Abstract:
Many solutions for scientific problems rely on finding the first (largest) eigenvalue and eigenvector of a particular matrix. We explore the distribution of the first eigenvector of a symmetric random sparse matrix. To analyze the properties of the first eigenvalue/vector, we employ a methodology based on the cavity method, a well-established technique in the statistical physics. A symmetric random sparse matrix in this paper can be regarded as an adjacency matrix for a network. We show that if a network is constructed by nodes that have two different types of degrees then the distribution of its eigenvector has fat tails such as the stable distribution ($\alpha < 2 $) under a certain condition; whereas if a network is constructed with nodes that have only one type of degree, the distribution of its first eigenvector becomes the Gaussian approximately. The cavity method is used to clarify these results.

Abstract:
Changes (returns) in stock index prices and exchange rates for currencies are argued, based on empirical data, to obey a stable distribution with characteristic exponent $ \alpha < 2 $ for short sampling intervals and a Gaussian distribution for long sampling intervals. In order to explain this phenomenon, an Ehrenfest model with large jumps (ELJ) is introduced to explain the empirical density function of price changes for both short and long sampling intervals.

Abstract:
The present paper introduces a majority orienting model in which the dealers' behavior changes based on the influence of the price to show the oscillation of stock price in the stock market. We show the oscillation of the price for the model by applying the van der Pol equation which is a deterministic approximation of our model.

Abstract:
The properties of the first (largest) eigenvalue and its eigenvector (first eigenvector) are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions. In principle, one should be able to accurately calculate them by solving a functional equation concerning auxiliary fields which come out in an analysis based on replica/cavity methods. However, the difficulty in analytically solving this equation makes an accurate calculation infeasible in practice. To overcome this problem, we develop approximation schemes on the basis of two exceptionally solvable examples. The schemes are reasonably consistent with numerical experiments when the statistical bias of positive matrix entries is sufficiently large, and they qualitatively explain why considerably large finite size effects of the first eigenvalue can be observed when the bias is relatively small.

Abstract:
Multi-dimensional density of states provides a useful description of complex frustrated systems. Recent advances in Monte Carlo methods enable efficient calculation of the density of states and related quantities, which renew the interest in them. Here we calculate density of states on the plane (energy, magnetization) for an Ising Model with three-spin interactions on a random sparse network, which is a system of current interest both in physics of glassy systems and in the theory of error-correcting codes. Multicanonical Monte Carlo algorithm is successfully applied, and the shape of densities and its dependence on the degree of frustration is revealed. Efficiency of multicanonical Monte Carlo is also discussed with the shape of a projection of the distribution simulated by the algorithm.

Abstract:
A methodology to analyze the properties of the first (largest) eigenvalue and its eigenvector is developed for large symmetric random sparse matrices utilizing the cavity method of statistical mechanics. Under a tree approximation, which is plausible for infinitely large systems, in conjunction with the introduction of a Lagrange multiplier for constraining the length of the eigenvector, the eigenvalue problem is reduced to a bunch of optimization problems of a quadratic function of a single variable, and the coefficients of the first and the second order terms of the functions act as cavity fields that are handled in cavity analysis. We show that the first eigenvalue is determined in such a way that the distribution of the cavity fields has a finite value for the second order moment with respect to the cavity fields of the first order coefficient. The validity and utility of the developed methodology are examined by applying it to two analytically solvable and one simple but non-trivial examples in conjunction with numerical justification.

Abstract:
We study, in random sparse networks, finite size scaling of the spin glass susceptibility $\chi_{\rm SG}$, which is a proper measure of the de Almeida-Thouless (AT) instability of spin glass systems. Using a phenomenological argument regarding the band edge behavior of the Hessian eigenvalue distribution, we discuss how $\chi_{\rm SG}$ is evaluated in infinitely large random sparse networks, which are usually identified with Bethe trees, and how it should be corrected in finite systems. In the high temperature region, data of extensive numerical experiments are generally in good agreement with the theoretical values of $\chi_{\rm SG}$ determined from the Bethe tree. In the absence of external fields, the data also show a scaling relation $\chi_{\rm SG}=N^{1/3}F(N^{1/3}|T-T_c|/T_c)$, which has been conjectured in the literature, where $T_c$ is the critical temperature. In the presence of external fields, on the other hand, the numerical data are not consistent with this scaling relation. A numerical analysis of Hessian eigenvalues implies that strong finite size corrections of the lower band edge of the eigenvalue distribution, which seem relevant only in the presence of the fields, are a major source of inconsistency. This may be related to the known difficulty in using only numerical methods to detect the AT instability.

Abstract:
This paper presents a detailed characterization of the trajectory of a single housefly with free range of a square cage. The trajectory of the fly was recorded and transformed into a time series, which was fully analyzed using an autoregressive model, which describes a stationary time series by a linear regression of prior state values with the white noise. The main discovery was that the fly switched styles of motion from a low dimensional regular pattern to a higher dimensional disordered pattern. This discovered exploratory behavior is, irrespective of the presence of food, characterized by anomalous diffusion.

A novel technique of lithotripsy was investigated with a
mid-infrared tunable pulsed laser using difference-frequency generation (DFG).
Human gallstone samples obtained from 24 patients were analyzed with their
infrared absorption spectra. It was found that the principal components of the
gallstones were different for the different patients and that the gallstone
samples used in this research could be classified into four groups, i.e., mixed stones, calcium bilirubinate
stones, cholesterol stones, and calcium carbonate stones. In addition, some
gallstone samples had different compositions within the single stone. The
mid-infrared laser tunable within a wavelength range of 5.5 - 10 μm was
irradiated to the cholesterol stones at two different wavelengths of 6.83 and
6.03 μm, where the cholesterol stones had relatively strong and weak absorption
peaks, respectively. As the result, the cholesterol stones were more
efficiently ablated at the wavelength of 6.83 μm with the strong absorption
peak. Therefore, it is suggested that the gallstones could be efficiently ablated
by tuning the wavelength of the laser to the strong absorption peak of the
gallstones. The higher efficiency of the ablation using the characteristic
absorption peaks should lead to the safer treatment without damage to the
surrounding normal tissues. In order to identify the composition of the
gallstones in the patients, endoscopic and spectroscopic diagnosis using the
DFG laser and an optical fiber probe made with two hollow optical fibers and a
diamond attenuation total reflection prism should be useful. The absorption
spectrum of the gallstones in the patients could be measured by measuring the
energy of the DFG laser transmitted through the optical fiber probe and by
scanning the wavelength of the DFG laser.

Since carbon dioxide laser is
excellent for incision, hemostasis, coagulation, and vaporization of soft
tissues, it has been widely applied in clinical treatments as the laser knife.
In these days, flexible thin hollow optical fibers transmitting mid-infrared
light have been developed, and the application of carbon dioxide laser to
endoscopic therapy has become possible. However, it is expected that the
irradiation effect is influenced by the change in the laser power at the tip of
the hollow optical fiber due to the change in the transmittance by the bending
loss. The purpose of this research is to quantitatively evaluate the change in
the output power and therapeutic effect by bending the hollow optical fiber in
a gastrointestinal endoscope. The change in the transmittance of the hollow
optical fiber due to the insertion of the fiber into the endoscope and bending
of the head of the endoscope was measured. Then, the relationship between the
irradiated laser power and the incision depth for a porcine stomach was
investigated. As the results, the most significant decrease in the
transmittance of the hollow optical fiber was caused by the insertion of the
fiber into the instrument channel of the endoscope, and
bending of the head of the endoscope with the angle of 90° decreased the output
laser power and incision depth by 10% and 25%, respectively. Therefore, it was
confirmed that the bending loss of the hollow optical fiber due to the bending
of the head of the endoscope had no significant influence on the endoscopic
therapy using the carbon dioxide laser.