Asymptotic Learning Curve and Renormalizable Condition in Statistical Learning Theory

Bayes statistics and statistical physics have the common mathematical structure, where the log likelihood function corresponds to the random Hamiltonian. Recently, it was discovered that the asymptotic learning curves in Bayes estimation are subject to a universal law, even if the log likelihood function can not be approximated by any quadratic form. However, it is left unknown what mathematical property ensures such a universal law. In this paper, we define a renormalizable condition of the statistical estimation problem, and show that, under such a condition, the asymptotic learning curves are ensured to be subject to the universal law, even if the true distribution is unrealizable and singular for a statistical model. Also we study a nonrenormalizable case, in which the learning curves have the different asymptotic behaviors from the universal law

Formation of hydrogen peroxide and water from the reaction of cold hydrogen atoms with solid oxygen at 10K

The reactions of cold H atoms with solid O2 molecules were investigated at 10 K. The formation of H2O2 and H2O has been confirmed by in-situ infrared spectroscopy. We found that the reaction proceeds very efficiently and obtained the effective reaction rates. This is the first clear experimental evidence of the formation of water molecules under conditions mimicking those found in cold interstellar molecular clouds. Based on the experimental results, we discuss the reaction mechanism and astrophysical implications.Comment: 12 pages, 3 Postscript figures, use package amsmath, amssymb, graphic

The Complexity of Kings

A king in a directed graph is a node from which each node in the graph can be reached via paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king [Lan53]. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of the semifeasible sets [HNP98,HT05] and in the study of the complexity of reachability problems [Tan01,NT02]. In this paper, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is known to belong to $\Pi_2^p$ [HOZZ]. We prove that this bound is optimal: We construct a succinctly specified tournament family whose king problem is $\Pi_2^p$-complete. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is $\Pi_2^p$-complete. We also obtain $\Pi_2^p$-completeness results for k-kings in succinctly specified j-partite tournaments, $k,j \geq 2$, and we generalize our main construction to show that $\Pi_2^p$-completeness holds for testing k-kingship in succinctly specified families of tournaments for all $k \geq 2$

Single-electron transistors in electromagnetic environments

The current-voltage (I-V) characteristics of single-electron transistors (SETs) have been measured in various electromagnetic environments. Some SETs were biased with one-dimensional arrays of dc superconducting quantum interference devices (SQUIDs). The purpose was to provide the SETs with a magnetic-field-tunable environment in the superconducting state, and a high-impedance environment in the normal state. The comparison of SETs with SQUID arrays and those without arrays in the normal state confirmed that the effective charging energy of SETs in the normal state becomes larger in the high-impedance environment, as expected theoretically. In SETs with SQUID arrays in the superconducting state, as the zero-bias resistance of the SQUID arrays was increased to be much larger than the quantum resistance R_K = h/e^2 = 26 kohm, a sharp Coulomb blockade was induced, and the current modulation by the gate-induced charge was changed from e periodic to 2e periodic at a bias point 0<|V|<2D_0/e, where D_0 is the superconducting energy gap. The author discusses the Coulomb blockade and its dependence on the gate-induced charge in terms of the single Josephson junction with gate-tunable junction capacitance.Comment: 8 pages with 10 embedded figures, RevTeX4, published versio

Transition from Band insulator to Bose-Einstein Condensate superfluid and Mott State of Cold Fermi Gases with Multiband Effects in Optical Lattices

We study two models realized by two-component Fermi gases loaded in optical lattices. We clarify that multi-band effects inevitably caused by the optical lattices generate a rich structure, when the systems crossover from the region of weakly bound molecular bosons to the region of strongly bound atomic bosons. Here the crossover can be controlled by attractive fermion interaction. One of the present models is a case with attractive fermion interaction, where an insulator-superfluid transition takes place. The transition is characterized as the transition between a band insulator and a Bose-Einstein condensate (BEC) superfluid state. Differing from the conventional BCS superfluid transition, this transition shows unconventional properties. In contrast to the one particle excitation gap scaled by the superfluid order parameter in the conventional BCS transition, because of the multi-band effects, a large gap of one-particle density of states is retained all through the transition although the superfluid order grows continuously from zero. A reentrant transition with lowering temperature is another unconventionality. The other model is the case with coexisting attractive and repulsive interactions. Within a mean field treatment, we find a new insulating state, an orbital ordered insulator. This insulator is one candidate for the Mott insulator of molecular bosons and is the first example that the orbital internal degrees of freedom of molecular bosons appears explicitly. Besides the emergence of a new phase, a coexisting phase also appears where superfluidity and an orbital order coexist just by doping holes or particles. The insulating and superfluid particles show differentiation in momentum space as in the high-Tc cuprate superconductors.Comment: 13 pages, 10 figure

Boolean Operations, Joins, and the Extended Low Hierarchy

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL_2, yet their join is in EL_2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL_ and prove that EL_2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL_2 lower bounds for certain notions generalizing Selman's P-selectivity, which may be regarded as an interesting result in its own right.Comment: 12 page

Quantum Valence Criticality as Origin of Unconventional Critical Phenomena

It is shown that unconventional critical phenomena commonly observed in paramagnetic metals YbRh2Si2, YbRh2(Si0.95Ge0.05)2, and beta-YbAlB4 is naturally explained by the quantum criticality of Yb-valence fluctuations. We construct the mode coupling theory taking account of local correlation effects of f electrons and find that unconventional criticality is caused by the locality of the valence fluctuation mode. We show that measured low-temperature anomalies such as divergence of uniform spin susceptibility \chi T^{-\zeta) with $\zeta~0.6$ giving rise to a huge enhancement of the Wilson ratio and the emergence of T-linear resistivity are explained in a unified way.Comment: 5 pages, 3 figures, to be published in Physical Review Letter