A partition of a finite poset into chains places a natural upper bound on the size of a union of k antichains. A chain partition is k-saturated if this bound is achieved. Greene and Kleitman [9] proved that, for each k, every finite poset has a simultaneously kand k + 1-saturated chain partition. West [15] showed that the Greene-Kleitman Theorem is best-possible in a strong sense by exhibiting,...

We provide a brief overview of the variational framework for obtaining deterministic approximations or upper bounds for the log-partition function. We also review some of the Monte Carlo based methods for estimating partition functions of arbitrary Markov Random Fields. We then develop an annealed importance sampling (AIS) procedure for estimating partition functions of restricted Boltzmann mac...

Let S be a two colored (red and blue) set of n points in the plane. A subset I of S is an island if there is a convex set C such that I = C∩S. The discrepancy of an island is the absolute value of the number of red minus the number of blue points it contains. A convex partition of S is a partition of S into islands, with disjoint convex hulls. The discrepancy of a convex partition is the discre...

It was recently proved using graph covers (Ruozzi, 2012) that the Bethe partition function is upper bounded by the true partition function for a binary pairwise model that is attractive. Here we provide a new, arguably simpler proof from first principles. We make use of the idea of clamping a variable to a particular value. For an attractive model, we show that summing over the Bethe partition ...

aqueous two-phase systems (atps) have  emerged as a powerful extraction method for the downstream processing of bio-molecules. the aim of this work was to investigate the possibility of utilizing atps for the separation of recombinant bacillus sphaericus phenylalanine dehydrogenase (phedh). polyethylene glycol (peg) and ammonium sulfate systems were selected for our experiment. the effect of di...

Because the partition coefficient is one of the most important parameters affecting microsegregation, the aim of this research is to experimentally analyse the partition coefficient in Al-Mg alloys. In order to experimentally measure the partition coefficient, a series of quenching experiments during solidification were carried out. For this purpose binary Al-Mg alloys containing 6.7 and 10.2 w...

The domatic number of a graph G, denoted dom(G), is the maximum possible cardinality of a family of disjoint sets of vertices of G, each set being a dominating set of G. It is well known that every graph without isolated vertices has dom(G) ≥ 2. For every k, it is known that there are graphs with minimum degree at least k and with dom(G) = 2. In this paper we prove that this is not the case if ...

We present bounds concerning the number of Hartmanis partitions of a finite set. An application of these inequalities improves the known asymptotic lower bound on the number of linear spaces on n points. We also present an upper bound for a certain class of these partitions which bounds the number of Steiner triple and quadruple systems. Recent work has extended the known numerical values for t...