Proof That a Dominant Endmember Formula Can Always Be Written for a Mineral or a Crystal Structure

An endmember formula must be: (1) conformable with the crystal structure of the mineral, (2) electroneutral (i.e., not carry a net electric charge), and (3) irreducible [i.e., not capable of being factored into components that have the same bond topology (atomic arrangement) as that of the original formula]. The stoichiometry of an endmember formula must match the “stoichiometry” of the sites in the structure; for ease of expression, I denote such a formula here as a chemical endmember. In order for a chemical endmember to be a true endmember, the corresponding structure must obey the valence-sum rule of bond-valence theory. For most minerals, the chemical endmember and the (true) endmember are the same. However, where local order would lead to strong deviation from the valence-sum rule for some local arrangements, such arrangements cannot occur and the (true) endmember differs from the chemical endmember. I present heuristic and algebraic proofs that a specific chemical formula can always be represented by a corresponding dominant endmember formula. That dominant endmember may be derived by calculating the difference between the mineral formula considered and all of the possible endmember compositions; the endmember formula which is closest to the mineral formula considered is the dominant endmember.

A MILP Model for a Byzantine Fault Tolerant Blockchain Consensus

Mixed-integer mathematical programming has been widely used to model and solve challenging optimization problems. One interesting feature of this technique is the ability to prove the optimality of the achieved solution, for many practical scenarios where a linear programming model can be devised. This paper explores its use to model very strong Byzantine adversaries, in the context of distributed consensus systems. In particular, we apply the proposed technique to find challenging adversarial conditions on a state-of-the-art blockchain consensus: the Neo dBFT. Neo Blockchain has been using the dBFT algorithm since its foundation, but, due to the complexity of the algorithm, it is challenging to devise definitive algebraic proofs that guarantee safety/liveness of the system (and adjust for every change proposed by the community). Core developers have to manually devise and explore possible adversarial attacks scenarios as an exhaustive task. The proposed multi-objective model is intended to assist the search of possible faulty scenario, which includes three objective functions that can be combined as a maximization problem for testing one-block finality or a minimization problem for ensuring liveness. Automated graphics help developers to visually observe attack conditions and to quickly find a solution. This paper proposes an exact adversarial model that explores current limits for practical blockchain consensus applications such as dBFT, with ideas that can also be extended to other decentralized ledger technologies.

Hidden Symmetries of Weighted Lozenge Tilings

We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this partition function is symmetric for large families of regions. We employ both combinatorial and algebraic proofs.

Detailed Balance $$=$$ Complex Balance $$+$$ Cycle Balance: A Graph-Theoretic Proof for Reaction Networks and Markov Chains

We further clarify the relation between detailed-balanced and complex-balanced equilibria of reversible chemical reaction networks. Our results hold for arbitrary kinetics and also for boundary equilibria. Detailed balance, complex balance, “formal balance,” and the new notion of “cycle balance” are all defined in terms of the underlying graph. This fact allows elementary graph-theoretic (non-algebraic) proofs of a previous result (detailed balance = complex balance + formal balance), our main result (detailed balance = complex balance + cycle balance), and a corresponding result in the setting of continuous-time Markov chains.

Congruence-based proofs of the recognizability theorems for free many-sorted algebras

We generalize several recognizability theorems for free single-sorted algebras to free many-sorted algebras and provide, in a uniform way and without using either regular tree grammars or tree automata, purely algebraic proofs of them based on congruences.