Observation of superfluidity in two- and one-dimensions

Even though there is no long-range-ordered state of a superfluid in dimensions lower than the three-dimension (3D) such as bulk ⁴He liquid, superfluidity has been observed for flat ⁴He films in 2D and recently for nanotubes of ⁴He in 1D by the torsional oscillator method. In the 2D state, in addition to the superfluid below the 2D Kosterlitz–Thouless transition temperature TKT, superfluidity is also observed in a normal fluid state above TKT, which depends strongly on the measurement frequency and the system size. In the 1D state of the nano-tubes, superfluidity is directly observed as a frequency shift in the torsional oscillator experiment. Some calcula-tions suggest a superfluidity of a 1D Bose fluid with a finite length, where thermal excitations of 2-phase winding play the main role for superfluid onset of each tube. Dynamics of the 1D superfluidity is also suggested by observing the dissipation in the torsional oscillator experiment

The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes

Valiant (SIAM J. Comput. 8 (1979) 410–421) showed that the problem of computing the number of simple s–t paths in graphs is #P-complete both in the case of directed graphs and in the case of undirected graphs. Welsh (Complexity: Knots, Colourings and Counting, Cambridge University Press, Cambridge, 1993, p. 17) asked whether the problem of computing the number of self-avoiding walks of a given length in the complete two-dimensional grid is complete for #P1, the tally-version of #P. This paper offers a partial answer to the question of Welsh: it is #P-complete to compute the number of self-avoiding walks of a given length in a subgraph of a two-dimensional grid. Several variations of the problem are also studied and shown to be #P-complete. This paper also studies the problem of computing the number of self-avoiding walks in a subgraph of a hypercube. Similar completeness results are shown for the problem. By scaling the computation time to exponential, it is shown that computing the number of self-avoiding walks in hypercubes is a complete problem for #EXP in the case when a subgraph of a hypercube is specified by its dimension and a boolean circuit that accepts the nodes. Finally, this paper studies the complexity of testing whether a given word over the four-letter alphabet {U,D,L,R} represents a self-avoiding walk in a two-dimensional grid. A linear-space lower bound is shown for nondeterministic Turing machines with a 1-way input head to make this test

Space-Efficient Recognition Of Sparse Self-Reducible Languages

. Mahaney and others have shown that sparse self-reducible sets have time-ecient algorithms, and have concluded that it is unlikely that NP has sparse complete sets. Mahaney's work, intuition, and a 1978 conjecture of Hartmanis notwithstanding, nothing has been known about the density of complete sets for feasible classes until now. This paper shows that sparse self-reducible sets have space-ecient algorithms, and in many cases, even have time-space-ecient algorithms. We conclude that NL, NC k , AC k , LOG(DCFL), LOG(CFL), and P lack complete (or even Turing-hard) sets of low density unless implausible complexity class inclusions hold. In particular, if NL (respectively P, k , or NP) has a polylog-sparse logspace-hard set, then NL SC (respectively P SC, k SC, or PH SC), and if P has subpolynomially sparse logspace-hard sets, then P 6= PSPACE. Subject classications. 68Q15, 03D15. 1. Introduction Complete sets are the quintessences of their complexity cla..

The Complexity of Counting Self-Avoiding Walks in Two-Dimensional Grid Graphs and in Hypercube Graphs

Valiant (SIAM Journal on Computing 8, pages 410--421) showed that the problem of computing the number of simple s-t paths in graphs is #P-complete both in the case of directed graphs and in the case of undirected graphs. Valiant then asked whether the self-avoiding walk problem on the two-dimensional grid, the problem of computing the number of self-avoiding walks of a given length in the two-dimensional grid is complete for #P 1 , the tally-version of #P. This paper offers a partial answer to the question of Valiant. It is shown that computing the number of self-avoiding walks of a given length in the two-dimensional grid graph is #P-complete. The paper also studies several variations of the prolem and shows that all of them are #P-complete. Thi

On Closure Properties of #P in the Context of PF∘#P

For any operatorτon integer-valued functions, we say that #P isclosed under τ in the context ofPF∘#P if, for everyf∈#P,τ[f] belongs to PF∘num;P. For several operatorsτ, it is shown that the closure properties of #P underτin the above sense is closely related to the relationships between P#P[1]and higher classes such as PHPPand PPPP