Linear Programming Relaxations for Goldreich's Generators over Non-Binary Alphabets

Goldreich suggested candidates of one-way functions and pseudorandom generators included in $\mathsf{NC}^0$. It is known that randomly generated Goldreich's generator using $(r-1)$-wise independent predicates with $n$ input variables and $m=C n^{r/2}$ output variables is not pseudorandom generator with high probability for sufficiently large constant $C$. Most of the previous works assume that the alphabet is binary and use techniques available only for the binary alphabet. In this paper, we deal with non-binary generalization of Goldreich's generator and derives the tight threshold for linear programming relaxation attack using local marginal polytope for randomly generated Goldreich's generators. We assume that $u(n)\in \omega(1)\cap o(n)$ input variables are known. In that case, we show that when $r\ge 3$, there is an exact threshold $\mu_\mathrm{c}(k,r):=\binom{k}{r}^{-1}\frac{(r-2)^{r-2}}{r(r-1)^{r-1}}$ such that for $m=\mu\frac{n^{r-1}}{u(n)^{r-2}}$, the LP relaxation can determine linearly many input variables of Goldreich's generator if $\mu>\mu_\mathrm{c}(k,r)$, and that the LP relaxation cannot determine $\frac1{r-2} u(n)$ input variables of Goldreich's generator if $\mu<\mu_\mathrm{c}(k,r)$. This paper uses characterization of LP solutions by combinatorial structures called stopping sets on a bipartite graph, which is related to a simple algorithm called peeling algorithm.Comment: 14 pages, 1 figur

Temperature-Dependent Cycloidal Magnetic Structure in GdRu2_{2}Al10_{10} Studied by Resonant X-ray Diffraction

We have performed resonant X-ray diffraction experiments on the antiferromagnet GdRu$_{2}$Al$_{10}$ and have clarified that the magnetic structure in the ordered state is cycloidal with the moments lying in the $bc$ plane and propagating along the $b$ axis. The propagation vector shows a similar temperature dependence to the magnetic order parameter, which can be interpreted as being associated with the gap opening in the conduction band and the resultant change in the magnetic exchange interaction. Although the $S=7/2$ state of Gd is almost isotropic, the moments show slight preferential ordering along the $b$ axis. The $c$ axis component in the cycloid develops with decreasing temperature through a tiny transition in the ordered phase. We also show that the scattering involves the $\sigma$-$\sigma'$ process, which is forbidden in normal $E1$-$E1$ resonance of magnetic dipole origin. We discuss the possibility of the $E1$-$E2$ resonance originating from a toroidal moment due to the lack of inversion symmetry at the Gd site. The spin-flop transition in a magnetic field is also described in detail.Comment: 8 pages, 11 figures, accepted for publication in J. Phys. Soc. Jp

Path-integral virial estimator for reaction rate calculation based on the quantum instanton approximation

The quantum instanton approximation is a type of quantum transition state theory that calculates the chemical reaction rate using the reactive flux correlation function and its low order derivatives at time zero. Here we present several path-integral estimators for the latter quantities, which characterize the initial decay profile of the flux correlation function. As with the internal energy or heat capacity calculation, different estimators yield different variances (and therefore different convergence properties) in a Monte Carlo calculation. Here we obtain a virial-type estimator by using a coordinate scaling procedure rather than integration by parts, which allows more computational benefits. We also consider two different methods for treating the flux operator, i.e., local-path and global-path approaches, in which the latter achieves a smaller variance at the cost of using second-order potential derivatives. Numerical tests are performed for a one-dimensional Eckart barrier and a model proton transfer reaction in a polar solvent, which illustrates the reduced variance of the virial estimator over the corresponding thermodynamic estimator.Comment: 23 pages, 5 figures, 1 tabl

Practical Preparation of Epilactose Produced with Cellobiose 2-Epimerase from Ruminococcus albus NE1

A practical purification method for a non-digestible disaccharide, epilactose (4-O-beta-galactosyl-D-mannose), was established. Epilactose was synthesized from lactose with cellobiose 2-epimerase and purified by the following procedure: (i) removal of lactose by crystallization, (ii) hydrolysis of lactose by beta-galactosidase, (iii) digestion of monosaccharides by yeast, and (iv) column chromatography with Na-form cation exchange resin. Epilactose of 91.1% purity was recovered at 42.5% yield

A visual feedback system for micromanipulation with stereoscopic microscope

A stereoscopic microscope is widely used in a micromanipulation such as to operate genes and to inspect integration circuits. As in these tasks the micromanipulation is handled and makes too heavy burden to operators, it is desirable to perform the micromanipulation automatically. In this paper, we propose a visual feedback system for micromanipulation with stereoscopic microscope. This system takes less time to control the manipulator by reducing searching area to detect an object </p