Representations of boolean functions as polynomials (over rings) have been used to establish lower bounds in complexity theory. Such representations were used to great effect by Smolensky, who established that MOD q / ∈ AC[MOD p] (for distinct primes p, q) by representing AC[MOD p] functions as low-degree multilinear polynomials over fields of characteristic p. Another tool which has yielded in...

We adapt the commutator theory of universal algebra to particular setting racks and quandles, exploiting a Galois connection between congruences certain normal subgroups displacement group. Congruence properties, such as abelianness centrality, are reflected by corresponding relative groups, global solvability nilpotence, properties whole To show new tool in action, we present three application...

Residually thin and nilpotent table algebras, which are abstractions of fusion rings adjacency algebras association schemes, defined investigated. A formula for the degrees basis elements in residually is established, yields an integrality result Gelaki Nikshych as immediate corollary; it shown that this holds only such algebras. These theorems specialize to new results schemes. Bi-anchored thi...

In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set stage, review notion of multiplet for general super Lie algebra, working in context BV BRST formalisms. Building on this, explain how can be viewed as constructing supermultiplet out input datum an equivariant graded module over ring functions nilpotence variet...

Abstract For a subring $R$ of the rational numbers, we study -localization functors in local homotopy theory simplicial presheaves on small site and then ${\mathbb {A}}^1$ -homotopy theory. To this end, introduce analyze two notions nilpotence for spaces theory, paying attention to future applications vector bundles. We show that behaves controlled fashion nilpotent consider. classifying space ...