Reluctant powers? Rising powers’ contributions to regional crisis management

The U.s. and Rising Powers

In 1999, in the wake of the Asian financial crisis, the meltdown of the Long-Term Capital Management hedge fund spooking the markets and economic crises in Russia and Brazil, the cover of Time magazine featured “The Committee to Save the World”: the chairman of the U.S. Federal Reserve and the secretary and deputy secretary of the U.S. Treasury. With the cold war over and with what U.S. Secreta...

Emerging Powers in Regional Architecture

Comparing Powers and Symbolic Powers of Ideals

We develop tools to study the problem of containment of symbolic powers I(m) in powers I for a homogeneous ideal I in a polynomial ring k[P ] in N + 1 variables over an arbitrary algebraically closed field k. We obtain results on the structure of the set of pairs (r, m) such that I(m) ⊆ I. As corollaries, we show that I2 contains I(3) whenever S is a finite generic set of points in P2 (thereby ...

Powers of Two Modulo Powers of Three

Since 2 is a primitive root of 3 for each positive integer m, the set of points {(n, 2 mod 3) : n > 0}, viewed as a subset of Z>0×Z>0 is bi-periodic, with minimal periods φ(3) (horizontally) and 3 (vertically). We show that if one considers the classes of n modulo 6, one obtains a finer structural classification. This result is presented within the context of the question of strong normality of...

Raising to powers revisited

We consider the theory of algebraically closed fields of characteristic zero with raising to powers operations. In an earlier paper we have described complete first-order theories of such a structures, provided that a diophantine conjecture CIT does hold. Here we get rid of this assumption. The theory of complex numbers with raising to real powers satisfies the description if Schanuel’s conject...