Computing A Glimpse of Randomness

A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly non-computable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 64 bits of a Chaitin Omega: 0000001000000100000110001000011010001111110010111011101000010000. Full description of programs and proofs will be given elsewhere.Comment: 16 pages; Experimental Mathematics (accepted

The Taylor Estimate of Recoverable Strains in Shape-Memory Polycrystals

Shape-memory behavior ls the ability of ccrwin materials to recover, on heating, apparently plastic deformation sustained below a critical temperature. Some materials have good shape-memory behavior as single crystals but little or none as polycrystals, while others have good shape-memory behavior even as polycrystals. Bhattacharya and Kohn (1996. 1997) have proposed a framework to understand this difference. They use energy minimization and the Taylor estimate to argue that the recoverable strains in a polycrystal depend not only on the texture of the polycrystal and the transformation, but critically on the change in symmetry during the underlying martensitic phase transformation. Their results agree with the experimental observations. Shu and Bhattacharya (1997) have also used the Taylor estimate to study the effect of texture in polycrys- tals of Nickel-Titanium and Copper based shape-memory alloys. The use of the Taylor estimate was evaluated in some detail in Bhattacharya and Kohn ( 1997) and more recently in Shu and Bhattacharya (1997) and Shu (1997). In this short report, we summarize the model of recoverable strain and discuss some results that allow us to evaluate the Taylor estimate

^{59}Co NMR evidence for charge ordering below T_{CO}\sim 51 K in Na_{0.5}CoO_2

The CoO$_{2}$ layers in sodium-cobaltates Na$_{x}$CoO$_{2}$ may be viewed as a spin $S=1/2$ triangular-lattice doped with charge carriers. The underlying physics of the cobaltates is very similar to that of the high $T_{c}$ cuprates. We will present unequivocal $^{59}$Co NMR evidence that below $T_{CO}\sim51 K$, the insulating ground state of the itinerant antiferromagnet Na$_{0.5}$CoO$_{2}$ ($T_{N}\sim 86 K$) is induced by charge ordering.Comment: Phys. Rev. Lett. 100 (2008), in press. 4 figure