Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations

A partial algebra construction of Gr\"atzer and Schmidt from "Characterizations of congruence lattices of abstract algebras" (Acta Sci. Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to a well-known fact that every finite distributive lattice is representable, seen as a special case of the Finite Lattice Representation Problem. The construction of this proof brings together Birkhoff's representation theorem for finite distributive lattices, an emphasis on boolean lattices when representing finite lattices, and a perspective based on inequalities of partially ordered sets. It may be possible to generalize the techniques used in this approach. Other than the aforementioned representation theorem only elementary tools are used for the two theorems of this note. In particular there is no reliance on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice to be representable (see William J. DeMeo), such as the closure method

Structures and proton-pumping strategies of mitochondrial respiratory enzymes

Enzymes of the mitochondrial respiratory chain serve as proton pumps, using the energy made available from electron transfer reactions to transport protons across the inner mitochondrial membrane and create an electrochemical gradient used for the production of ATP. The ATP synthase enzyme is reversible and can also serve as a proton pump by coupling ATP hydrolysis to proton translocation. Each of the respiratory enzymes uses a different strategy for performing proton pumping. In this work, the strategies are described and the structural bases for the action of these proteins are discussed in light of recent crystal structures of several respiratory enzymes. The mechanisms and efficiency of proton translocation are also analyzed in terms of the thermodynamics of the substrate transformations catalyzed by these enzymes

Modeling Braids with Space-Varying and Time-Varying Stranded Cellular Automata

Braids in a traditional sense and braids in a mathematical sense are wildly different outlooks on the same concept. Using cellular automata to represent and analyze braids is a way to bridge the gap between them. Joshua and Lana Holden and Hao Yang have previously worked on developing and expanding upon a Stranded Cellular Automata (SCA) model capable of representing many different braids and weaves. Continuing their work, we were able to devise a more user-friendly method for interacting with the model such that even those without a mathematical background can construct and analyze braids of their own. This paper will also discuss the addition of space-varying and time-varying rulesets to expand upon the types of braids and weaves the SCA model is able to represent