We introduce the Jordan product associated with the second-order cone K into the real Hilbert space H, and then define a one-parametric class of complementarity functions t on H H with the parameter t[0, 2). We show that the squared norm of t with t(0, 2) is a continuously F (rchet)-differentiable merit function. By this, the second-order cone complementarity problem (SOCCP) in H can be converted into an unconstrained smooth minimization problem involving this class of merit functions, and furthermore, under the monotonicity assumption, every stationary point of this minimization problem is shown to be a solution of the SOCCP.

We establish the no trade principle, ie, the no trade theorem and its converse, for any dual pair of bet and extended belief spaces, defined on a given measurable space. A key condition is that, except perhaps one of the agents, everyone else has (weak*) compact sets of beliefs. We find out that in most of the models of uncertainty adopted in the economic literature, roughly speaking, the epistemic statement that an agent has a compact set of beliefs is equivalent to the economic statement that he has an open cone of positive bets. This improves our understanding of what compactness actually means within an economic context.

Using elementary arguments, we shall show that every bounded disjointness preserving linear functional of the Wiener ring L1 (Z) assumes the form (f)=

In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks operators will be demonstrated.

Let A , A be two regular commutative unital Banach algebras such that A is integral over A . In 2003, Dawson and Feinstein showed that the topological stable rank A whenever A . In this note, we investigate whether we will have A in general. For instance, when A is a commutative unital A -algebra, we show that A , and the equality holds at least when the integral extension is separable. In general, A and A have the same Bass stable ranks A .