Entropy-Based Greedy Algorithm for Decision Trees Using Hypotheses

In this paper, we consider decision trees that use both conventional queries based on one attribute each and queries based on hypotheses of values of all attributes. Such decision trees are similar to those studied in exact learning, where membership and equivalence queries are allowed. We present greedy algorithm based on entropy for the construction of the above decision trees and discuss the results of computer experiments on various data sets and randomly generated Boolean functions.     

Decision Rules Derived from Optimal Decision Trees with Hypotheses

Conventional decision trees use queries each of which is based on one attribute. In this study, we also examine decision trees that handle additional queries based on hypotheses. This kind of query is similar to the equivalence queries considered in exact learning. Earlier, we designed dynamic programming algorithms for the computation of the minimum depth and the minimum number of internal nodes in decision trees that have hypotheses. Modification of these algorithms considered in the present paper permits us to build decision trees with hypotheses that are optimal relative to the depth or relative to the number of the internal nodes. We compare the length and coverage of decision rules extracted from optimal decision trees with hypotheses and decision rules extracted from optimal conventional decision trees to choose the ones that are preferable as a tool for the representation of information. To this end, we conduct computer experiments on various decision tables from the UCI Machine Learning Repository. In addition, we also consider decision tables for randomly generated Boolean functions. The collected results show that the decision rules derived from decision trees with hypotheses in many cases are better than the rules extracted from conventional decision trees.     

基于机器学习的合金催化剂表面吸附能预测

近年来,机器学习方法逐渐成为多相催化中的一种关键研究手段. 二元合金材料作为重要的催化剂之一,在双功能催化剂的筛选中受到了广泛的关注. 本文提出了一个将机器学习方法应用在预测催化性质上的整体框架,从而快速预测原子、分子在金属和二元合金表面的吸附能. 通过测试不同的机器学习方法来评估它们对于该问题的适用性,并将树集成的方法与压缩感知方法相结合,利用约6×10⁴个吸附能数据构建了预测模型. 相对于线性比例关系,该方法可以更准确地预测大量合金上的吸附能(预测的均方根误差降低一半),并且更通用地预测各种吸附物的能量,为发现新的双金属催化剂铺平了道路.     

VIBRATIONS OF NON-UNIFORM CONTINUOUS BEAMS UNDER MOVING LOADS

The dynamic behavior of multi-span non-uniform beams transversed by a moving load at a constant and variable velocity is investigated. The continuous beam is modelled using Bernoulli-Euler beam theory. The solution is obtained by using both the modal analysis method and the direct integration method. The natural frequencies and mode shapes used in the solution of this problem are obtained exactly by deriving the exact dynamic stiffness matrices for any polynomial variation of the cross-section along the beam using the exact element method. The mode shapes are expressed as infinite polynomial series. Using the exact mode shapes yields the exact solution for general variation of the beam section in case of constant and variable velocity. Numerical examples are presented in order to demonstrate the accuracy and the effectiveness of the present study, and the results are compared to previously published results.     

Disordered Dirac fermions: Multifractality termination and logarithmic conformal field theories

We reexamine in detail the problem of fermions interacting with a non-Abelian random vector potential. Without resorting to the replica or supersymmetry approaches, we show that in the limit of infinite disorder strength the theory possesses an exact solution which takes the form of a logarithmic conformal field theory. We show that the proper treatment of the locality conditions in the SU(2) theory leads to the termination of the multifractal spectrum, or in other words to the termination of the infinite hierarchies of negative-dimensional operators that were thought to occur. Based on arguments of logarithmic degeneracies, we conjecture that such a termination mechanism should be present for general SU(N). Moreover, our results lead to the conclusion that the previous replica solution of this problem yields incorrect results.     

Inhomogeneous swelling in filled elastomers

An exact formulation is given for the inhomogeneous swelling of an elastomeric matrix containing a spherical inclusion, and is presented in a form amenable to solution for arbitrary free energy of mixing and network strain energy functions, rigid or soft inclusions, and finite or infinite matrix size. Specific examples of the deformation field, stress field, and composition variation are given as functions of distance from a rigid inclusion imbedded in an infinite matrix displaying Flory-Huggins/Gaussian swelling behavior. Interfacial stresses and extension ratios are given as functions of cross-link density and Flory-Huggins X parameter.

In a related study published elsewhere, Kotani and Sternstein have obtained experimental verification of the theory using birefringence techniques on model-filled elastomers. Stein and co-workers have found similar agreement of the theory with their light scattering measurements.     

Probability Distribution on Full Rooted Trees

The recursive and hierarchical structure of full rooted trees is applicable to statistical models in various fields, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is not a random variable; as such, model selection to avoid overfitting is problematic. One method to solve this problem is to assume a prior distribution on the full rooted trees. This enables the optimal model selection based on Bayes decision theory. For example, by assigning a low prior probability to a complex model, the maximum a posteriori estimator prevents the selection of the complex one. Furthermore, we can average all the models weighted by their posteriors. In this paper, we propose a probability distribution on a set of full rooted trees. Its parametric representation is suitable for calculating the properties of our distribution using recursive functions, such as the mode, expectation, and posterior distribution. Although such distributions have been proposed in previous studies, they are only applicable to specific applications. Therefore, we extract their mathematically essential components and derive new generalized methods to calculate the expectation, posterior distribution, etc.     

Optimizing Age Penalty in Time-Varying Networks with Markovian and Error-Prone Channel State

In this paper, we consider a scenario where the base station (BS) collects time-sensitive data from multiple sensors through time-varying and error-prone channels. We characterize the data freshness at the terminal end through a class of monotone increasing functions related to Age of information (AoI). Our goal is to design an optimal policy to minimize the average age penalty of all sensors in infinite horizon under bandwidth and power constraint. By formulating the scheduling problem into a constrained Markov decision process (CMDP), we reveal the threshold structure for the optimal policy and approximate the optimal decision by solving a truncated linear programming (LP). Finally, a bandwidth-truncated policy is proposed to satisfy both power and bandwidth constraint. Through theoretical analysis and numerical simulations, we prove the proposed policy is asymptotic optimal in the large sensor regime.     

Asymmetric and axisymmetric vibrations of finite transversely isotropic circular cylinders

This paper presents an approach for obtaining the exact frequency equations of axisymmetric and asymmetric free vibrations of transversely isotropic circular cylinders. The solution method is based on the three dimensional theory of linear elasticity and uses potential functions. Using this approach, the frequency spectra and vibration mode shapes are plotted for a number of transversely isotropic cylinders. The proposed approach introduces a number of merits compared to earlier approximate and exact solution methods. First, unlike numerically complicated series methods that provide approximate solutions, the proposed approach is exact. Second, combination of scalar functions employed for representing the displacement field is consistent with the physics of the problem. One scalar potential function has been considered for each component of the wave field inside the elastic cylinder. As a result, the solution is systematically divided into coupled and decoupled equations. In addition, by using this approach, there is no need to guess the final of the solution a priori. These merits make the proposed approach suitable for other vibration problems of anisotropic materials.     

Exact Learning Augmented Naive Bayes Classifier

Earlier studies have shown that classification accuracies of Bayesian networks (BNs) obtained by maximizing the conditional log likelihood (CLL) of a class variable, given the feature variables, were higher than those obtained by maximizing the marginal likelihood (ML). However, differences between the performances of the two scores in the earlier studies may be attributed to the fact that they used approximate learning algorithms, not exact ones. This paper compares the classification accuracies of BNs with approximate learning using CLL to those with exact learning using ML. The results demonstrate that the classification accuracies of BNs obtained by maximizing the ML are higher than those obtained by maximizing the CLL for large data. However, the results also demonstrate that the classification accuracies of exact learning BNs using the ML are much worse than those of other methods when the sample size is small and the class variable has numerous parents. To resolve the problem, we propose an exact learning augmented naive Bayes classifier (ANB), which ensures a class variable with no parents. The proposed method is guaranteed to asymptotically estimate the identical class posterior to that of the exactly learned BN. Comparison experiments demonstrated the superior performance of the proposed method.     

A theory of the diffraction of an electromagnetic wave by a plane screen

The problem of the diffraction of an arbitrarily polarized electromagnetic plane wave by a perfectly conducting plane screen of infinitesimal thickness is reduced to scalar problems by the method known from earlier papers. The wave field diffracted either by a screen of infinite extent which is perforated by an aperture of arbitrary size and shape or by a complementary screen is then described by means of the bra-vectors, ket-vectors and linear operators introduced by the present author in one of his preceding papers.     

Infinite hierarchies of exact solutions of the einstein and einstein-maxwell equations for interacting waves and inhomogeneous cosmologies

For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein and Einstein-Maxwell equations as represented by their Ernst potentials. This hierarchy contains three arbitrary rational functions of an auxiliary complex parameter. They are constructed using the so-called "monodromy transform" approach and our new method for the solution of the linear singular integral equation form of the reduced Einstein equations. The solutions presented, which describe inhomogeneous cosmological models or gravitational and electromagnetic waves and their interactions, include a number of important known solutions as particular cases.     

GREEN FUNCTION THEORY AND ITS GLOBAL STRUCTURE

The Green function theory is successful in many fields of theoretical physics and the Bogoliubov-Gor'kov theory system is its important branch that involves many theoretical methods. In the last twenty years many papers have been published, but as all the equations of motion are coupled and their number is infinite, its global structure problem has not been investigated as yet. This paper is devoted to the study of its global structure, the exact decoupling problem, and the uniqueness and completeness problems. Some higher order spectral representation theorems and exact relationships between higher and lower order Green functions are obtained. Thus the equations of motion are decoupled exactly. In this paper it is proved that after cutting off the equations of motion are decoupled exactly and the solution of these equationsystemsmay not be unique. It is also proved that if there is one solution satisfying all the Green function's equations, then there must be a solution set, all of them satisfy all the equations and its number is infinite and the differences between them are arbitrary. By adding some limitations to ImG(x) at the regal axis a uniqueness theorem is proved.     

Finite-size scaling for correlations of quantum spin chains at criticality

We study the finite-size scaling behavior of two-point correlation functions of translationally invariant many-body systems at criticality. We propose an efficient method for calculating the two-point correlation functions in the thermodynamic limit from numerical data of finite systems. Our method is most effective when applied to a two-dimensional (classical) system which possesses a conformal invariance. By using this method with numerical data obtained from exact diagonalizations and Monte Carlo simulations, we study the spin-spin correlations of the quantum spin-1/2 and-3/2 antifierromagnetic chains. In particular, the logarithmic corrections to power-law decay of the correlation of the spin-1/2 isotropic Heisenberg antiferromagnetic chain are studied thoroughly. We clarify the cause of the discrepancy in previous calculations for the logarithmic corrections. Our result strongly supports the field-theoretic prediction based on the mappings to the Wess-Zumino-Witten nonlinear -model or the sine-Gordon model. We also treat logarithmic corrections and crossover phenomena in the spin-spin correlation of the spin-3/2 isotropic Heisenberg antiferromagnetic chain. Our results are consistent with the Affleck-Haldane prediction that the correlation of the spin-3/2 chain exhibits a crossover to the same asymptotic behavior as in the spin-1/2 chain.     

Exact entanglement entropy of the XYZ model and its sine-Gordon limit

We obtain the exact expression for the Von Neumann entropy for an infinite bipartition of the XYZ model, by connecting its reduced density matrix to the corner transfer matrix of the eight vertex model. Then we consider the anisotropic scaling limit of the XYZ chain that yields the (1+1)-dimensional sine-Gordon model. We present the formula for the entanglement entropy of the latter, which has the structure of a dominant logarithmic term plus a constant, in agreement with what is generally expected for a massive quantum field theory.     

Solution of the Ornstein-Zernike equation for a softcore Yukawa fluid. II. Numerical results

Numerical calculations are reported for the simplest case of the soft-core Yukawa fluid introduced in an earlier paper. Attention is given to the thermodynamic behavior, the correlation functions, and the interparticle potentials found by inverting the structural information using Percus-Yevick and hypernetted chain integration equation approximations.Supported by ARGC grant No. B7715646R.     

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach(SA) is proposed to find an analytic exact solution of nonlinear differential difference equations.In this manner we avoid solving the nonlinear problem directly.By extension of sensitivity approach for differential difference equations(DDEs),the nonlinear original problem is transformed into infinite linear differential difference equations,which should be solved in a recursive manner.Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained.Numerical examples are employed to show the effectiveness of the proposed approach.     

Periodic contact between piezoelectric materials and a rigid body with a wavy surface

An exact analysis is conducted for periodic, two-dimensional (2D) contact of piezoelectric materials in contact with a rigid body with a wavy surface pressed by uniform stresses at infinity. For three cases of eigenvalue distribution, three harmonic functions automatically satisfying the periodicity conditions are carefully constructed to facilitate the derivation of the solution of the considered problem. The stresses and electric displacements are obtained as infinite series. It is found that for the full contact case, the disturbance stress and electric displacement fields remain only the first harmonic which has the slowest decay in the y-direction. The convergence behaviours of the infinite series are checked, which shows that the external loading p and different positions have a great effect on the convergence behaviours of the infinite series and 400 terms are enough to get accurate solution at each position. Numerical results are presented to justify the validity of the present derivation and show the effect of the external loading on the contact behaviours.     

The depletion potential in one, two and three dimensions

We study the behavior of the depletion potential in binary mixtures of hard particles in one, two, and three dimensions within the framework of a general theory for depletion potential using density functional theory. By doing so we extend earlier studies of the depletion potential in three dimensions to the cases ofd = 1 and 2 about which little is known, despite their importance for experiments. We also verify scaling relations between depletion potentials in sphere-sphere and wall-sphere geometries ind = 3 and in disk-disk and wall-disk geometries ind = 2, which originate from geometrical considerations.     

Density of states of a one-dimensional disordered photonic crystal

An analytic theory of the density of states in one-dimensional disordered photonic crystals is proposed. It is shown that the problem of the density of optical modes can be reduced in the small dielectric contrast approximation to solving a generalized Fokker-Planck equation for the distribution function of the logarithmic derivative of the electric field (the wave phase). The exact analytic solution and density-of-states asymptotics deep in the band gap of the photonic crystal and close to the band gap edge are derived. The results obtained agree well with the empirical relations derived earlier from numerical experiments.