On Volume Functions of Special Flow Polytopes Associated to the Root System of Type $A$

In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type $A$.

Minimum sample size for fitting compatible taper-volume functions for three pine species in Chihuahua

Introduction: The choice of sample size is an important decision in the development of volume models and taper functions. Objective: To calculate the minimum sample size required for fitting compatible taper-volume functions for Pinus arizonica Engelm., P. durangensis Martínez and P. engelmannii Carr. in Chihuahua. Materials and methods: The methodology was divided into three phases: (i) fitting of a linear regression model to the diameter-height data of 50 trees of each species in the three forest regions; (ii) calculation of the minimum sample size required, and (iii) comparison of the goodness of fit of the taper-volume function using both sample sizes. Results and discussion: The minimum number of trees calculated ranged from 53 (Pinus durangensis) to 88 (P. engelmannii) and it is located in the interval reported in studies carried out to estimate the optimal sample size for the development of taper functions. No significant differences were observed in the goodness of fit (α = 0.05) in terms of the R 2 and the root mean square error, using the full sample size and the calculated minimum sample size; no significant effect was observed in the stem volume estimates. Conclusion: The use of small samples in the fit of taper-volume models generates accurate estimates if adequate representation of the study population is ensured.

Book Review: Canvas Detroit

The title Canvas Detroit fulfills a promise long-understood in the Southeastern Michigan community: the Motor City, despite its rough and tumble journey into the 21st century, is home to a vibrant and flourishing arts scene. Far from a standard coffee table book, though, this rich—yet surprisingly accessible—volume functions as a visual ethnography of the artists who define that scene. Hundreds of beautiful images complement dozens of profiles of natives, adopted locals, and famous visitors, who collectively shape Detroit’s culture while representing the resilience of a population indelibly linked to the city’s Rust Belt origins.

The growth of the first non-Euclidean filling volume function of the quaternionic Heisenberg group

The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n + 1, which is strictly faster than the growth rate of the (n + 1)-dimensional filling volume function of the Euclidean space.