores connections among central factorial numbers, the Stirling numbers, and specific matrix inverses, and derives several closed-form formulas and inequalities. Additionally, this paper reveals new insights into the properties of these mathematical objects, including logarithmic convexity, explicit expressions for certain quantities, and identities involving the Bell polynomials of the second kind.

Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Pólya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Döbler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.