In this paper, we present and prove the first closed formula bounding the degree of regularity of an HFE system over an arbitrary finite field. Though these bounds are not necessarily optimal, they can be used to deduce 1. if D, the degree of the corresponding HFE polynomial, and q, the size of the corresponding finite field, are fixed, inverting HFE system is polynomial for all fields; 2. if D...

The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for pm(n) for any desired m. We do this to demonstrate the power of “rigorous guessing” as facilitated by the quasi-polynomial ansatz. ∗E-mail addresses: ASills@GeorgiaSouth...

A new family of limited-memory variationally-derived variable metric or quasi-Newton methods for unconstrained minimization is given. The methods have quadratic termination property and use updates, invariant under linear transformations. Some encouraging numerical experience is reported.

We define theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth threshold circuits of polynomial and quasipolynomial size. Then we define certain second-order theories and show that they characterize the functions in the Counting Hierarchy. Finally we show that the former theories are isomorphic to the latter via the socalled RSUV -isomorphism.

In this note we study the higher direct image functors Rf∗(−) and the higher coinverse image functors Rf (−) which will play a role in our study of Serre duality. The main theorem is the proof that if F is quasi-coherent then so is Rf∗(F ), which we prove first for noetherian schemes and then more generally for quasi-compact quasi-separated schemes. Most proofs are from either Hartshorne’s book...

We discuss recent results on the propositional proof complexity of Frege proof systems, including some recently discovered quasipolynomial size proofs for the pigeonhole principle and the Kneser-Lovász theorem. These are closely related to formalizability in bounded arithmetic.

A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points x in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness functions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi...

Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly ...

We consider a q-ary quasi-cyclic code C of length m` and index `, where both m and ` are relatively prime to q. If the constituents of C are cyclic codes, we show that C can also be viewed as a 2-D cyclic code of size m × ` over Fq. If we further assume that m and ` are also coprime to each other, then we easily observe that the code C must be equivalent to a cyclic code. The last fact was prov...

An elementary argument constructs, for each cardinal α, a topological space whose subspaces, ordered by homeomorphic embeddability, can model every partial order on α-many points. We show how to modify this procedure to deal also with quasi-orders (where the antisymmetry condition may fail), obtaining an initial estimate of the cardinality of the space then required.