We apply the Ginzburg criterion to the dimer problem and we solve the apparent contradiction of a system with mean field α = 12 , the typical value of tricritical systems, and upper critical dimension Dcr = 6. We find that the system has upper critical dimensionDcr = 6 , while for D ≤ 4 it should undergo a first order phase transition. We comment on the latter wrong result examining the approxi...

In 1918, Hardy and Ramanujan wrote their landmark paper deriving the asymptotic formula for the partition function. The paper however was fundamental for another reason, namely for introducing the circle method in questions of additive number theory. Though this method is powerful, it is often difficult and technically complicated to employ. In 2011, Bruinier and Ono discovered a new algebraic ...

Stanley defined a partition function t(n) as the number of partitions λ of n such that the number of odd parts of λ is congruent to the number of odd parts of the conjugate partition λ modulo 4. We show that t(n) equals the number of partitions of n with an even number of hooks of even length. We derive a closed-form formula for the generating function for the numbers p(n)− t(n). As a consequen...

Background and Aims: Whole body vibration exercise by affecting the neuromuscular system and mechanical mechanisms increase muscles strength. This study investigated the effect of whole body vibration training on neuromuscular function of elderly men. Methods: Based on a randomized controlled trial design, 72 retired elderly men with a mean age of 73.00 ± 2.67 years participated in thisstudy. A...