Trace norm estimates for products of integral operators and diffusion semigroups

We give trace norm estimates for products of integral operators and for diffusion semigroups. These are applied to differences of heat semigroups. A natural example of an integral operator with finite trace which is not trace class is given.     

Strong asymptotics in Lagrange interpolation with equidistant nodes

In this paper we prove three conjectures of Revers on Lagrange interpolation for f_λ(t)=|t|^λ,λ>0, at equidistant nodes. In particular, we describe the rate of divergence of the Lagrange interpolants L_N( f_λ,t) for 0<|t|<1, and discuss their convergence at t=0. We also establish an asymptotic relation for max_|t|1| |t|^λ−L_N( f_λ,t)|. The proofs are based on strong asymptotics for |t|^λ−L_N( f_λ,t), 0|t|<1.     

Avoider-Enforcer games

Let p and q be positive integers and let H be any hypergraph. In a (p,q,H) Avoider-Enforcer game two players, called Avoider and Enforcer, take turns selecting previously unclaimed vertices of H. Avoider selects p vertices per move and Enforcer selects q vertices per move. Avoider loses if he claims all the vertices of some hyperedge of H; otherwise Enforcer loses. We prove a sufficient condition for Avoider to win the (p,q,H) game. We then use this condition to show that Enforcer can win the (1,q) perfect matching game on K2n for every q?cn/logn for an appropriate constant c, and the (1,q) Hamilton cycle game on Kn for every q?cnloglogloglogn/lognlogloglogn for an appropriate constant c. We also determine exactly those values of q for which Enforcer can win the (1,q) connectivity game on Kn. This result is quite surprising as it substantially differs from its Maker-Breaker analog. Our method extends easily to improve a result of Lu [X. Lu, A note on biased and non-biased games, Discrete Appl. Math. 60 (1995) 285-291], regarding forcing an opponent to pack many pairwise edge disjoint spanning trees in his graph.     

A Modified Szilard Engine: Measurement, Information, and Maxwell's Demon

Using an isolated measurement process, we calculate the effect measurement has on entropy for the multi-cylinder Szilard engine. We find that the system of cylinders possesses an entropy associated with cylinder total energy states, and that it records information transferred at measurement. Contrary to other's results, we find that the apparatus loses entropy due to measurement. The Second Law of Thermodynamics may be preserved if Maxwell's demon gains entropy moving the engine partition.