where a = 0 or 1, are among the most famous q-series identities in partitions and combinatorics. Since their discovery the Rogers-Ramanujan identities have been proved and generalized in various ways (see [2, 4, 5, 13] and the references cited there). In [13], by adapting a method of Macdonald for calculating partial fraction expansions of symmetric formal power series, Stembridge gave an unusu...

We investigate differences in the elementary theories of Rogers semilattices of arith-metical numberings, depending on structural invariants of the given families of arithmetical sets. It is shown that at any fixed level of the arithmetical hierarchy there exist infinitely many families with pairwise elementary different Rogers semilattices. For unexplained terminology and notations relative to...

In an article recently published in Thermochimica Acta, Ray Rogers claims that certain results and observations from procedures that he performed on Shroud samples “prove that the radiocarbon sample was not part of the original cloth of the Shroud of Turin.” While I welcome the relative increase in publicity regarding the authenticity and study of the Shroud of Turin which his article generated...

if global trade were fair, it is argued, then international aid would be unnecessary and inequalities inherent to the economic system would be justifiable. here, we argue that while global trade is unfair, in part because richer countries set the rules, we believe that additional interventions must go beyond trade regulation and short-term aid to redress inequalities among countries that will p...

Let P be a linear, elliptic second order symmetric operator, with an associated quadratic form q, and let W be a potential such that the Hardy inequality λ0 ∫ Ω Wu 2 ≤ q(u) holds with (non-negative) best constant λ0. We give sufficient conditions so that the spectrum of the operator 1 W P is [λ0,∞). In particular, we apply this to several well-known Hardy inequalities: (improved) Hardy inequali...

We find sufficient conditions for a probability measure μ to satisfy an inequality of the type ∫ Rd fF ( f ∫ Rd f 2 dμ ) dμ ≤ C ∫ Rd f2c∗ ( |∇f | |f | ) dμ + B ∫ Rd f dμ, where F is concave and c (a cost function) is convex. We show that under broad assumptions on c and F the above inequality holds if for some δ > 0 and ε > 0 one has ∫ ε 0 Φ ( δc [ tF ( 1t ) Iμ(t) ]) dt <∞, where Iμ is the isop...

Given an arbitrary Petri net structure which may have uncontrollable and unobservable transitions and may be unbounded, the procedure described in this paper generates a supervisor for liveness enforcement. The supervisor is specified as a conjunction of linear marking inequalities. For all initial markings satisfying the linear marking inequalities, the supervised Petri net is live. Moreover, ...