Mitochondria respond to metabolic demands of the cell and to incremental damage, in part, through dynamic structural changes that include fission (fragmentation), fusion (merging of distinct mitochondria), autophagic degradation (mitophagy), and biogenic interactions with the endoplasmic reticulum (ER). High resolution study of mitochondrial structural and functional relationships requires rapid preservation of specimens to reduce technical artifacts coupled with quantitative assessment of mitochondrial architecture. A practical approach for assessing mitochondrial fine structure using two dimensional and three dimensional high‐resolution electron microscopy is presented, and a systematic approach to measure mitochondrial architecture, including volume, length, hyperbranching, cristae morphology, and the number and extent of interaction with the ER is described. These methods are used to assess mitochondrial architecture in cells and tissue with high energy demand, including skeletal muscle cells, mouse brain tissue, and Drosophila muscles. The accuracy of assessment is validated in cells and tissue with deletion of genes involved in mitochondrial dynamics.

A numerical semigroup $S$ is a cofinite, additively-closed subset of the nonnegative integers that contains $0$. In this paper, we initiate the study of atomic density, an asymptotic measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras. It is known that the atomic density of the polynomial ring $\mathbb{F}_q[x]$ is zero for any finite field $\mathbb{F}_q$; we prove that the numerical semigroup algebra $\mathbb{F}_q[S]$ also has atomic density zero for any numerical semigroup~$S$. We also examine the particular algebra $\mathbb{F}_2[x^2,x^3]$ in more detail, providing a bound on the rate of convergence of the atomic density as well as a counting formula for irreducible polynomials using Möbius inversion, comparable to the formula for irreducible polynomials over a finite field $\mathbb{F}_q$.

This paper is motivated by a practical problem: many U.S. states have public hearings on "communities of interest" as part of their redistricting process, but no state has as yet adopted a concrete method of spatializing and aggregating community maps in order to take them into account in the drawing of new boundaries for electoral districts. Below, we describe a year-long project that collected and synthesized thousands of community maps through partnerships with grassroots organizations and/or government offices. The submissions were then aggregated by geographical clustering with a modified Hausdorff distance; then, the text from the narrative submissions was classified with semantic labels so that short runs of a Markov chain could be used to form semantic sub-clusters. The resulting dataset is publicly available, including the raw data of submitted community maps as well as post-processed community clusters and a scoring system for measuring how well districting plans respect the clusters. We provide a discussion of the strengths and weaknesses of this methodology and conclude with proposed directions for future work.

Abstract In the context of factorization in monoid rings with zero divisors, we study associate relations and the resulting notions of irreducibility and factorization length. Building upon these facts, we determine necessary and sufficient conditions for broad classes of monoid rings to satisfy the ascending chain condition on principal ideals or be a bounded factorization ring. Along the way, we completely characterize the monoid rings that are présimplifiable or domainlike, building upon the results of Ghanem/Wyn-Jones/Al-Ezeh and Anderson/Al-Mallah about group rings with these properties.

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

This article defines amicable pairs in the complex numbers and finds that some amicable pairs in the natural numbers are also amicable in the complex numbers. Unlike the case in the natural numbers, it is proved that no (2, 1) pairs made up of natural numbers where the common factor is a power of 2 exist as Gaussian amicable pairs. Many pairs are found with complex parts using the Divisor-Sigma function in Mathematica. The factorizations into primes is given so that the type of pair might be determined.