KKM—A Topological Approach For Trees

The Knaster–Kuratowski–Mazurkiewicz (KKM) theorem is a powerful tool in many areas of mathematics. In this paper we introduce a version of the KKM theorem for trees and use it to prove several combinatorial theorems.A 2-tree hypergraph is a family of nonempty subsets of T R (where T and R are trees), each of which has a connected intersection with T and with R. A homogeneous 2-tree hypergraph is a family of subsets of T each of which is the union of two connected sets.For each such hypergraph H we denote by (H) the minimal cardinality of a set intersecting all sets in the hypergraph and by (H) the maximal number of disjoint sets in it.In this paper we prove that in a 2-tree hypergraph (H)2(H) and in a homogeneous 2-tree hypergraph (H)3(H). This improves the result of Alon [3], that (H)8(H) in both cases.Similar results are proved for d-tree hypergraphs and homogeneous d-tree hypergraphs, which are defined in a similar way. All the results improve the results of Alon [3] and generalize the results of Kaiser [1] for intervals.     

Noncommutative Rigidity

Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metric and the Poisson structure that describes the noncommutativity. I illustrate this by computing the obstructions for well known examples of noncommutative geometries and quantum groups. These rigid conditions may cast doubt on the idea of noncommutatively deformed space-time.     

Kinetics of phase transformation on a Bethe lattice in the presence of spin exchange

Kinetics of phase transformation on a Bethe lattice governed by single-spin-flip Glauber and spin-exchange Kawasaki dynamics is examined. For a general Glauber dynamics for which all processes (splitting and coagulation, growth and decay of clusters, as well as creation and annihilation of single-spin clusters) take place, the addition of the Kawasaki dynamics accelerates the transformation process without changing the qualitative behavior. In the growth-decay regime of the Glauber dynamics, regime in which the splitting and coagulation, and creation and annihilation processes due to single-spin flips are negligible, the Kawasaki dynamics strongly increases the fraction of transformed phase because of the splitting and coagulation of clusters induced by the spin-exchange processes. Acting alone, the Kawasaki dynamics leads to the growth of the clusters of each of the phases after the quenching of the temperature to a lower value. When the final temperature T(f) is smaller than a certain temperature T(f0), the average cluster radius grows linearly with time during both the initial and intermediate stages of the kinetic process, and diverges as log(2)(t(d)-t)(-1) when the time t approaches the value t(d) at which infinite clusters arise. It is shown that, among the various spin-exchange processes involved in Kawasaki dynamics, the main contribution is provided by those which decrease or increase the number of clusters by unity.     

Phase transformation in a lattice system in the presence of spin-exchange dynamics

A joint action of the Glauber single-spin-flip and the Kawasaki spin-exchange mechanisms upon the processes of phase transformation is examined in the framework of the one-dimensional kinetic Ising model. It is shown that the addition of the Kawasaki dynamics to that of Glauber accelerates the process of phase transformation in the initial stage, but slows it down in later stages. For the truncated form of Glauber dynamics, which excludes the processes of splitting and coagulation of clusters, the addition of the Kawasaki dynamics always accelerates the phase transformation process. Acting alone, the Kawasaki mechanism provides a cluster growth proportional to t(1/2) (where t is the time) in the initial stage and proportional to t(1/3) (Lifshitz-Slyozov-Wagner law) in the intermediate stage. In the final stage, a cluster size approaches exponentially its equilibrium value.     

In vivo optical properties of normal canine prostate at 732 nm using motexafin lutetium-mediated photodynamic therapy

The optical properties (absorption [mu(a)], transport scattering [mu('s)] and effective attenuation [mu(eff)] coefficients) of normal canine prostate were measured in vivo using interstitial isotropic detectors. Measurements were made at 732 nm before, during and after motexafin lutetium (MLu)-mediated photodynamic therapy (PDT). They were derived by applying the diffusion theory to the in vivo peak fluence rates measured at several distances (3, 6, 9, 12 and 15 mm) from the central axis of a 2.5 cm cylindrical diffusing fiber (CDF). Mu(a) and mu('s) varied between 0.03-0.58 and 1.0-20 cm(-1), respectively. Mu(a) was proportional to the concentration of MLu.Mu(eff) varied between 0.33 and 4.9 cm(-1) (mean 1.3 +/- 1.1 cm(-1)), corresponding to an optical penetration depth (8 = 1/(mu(eff)) of 0.5-3 cm (mean 1.3 +/- 0.8 cm). The mean light fluence rate at 0.5 cm from the CDF was 126 +/- 48 mW/cm2 (N = 22) when the total power from the fiber was 375 mW (150 mW/cm). This study showed significant inter- and intraprostatic differences in the optical properties, suggesting that a real-time dosimetry measurement and feedback system for monitoring light fluences during treatment should be advocated for future PDT studies. However, no significant changes were observed before, during and after PDT within a single treatment site.     

Preferential attachment in the protein network evolution

The Saccharomyces cerevisiae protein-protein interaction map, as well as many natural and man-made networks, shares the scale-free topology. The preferential attachment model was suggested as a generic network evolution model that yields this universal topology. However, it is not clear that the model assumptions hold for the protein interaction network. Using a cross-genome comparison, we show that (a) the older a protein, the better connected it is, and (b) the number of interactions a protein gains during its evolution is proportional to its connectivity. Therefore, preferential attachment governs the protein network evolution. Evolutionary mechanisms leading to such preference and some implications are discussed.     

Quasifactors of ergodic systems with positive entropy

The relation between the two notions, quasifactors and joinings, is investigated and the notion of a joining quasifactor is introduced. We clarify the close connection between quasifactors and symmetric infinite selfjoinings which arises from de Finetti-Hewitt-Savage theorem. Unlike the zero-entropy case where quasifactors seems to preserve some properties of their parent system, it is shown that any ergodic system of positive entropy admits all ergodic systems of positive entropy as quasifactors. A restricted version of this result is obtained for joining quasifactors.