To say that n is the rank of Q is to assert that Q : An → A. So suppose that n and m are natural numbers and Q has rank m as well as rank n. We need to prove that n = m. It follows that An = Am since both An and Am turn out to be the domain of Q. Now in general AX is the set of all functions from X into A. Recalling that n = {0, 1, . . . , n−1} and m = {0, 1, . . . ,m−1} we can understand more ...

Through three examples we illustrate some of the concepts and ingredients required for pattern formation at mesoscopic scales. Two examples build on microscopic models where mesoscopic patterns emerge from homogeneous ground states driven into instability by external forcing. In contrast, the third example builds on a mesoscopic phenomenological Ginzburg-Landau type model of solid-solid structu...

Three seasonal ornamental plants, namely Petunia hybrida, Callistephus chinensis and Impatiens balsamina, were tested for their response to inoculation with mixed indigenous AM culture when grown on a marginal wasteland amended with organic matter. Subsequently, the treatments consisting of AM inoculations were compared with those consisting of recommended chemical fertilizers in terms of growt...

BACKGROUND Recently, anatomic anterior cruciate ligament (ACL) reconstruction is emphasized to improve joint laxity and to potentially avert initiation of cartilage degeneration. There is a paucity of information on the efficacy of ACL reconstructions by currently practiced tunnel creation techniques in restoring normal joint laxity. STUDY DESIGN Controlled laboratory study. HYPOTHESIS Ante...

Background The purpose of this study is to evaluate the intraoperative tension change pattern of each anteromedial (AM) graft and posterolateral (PL) graft and to investigate the optimal femoral tunnel position in double bundle (DB) anterior cruciate ligament reconstruction (ACLR) by comparing postoperative outcomes with each combination of graft tension change pattern. Methods Eighty-four un...

Faced with extreme demands, hypothetical thinking runs the danger of total failure. Paradoxical propositions such as the LIAR (‘I am lying’) provide an opportunity to test it to its limits. Embedded in conditionals, they tended to occasion a breakdown of probabilistic inference (0% True-0% False pattern) demonstrating the vulnerability of hypothetical thinking when taxed by embedded supposition...

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