Pricing European options with a log Student’s t-distribution: A Gosset formula

The distributions of returns for stocks are not well described by a normal probability density function (pdf). Student’s t-distributions, which have fat tails, are known to fit the distributions of the returns. We present pricing of European call or put options using a log Student’s t-distribution, which we call a Gosset approach in honour of W.S. Gosset, the author behind the nom de plume Student. The approach that we present can be used to price European options using other distributions and yields the Black-Scholes formula for returns described by a normal pdf.     

Physical consideration of an image in image restoration using Bayes’ formula

We consider image restoration by Bayes’ formula and investigate the relationship between an image and a prior probability from the following two viewpoints: hyperparameter estimation and the accuracy of a restored image. The Q-Ising model is adopted as a prior probability in Bayes’ formula. Not the Q-Ising energy, but the Potts energy plays an important role in the hyperparameter estimation. From the viewpoint of the hyperparameter estimation, the relationship between a natural image and a prior probability is characterized through the Potts energy and magnetization of an image. The Potts energy and magnetization of an image are defined by a set of pixels’ state of an image. The closer to the average Potts energy and magnetization over a prior probability the Potts energy and magnetization of a natural image is, the closer to the true value of a hyperparameter the estimated value of a hyperparameter from a degraded image is. For the accuracy of a restored image, the image which has a smaller Q-Ising energy is better restored by Bayes’ formula composed of the Q-Ising prior. The consideration for the relationship between an image and a prior probability is expected to be valid for a more complicated prior probability.     

Probability density in the complex plane

The correspondence principle asserts that quantum mechanics resembles classical mechanics in the high-quantum-number limit. In the past few years, many papers have been published on the extension of both quantum mechanics and classical mechanics into the complex domain. However, the question of whether complex quantum mechanics resembles complex classical mechanics at high energy has not yet been studied. This paper introduces the concept of a local quantum probability density ρ(z) in the complex plane. It is shown that there exist infinitely many complex contours C of infinite length on which ρ(z) dz is real and positive. Furthermore, the probability integral is finite. Demonstrating the existence of such contours is the essential element in establishing the correspondence between complex quantum and classical mechanics. The mathematics needed to analyze these contours is subtle and involves the use of asymptotics beyond all orders.     

Properties of the Ising magnet confined in a corner geometry

The properties of Ising square lattices with nearest neighbor ferromagnetic exchange confined in a corner geometry, are studied by means of Monte Carlo simulations. Free boundary conditions at which boundary magnetic fields ±h are applied, i.e., at the two boundary rows ending at the lower left corner a field +h acts, while at the two boundary rows ending at the upper right corner a field −h acts. For temperatures T less than the critical temperature T_c of the bulk, this boundary condition leads to the formation of two domains with opposite orientation of the magnetization direction, separated by an interface which for T larger than the filling transition temperature T_f(h) runs from the upper left corner to the lower right corner, while for T<T_f(h) this interface is localized either close to the lower left corner or close to the upper right corner. It is shown that for T=T_f(h) the magnetization profile m(z) in the z-direction normal to the interface simply is linear and the interfacial width scales as w∝L, while for T>T_f(h) it scales as . The distribution P(?) of the interface position ? (measured along the z-direction from the corners) decays exponentially for T<T_f(h) from either corner, is essentially flat for T=T_f(h), and is a Gaussian centered at the middle of the diagonal for T>T_f(h). Unlike the findings for critical wetting in the thin film geometry of the Ising model, the Monte Carlo results for corner wetting are in very good agreement with the theoretical predictions.     

Tsallis’ statistics in the variability of El Niño/Southern Oscillation during the Holocene epoch

Analysis of the Tsallis q-triplet for the variability of El Niño Southern Oscillation (ENSO) index during the Holocene epoch (last 11,000 years) is presented. Three periods are analyzed, 0-7000, 7000-9700, 9700-11,000 years before the present. During the first and the third periods, the q-index values have the expected usual relations between them (q_sens<1<q_stat<q_rel), and in the second one there is an inversion between q_stat and q_rel (q_stat>q_rel).     

Nonlinear statistical coupling

By considering a nonlinear combination of the probabilities of a system, a physical interpretation of Tsallis statistics as representing the nonlinear coupling or decoupling of statistical states is proposed. The escort probability is interpreted as the coupled probability, with Q=1−q defined as the degree of nonlinear coupling between the statistical states. Positive values of Q have coupled statistical states, a larger entropy metric, and a maximum coupled-entropy distribution of compact-support coupled-Gaussians. Negative values of Q have decoupled statistical states and for −2<Q<0 a maximum coupled-entropy distribution of heavy-tail coupled-Gaussians. The conjugate transformation between the heavy-tail and compact-support domains is shown to be for coupled-Gaussian distributions. This conjugate relationship has been used to extend the generalized Fourier transform to the compact-support domain and to define a scale-invariant correlation structure with heavy-tail limit distribution. In the present paper, we show that the conjugate is a mapping between the source of nonlinearity in non-stationary stochastic processes and the nonlinear coupling which defines the coupled-Gaussian limit distribution. The effects of additive and multiplicative noise are shown to be separable into the coupled-variance and the coupling parameter Q, providing further evidence of the importance of the generalized moments.     

Organization of networks with tagged nodes and biased links: A priori distinct communities: The case of intelligent design proponents and Darwinian evolution defenders

Among the topics of opinion formation it is of interest to observe the characteristics of networks with a priori distinct communities. The citation network(s) between selected members of the Neocreationist and Intelligent Design and the Darwinian Evolution communities are unfolded through the available internet citations. The resulting adjacency matrix is not symmetric. A generalization of considerations pertaining to the case of networks with tagged nodes and biased links, directed or undirected, is presented. The main characteristic coefficients describing the structure of such networks are outlined. The structural features are discussed searching for statistical aspects of the communities. The degree distributions, each network’s assortativity, specific global and local clustering coefficients and the Average Overlap Indices are especially calculated since the distribution of elements in the rectangular submatrices represent inter-community connections. The various closed and open triangles made from nodes, distinguishing the community, are listed. The z-scores of patterns are calculated. One can distinguish between opinion leaders, followers and main rivals and briefly interpret their relationships through intuitively expected behavior in defence of an opinion. Suggestions for more elaborate models describing such communities and their subsequent structures are found in conclusions.     

Modal series solution for an epidemic model

In this article, we generalize a recently proposed method to obtain an exact general solution for the classical Susceptible, Infected, Recovered and Susceptible (SIRS) epidemic mathematical model. This generalization is based upon the nonlinear coupling of two frequencies in an infinite modal series solution. It is shown that these series provide a nonstandard approach in order to obtain an accurate analytical solution for the classical SIRS epidemic model. Numerical results of the SIRS epidemic model for real and complex frequencies are included in order to test the validity and reliability of the method. This method could be applied to a wide class of models in physics, chemistry or engineering.     

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈10⁶) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos.     

Intense laser effects on donor impurity in a cylindrical single and vertically coupled quantum dots under combined effects of hydrostatic pressure and applied electric field

Using the effective mass and parabolic band approximations and a variational procedure we have calculated the combined effects of intense laser radiation, hydrostatic pressure, and applied electric field on shallow-donor impurity confined in cylindrical-shaped single and double GaAs-Ga_1−xAl_xAs QD. Several impurity positions and inputs of the heterostructure dimensions, hydrostatic pressure, and applied electric field have been considered. The laser effects have been introduced by a perturbative scheme in which the Coulomb and the barrier potentials are modified to obtain dressed potentials. Our findings suggest that (1) for on-center impurities in single QD the binding energy is a decreasing function of the dressing parameter and for small dot dimensions of the structures (lengths and radius) the binding energy is more sensitive to the dressing parameter, (2) the binding energy is an increasing/decreasing function of the hydrostatic pressure/applied electric field, (3) the effects of the intense laser field and applied electric field on the binding energy are dominant over the hydrostatic pressure effects, (4) in vertically coupled QD the binding energy for donor impurity located in the barrier region is smaller than for impurities in the well regions and can be strongly modified by the laser radiation, and finally (5) in asymmetrical double QD heterostructures the binding energy as a function of the impurity positions follows a similar behavior to the observed for the amplitude of probability of the noncorrelated electron wave function.     

Mean wind: Its velocity and temperature fluctuation in low-Prandtl-number thermal convection

Large-scale circulation in the confined thermal turbulence of mercury is studied experimentally. The instantaneous velocity profile at 128 points is directly measured using ultrasonic velocimetry and the temperature fluctuation is simultaneously obtained by a small thermistor at the cell center. The periodic velocity oscillation is observed for aspect ratios Γ=1,2, but for Γ=0.5. The temperature oscillation is confirmed only in the Γ=2 cell. It is found that the flow patterns depending on the aspect ratio are characterized sufficiently by the joint probability density function up to Ra=4×10⁸.     

A psychophysical model of decision making

Herein we develop a psychophysical model of decision making based on the difference between objective clock time and the human brain’s perception of time. In this model the utility function is given by the survival probability, which is shown to be a generalized hyperbolic distribution. The parameters of the utility function are fit to intertemporal choice model experimental data and decision making is determined to be a 1/f-noise process.     

Assessing symmetry of financial returns series

Testing symmetry of a probability distribution is a common question arising from applications in several fields. Particularly, in the study of observables used in the analysis of stock market index variations, the question of symmetry has not been fully investigated by means of statistical procedures. In this work a distribution-free test statistic T_n for testing symmetry, derived by Einmahl and McKeague, based on the empirical likelihood approach, is used to address the study of symmetry of financial returns. The asymptotic points of the test statistic T_n are also calculated and a procedure for assessing symmetry for the analysis of the returns of stock market indices is presented.     

An extended clique degree distribution and its heterogeneity in cooperation-competition networks

After Xiao et al. [W.-K. Xiao, J. Ren, F. Qi, Z.W. Song, M.X. Zhu, H.F. Yang, H.Y. Jin, B.-H. Wang, Tao Zhou, Empirical study on clique-degree distribution of networks, Phys. Rev. E 76 (2007) 037102], in this article we present an investigation on so-called k-cliques, which are defined as complete subgraphs of k (k>1) nodes, in the cooperation-competition networks described by bipartite graphs. In the networks, the nodes named actors are taking part in events, organizations or activities, named acts. We mainly examine a property of a k-clique called “k-clique act degree”, q, defined as the number of acts, in which the k-clique takes part. Our analytic treatment on a cooperation-competition network evolution model demonstrates that the distribution of k-clique act degrees obeys Mandelbrot distribution, P(q)∝(q+α)^−γ. To validate the analytical model, we have further studied 13 different empirical cooperation-competition networks with the clique numbers k=2 and k=3. Empirical investigation results show an agreement with the analytic derivations. We propose a new “heterogeneity index”, H, to describe the heterogeneous degree distributions of k-clique and heuristically derive the correlation between H and α and γ. We argue that the cliques, which take part in the largest number of acts, are the most important subgraphs, which can provide a new criterion to distinguish important cliques in the real world networks.     

A local-world heterogeneous model of wireless sensor networks with node and link diversity

In this paper, we present a novel local-world model of wireless sensor networks (WSN) with two kinds of nodes: sensor nodes and sink nodes, which is different from other models with identical nodes and links. The model balances energy consumption by limiting the connectivity of sink nodes to prolong the life of the network. How the proportion of sink nodes, different energy distribution and the local-world scale would affect the topological structure and network performance are investigated. We find that, using mean-field theory, the degree distribution is obtained as an integral with respect to the proportion of sink nodes and energy distribution. We also show that, the model exhibits a mixed connectivity correlation which is greatly distinct from general networks. Moreover, from the perspective of the efficiency and the average hops for data processing, we find some suitable range of the proportion p of sink nodes would make the network model have optimal performance for data processing.     

Intra-Nuclear Cascade (INC) code for simulation of meson photoproduction in nuclei

η meson photoproduction in the ¹⁴N nucleus was studied using simulations done with INC (Intra-Nuclear Cascade) code. The probability for mesons and nucleons to escape the nucleus on different steps of the reaction was evaluated. These results predict the possibility of studying elastic and inelastic η meson interactions with intra-nuclear nucleons using the recoil proton as a tag for the case of slow η meson and fast recoil proton.     

Magnetization and characteristic fields in Nd0.8Ba0.2FeAsO0.6F0.4 with binary doping

In the present work, the iron oxypnictide superconductor of the namely Nd_0.8Ba_0.2FeAsO_0.6F_0.4 is prepared with respect to the observation of distinct magnetization characteristics arising from both electron and hole doping. A magnetothermal phase diagram is given for the present iron oxypnictide system, based on the irreversibility fields (H_irr) and the upper critical fields (H_c2) obtained from magnetotransport measurements, as well as the lower critical fields (H_c1) evaluated by magnetization loops at various temperatures. High H_c2(0) are revealed at low temperature range, which is in consistent with the observations of comparatively high critical current (J_c) and the temperature dependent peak effect in the J_c vs H curves.     

Transport of waves in disordered waveguides: A potential model

We study the statistical properties of wave transport in a disordered waveguide. We first derive the properties of a “building block” (BB) of length δL starting from a potential model consisting of thin potential slices. We then find a diffusion equation—in the space of transfer matrices that describe our system—which governs the evolution with the length L of the disordered waveguide of the transport properties of interest. The latter depend only on the mean free paths and on no other property of the slice distribution. The universality that arises demonstrates the existence of a generalized central-limit theorem. We have developed a numerical simulation in which the universal statistical properties of the BB found analytically are first implemented numerically, and then the various BBs are combined to construct the full waveguide. The reported results thus obtained are in good agreement with microscopic calculations, for both bulk and surface disorder.     

Analytical results of the generalized mean-field theory for diluted magnetic semiconductors with RKKY interaction

Analytical results have been obtained in the framework of the generalized mean-field theory for diluted semiconductors with RKKY interaction. That theory accounts for the non-equivalency of different lattice sites by introducing the distribution function of local effective magnetic fields for non-regular (random) systems with magnetic interaction. The procedure is described that permits to deduce the analytical expression for that function. Corresponding improvement of the traditional mean-field theory could be observed by comparing results of such a generalized analytical model with exact results known for some simple cases, with numerical results of different authors considering the disorder of magnetic impurities’ arrangement, and with experimental data, as well.     

Structural modifications of diamond like carbon films induced by MeV nitrogen ion irradiation

Diamond-like carbon (DLC) films were deposited on Si(1 0 0) substrates using plasma deposition technique. The deposited films were irradiated using 2 MeV N⁺ ions at fluences of 1×10¹⁴, 1×10¹⁵ and 5×10¹⁵ ions/cm². Samples have been characterized by using Raman spectroscopy, X-ray photoelectron spectroscopy (XPS) and high-resolution transmission electron microscopy (HRTEM). Analysis of Raman spectra shows a gradual shift of both D and G band peaks towards higher frequencies along with an increase of the intensity ratio, I(D)/I(G), with increasing ion fluence in irradiation. These results are consistent with an increase of sp² bonding. XPS results also show a monotonic increase of sp²/sp³ hybridization ratio with increasing ion fluence. Plan view TEM images show the formation of clusters in the irradiated DLC films. HRTEM micrographs from the samples irradiated at a fluence of 5×10¹⁵ ions/cm² show the lattice image with an average interplanar spacing of 0.34 nm, revealing that the clusters are graphite clusters. The crystallographic planes in these clusters are somewhat distorted compared to the perfect graphite structure.