RANDOMNESS NOTIONS AND REVERSE MATHEMATICS

We investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal.

The Limits of Derivative Actions: The Application of Limitation Periods to Derivative Actions

Limitation periods are an integral and significant aspect of the litigation process in Canada. Although the application of limitation periods may often seem harsh, they are generally considered to be beneficial by bringing stability to society and by providing an incentive to plaintiffs not to “sleep on their rights.” However, in corporate derivative actions (actions brought by a shareholder against directors or officers of the corporation on the corporation’s behalf), the application of a limitation period presents certain issues that could result in such goals not being advanced. Specifically, two main issues arise, namely; who is the claimant for the purposes of limitation periods, and how do limitation periods apply to leave applications? The authors propose that the Canadian judiciary should adopt the adverse domination doctrine, applying the majority test, and explicitly hold that the filing of the leave application is sufficient to bring the derivative action within the limitation period. This approach would be consistent with the separate corporate existence principle and the purposes underlying limitation periods, as well as providing certainty and predictability to the adjudication of derivative action claims.

A note on a dirichlet problem with concave-convex nonlinearity

We prove an existence principle that would apply for elliptic problems with nonlinearity separating into a difference of derivatives of two convex functions in the case when the growth conditions are imposed only on the minuend term. We present abstract result and its application. We modify the so called dual variational method.