A nonlocal theory for brittle fracture

In this paper, a nonlocal theory of fracture for brittle materials has been systematically developed, which is composed of the nonlocal elastic stress fields of Griffith cracks of mode-I, II and III, the asymptotic forms of the stress fields at the neighborhood of the crack tips, and the maximum tensile stress criterion for brittle fracture. As an application of the theory, the fracture criteria of cracks of mode-I, II, III and mixed mode I–II, I–III are given in detail and compared with some experimental data and the theoretical results of minimum strain energy density factor.     

A two-body continuous-discrete filter

In the theory of classical mechanics, the two-body central forcing problem is formulated as a system of the coupled nonlinear second-order deterministic differential equations. The uncertainty introduced by the small, unmodeled stochastic acceleration is not assumed in the particle dynamics. The small, unmodeled stochastic acceleration produces an additional random force on a particle. Estimation algorithms for a two-body dynamics, without introducing the stochastic perturbation, may cause inaccurate estimation of a particle trajectory. Specifically, this paper examines the effect of the stochastic acceleration on the motion of the orbiting particle, and subsequently, the stochastic estimation algorithm is developed by deriving the evolutions of conditional means and conditional variances for estimating the states of the particle-earth system. The theory of the nonlinear filter of this paper is developed using the Kolmogorov forward equation “between the observations" and a functional difference equation for the conditional probability density “at the observation." The effectiveness of the nonlinear filter is examined on the basis of its ability to preserve perturbation effect felt by the orbiting particle and the signal-to-noise ratio. The Kolmogorov forward equation, however, is not appropriate for the numerical simulations, since it is the equation for the evolution of “the conditional probability density." Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions and velocities of the orbiting body. Even these equations are not appropriate for the numerical implementations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher order moments. Hence, we consider the approximations to these moment evolution equations. This paper makes a connection between classical mechanics, statistical mechanics and the theory of the nonlinear stochastic filtering. The results of this paper will be of use to astrophysicists, engineers and applied mathematicians, who are interested in applications of the nonlinear filtering theory to the problems of celestial and satellite mechanics. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed, in this paper.     

未熔透尺寸对十字焊接接头疲劳性能影响规律试验研究

针对某厂一压力机架的焊接结构，采用极大似然方法，进行了部分熔透焊接十字接头的拉—拉疲劳试验，得到了该类结构的Ｐ－Ｓ－Ｎ对数曲线。研究了未熔透尺寸对接头的应力集中系数、疲劳强度和试件疲劳破坏形式的影响规律，结果表明，在焊缝与母材等强的情况下，当未熔透尺寸２ａ／Ｔ＜０．５时，其疲劳性能无明显减弱；而当未熔透尺寸２ａ／Ｔ＞０．５时，则接头的抗疲劳性能有显著的改变。     

二自由度耦合非线性随机系统的最大Lyapunov指数和稳定性

利用摄动方法讨论了一类耦合二自由度非线性系统，在小强度白噪声参数激励下系统运动模态的稳定性，获得了系统扩散过程的稳态概率密度的渐近表达式，由此获得了系统运动模态几乎必然稳定的充分必要条件。     

Buoyancy-driven convection of water near its density maximum with time periodic partially active vertical walls

The effect of density inversion on transient natural convection heat transfer of cold water in a square cavity with partially active vertical walls is studied numerically. The governing equations are solved by control volume method with power law scheme. In the hot location the temperature is varied sinusoidally and in the cold location uniform temperature is maintained. Nine different positions of the active zones are considered. Results are discussed for various values of the amplitude, period and different Grashof numbers and presented graphically in the form of isotherms, streamlines, mid-height velocity profile and average Nusselt number. It is found that density inversion of water affects natural convection flow and heat transfer. Heat transfer rate is enhanced upto 80% when the heating location is in the middle of the hot wall.     

Calibration Invariance of the MaxEnt Distribution in the Maximum Entropy Principle

The maximum entropy principle consists of two steps: The first step is to find the distribution which maximizes entropy under given constraints. The second step is to calculate the corresponding thermodynamic quantities. The second part is determined by Lagrange multipliers’ relation to the measurable physical quantities as temperature or Helmholtz free energy/free entropy. We show that for a given MaxEnt distribution, the whole class of entropies and constraints leads to the same distribution but generally different thermodynamics. Two simple classes of transformations that preserve the MaxEnt distributions are studied: The first case is a transform of the entropy to an arbitrary increasing function of that entropy. The second case is the transform of the energetic constraint to a combination of the normalization and energetic constraints. We derive group transformations of the Lagrange multipliers corresponding to these transformations and determine their connections to thermodynamic quantities. For each case, we provide a simple example of this transformation.     

Exact Recovery of Stochastic Block Model by Ising Model

In this paper, we study the phase transition property of an Ising model defined on a special random graph—the stochastic block model (SBM). Based on the Ising model, we propose a stochastic estimator to achieve the exact recovery for the SBM. The stochastic algorithm can be transformed into an optimization problem, which includes the special case of maximum likelihood and maximum modularity. Additionally, we give an unbiased convergent estimator for the model parameters of the SBM, which can be computed in constant time. Finally, we use metropolis sampling to realize the stochastic estimator and verify the phase transition phenomenon thfough experiments.     

Minimum Divergence Estimators,Maximum Likelihood and the Generalized Bootstrap

This paper states that most commonly used minimum divergence estimators are MLEs for suited generalized bootstrapped sampling schemes. Optimality in the sense of Bahadur for associated tests of fit under such sampling is considered.     

Estimation of Feeding Composition of Industrial Process Based on Data Reconciliation

For an industrial process, the estimation of feeding composition is important for analyzing production status and making control decisions. However, random errors or even gross ones inevitably contaminate the actual measurements. Feeding composition is conventionally obtained via discrete and low-rate artificial testing. To address these problems, a feeding composition estimation approach based on data reconciliation procedure is developed. To improve the variable accuracy, a novel robust M-estimator is first proposed. Then, an iterative robust hierarchical data reconciliation and estimation strategy is applied to estimate the feeding composition. The feasibility and effectiveness of the estimation approach are verified on a fluidized bed roaster. The proposed M-estimator showed better overall performance.     

The maximum number of columns in supersaturated designs with

Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows () and a maximum in absolute value correlation between any two columns (). In particular, they proved that for (mod ) and . However, the only known SSD satisfying this upper bound is when . By utilizing a computer search, we prove that for , and . These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the lower bound for SSDs with rows and columns, for , and . Finally, we show that a skew‐type Hadamard matrix of order can be used to construct an SSD with rows and columns that proves . Hence, we establish for and for all (mod ) such that . Our result also implies that when is a prime power and (mod ). We conjecture that for all and (mod ), where is the maximum number of equiangular lines in with pairwise angle .