Three person baccarat

Baccarat and the closely related game Chemin-de-Fer are played for high stakes in casinos around the world. Optimal strategies for the player and the banker in the two person game have been developed by Kemeny and Snell. Similar strategies are now the rules of play in Nevada. Thorp and Walden developed card counting strategies to make certain side bets profitable but these bets are no longer allowed. Hence, the game is essentially a Bernoulli trial with a banker edge of 1.24%. For the three person game we determine the banker's strategy that optimally counters the two players fixed rules of play. It is optimal for the players to collude and place indentical bets in which case the game becomes a Bernoulli trial with banker's edge of 0.82%.     

Asymptotic normality of cumulant spectral estimates

The paper considers higher-order cumulant spectral estimates obtained by directly Fourier transforming weighted cumulant estimates. Such estimates computationally are different from those based on the finite Fourier transform. These estimates can be looked at continuously as well as directly on submanifolds. The estimates of cumulants are based on unbiased moment estimates. Asymptotic normality is obtained for these estimates and is based on a strong mixing condition and only a finite number of cumulant summability conditions.     

No random reals in countable support iterations

We prove a preservation theorem for limit steps of countable support iterations of proper forcing notions whose particular cases are preservations of the following properties on limit steps: “no random reals are added”, “μ(Random(V))≠1”, “no dominating reals are added”, “Cohen(V) is not comeager”. Consequently, countable support iterations of σ-centered forcing notions do not add random reals. The work was supported by BRF of Israel Academy of Sciences and by grant GA SAV 365 of Slovak Academy of Sciences.     

Synthesis of polyformals from nitro- and fluordiols. Substituent and chainlength effects

A series of primary and secondary diols carrying nitroalkyl, nitro, and fluoro substituents was reacted with formaldehyde under conditions of acid catalysis. Short-chain primary diols gave cyclic formals while predominantly linear polymers were obtained beginning with pentane-1,5-diols and hexane-1,6-diols, depending upon the nature of the substituents. No polymers were formed from secondary diols with 1-nitroalkyl substituents. The effect of monomer structure on polymer formation is discussed. The polymers obtained were characterized by GPC and end group analysis.     

Polar anchoring strength of a tilted nematic: confirmation of the dual easy axis model

The polar anchoring strength coefficient W and polar pretilt angle theta0 were measured simultaneously for the liquid crystal pentylcyanobiphenyl at a rubbed polyimide alignment layer that is ordinarily used for vertical alignment. It was found that W proportional theta(2)0 over the range 0 degrees < or =theta0 less or similar to 35 degrees . The results provide a confirmation of the dual easy axis model, wherein the liquid crystal director adopts an equilibrium orientation theta0 at the substrate that is determined by competition between a pair of preferred orientation directions.     

Projections onto translation—Invariant subspaces of L1(G)

Let G be a locally compact abelian group. A translation-invariant subspace in L₁(G) may or may not be complemented depending on the structure of its hull in ?. Techniques for deciding this complementation problem in a variety of situations are developed and illustrated with examples. A complete characterization is obtained for those ideals with a discrete hull.