Function S-rough sets and mining-discovery of rough law in systems

Function S-rough sets （function singular rough sets） is defined on a -function equivalence class [u]. Function S-rough sets is the extension form of S-rough sets. By using the function S-rough sets, this paper gives rough law generation model of a-function equivalence class, discussion on law mining and law discovery in systems, and application of law mining and law discovery in communication system. Function S-rough sets is a new theory and method in law mining research.

Interaction of elementary waves of scalar conservation laws with discontinuous flux function

In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simulations were given.

Function P-sets and Mining-discovery Theorems of Unknown Information Law

Function P-sets are a set pair which are composed of function internal packetsets S F and function outer packet sets S F or (S F , S F ) are function P-sets. Function P-sets have dynamic characteristics and law characteristics. Based on function P-sets, P-information law discovery theorems which has attribute supplementing-deleting are given. Mining-discovery and mining-discovery theorems of unknown information law are proposed. Function P-sets are a new theory and new method to research dynamic information laws for information systems.

Experimental researches for scaling laws of low-order structure function in boundary layer

Scaling laws are addressed by analysing moments of velocity increments which obtained by Particle-image Velocimetry(PIV)system in the boundary layer of a flat plate.In the paper,we measure the moments of increments of upstream velocity(u'),longitudinal velocity(v')and ponderance of vorticity(dv'/dx)at Reθ=2167 in different wall distance and verify the anomaly of the scaling exponents of high-order structure functions with the increasing order of the moments,discuss the scaling of non-integer moments of order between+2 and-1.The difference of scaling exponents of low-order structure functions between the experimental data and Kolmogorov's,SL's(She & Leveque)prediction increases as the moment order decreases toward-1,which shows that the anomaly is manifested in low-oeder moments as well.However,for same order structure functions,the scaling exponents of v' and dv'/dx are not changeable in different wall distance.

The Effectiveness of the Squared Error and Higgins-Tsokos Loss Functions on the Bayesian Reliability Analysis of Software Failure Times under the Power Law Process

Reliability analysis is the key to evaluate software’s quality. Since the early 1970s, the Power Law Process, among others, has been used to assess the rate of change of software reliability as time-varying function by using its intensity function. The Bayesian analysis applicability to the Power Law Process is justified using real software failure times. The choice of a loss function is an important entity of the Bayesian settings. The analytical estimate of likelihood-based Bayesian reliability estimates of the Power Law Process under the squared error and Higgins-Tsokos loss functions were obtained for different prior knowledge of its key parameter. As a result of a simulation analysis and using real data, the Bayesian reliability estimate under the Higgins-Tsokos loss function not only is robust as the Bayesian reliability estimate under the squared error loss function but also performed better, where both are superior to the maximum likelihood reliability estimate. A sensitivity analysis resulted in the Bayesian estimate of the reliability function being sensitive to the prior, whether parametric or non-parametric, and to the loss function. An interactive user interface application was additionally developed using Wolfram language to compute and visualize the Bayesian and maximum likelihood estimates of the intensity and reliability functions of the Power Law Process for a given data.

A LAW OF THE ITERATED LOGARITHM FOR NEAREST NEIGHBOR ESTIMATION OF MULTIVARIATE DENSITY FUNCTION

Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we established the law of the iterated logarithm of f(n) for general case of d greater-than-or-equal-to 1, which gives the exact pointwise strong convergence rate of f(n).

F-rough law and the discovery of rough law

By using function one direction S-rough sets （function one direction singular rough sets）, this article presents the concepts of F-law, F-rough law, and the relation metric of rough law; by using these concepts, this article puts forward the theorem of F-law relation metric, two orders theorem of F-rough law relation metric, the attribute theorem of F-rough law band, the extremum theorem of F-rough law relation metric, the discovery principle of F-rough law and the application of F-rough law.

MULTI-DIMENSIONAL RIEMANN PROBLEM OF SCALAR CONSERVATION LAW

This paper considers multi-dimensional Riemann problem in another kind of view. The author gets solution of (1.1)(1.2) in Theorem 3.4 and proves itu uniqueness. A new method of solution constructing is applied, which is different from the usual self-similar transformation. The author also discusses some generalized concepts in multi-dimensional situation (such as 'convex condition', 'left value' and 'right value', etc). An example is finally given to demonstrate that rarefaction wave solution of (1.1)(1.2) is not self-similar.

Residual Fatigue Life Prediction of Ball Bearings Based on Paris Law and RMS

Paris law can reflect the failure mechanism of materials and is usually used to be a method to predict fatigue life or residual fatigue life.But the variable which can represent the health of machine is hardly measured on line.To a degree,the difficulty of on-line application restricts the scope of application of Paris law.The relationship between characteristic values of vibration signals and the variable in the Paris equation which can describe the health of machine is investigated by taking ball bearings as investigative objects.Based on 6205 deep groove ball bearings as a living example,historical lives and vibration signals are analyzed.The feasibility of describing that variable in the Paris equation by the characteristic value of vibration signals is inspected.After that vibration signals decomposed by empirical mode decomposition（EMD）,root mean square（RMS） of intrinsic mode function（IMF） involving fault characteristic frequency has a consistent trend with the diameter of flaws.Based on the trend,two improved Paris models are proposed and the scope of application of them is inspected.These two Paris Models are validated by fatigue residual life data from tests of rolling element bearings and vibration signals monitored in the process of operation of rolling element bearings.It shows that the first improved Paris Model is simple and plain and it can be easily applied in actual conditions.The trend of the fatigue residual life predicted by the second improved Paris model is close to the actual conditions and the result of the prediction is slightly greater than the truth.In conclusion,after the appearance of detectable faults,these improved models based on RMS can predict residual fatigue life on line and a new approach to predict residual fatigue life of ball bearings on line without disturbing the machine running is provided.

F-generation law and recognition of system law

If a system is not disturbed （or invaded） by some law, there is no doubt that each system will move according to the expected law and keep stable. Although such a fact often appears, some unknown law breaks into the system and leads it into turbulence. Using function one direction S-rough sets, this article gives the concept of the F-generation law in the system, the generation model of the F-generation law and the recognition method of the system law. Function one direction singular rough sets is a new theory and method in recognizing the disturbance law existing in the system and recognizing the system law.

Rough law energy and measurement of ■-decomposition rough law

By employing function one-direction S-rough sets and rough law generation method based on function S-rough sets, ^-f-decomposition law and ^-F-decomposition rough law are proposed, and the measurement of rough law variation in the process of rough law ^-F-decomposition is researched. The concepts of law energy and attdbute ^-f-interference degree are presented, which make the variation of rough law become measurable. ^-f-decomposition law energy characteristic theorem, ^-f- decomposition law energy inequality theorem, ^-F-decomposition rough law energy characteristic theorem, and ^-f-decomposition law energy mean value theorem are presented.

Hiding dependence-discovery of F-hiding laws and system laws

Function one direction S-rough sets have dynamic characteristics and law characteristics. By using the function one direction S-rough sets, this article presents the concepts of the f-hiding law, F-hiding law, f-hiding law dependence and F-hiding law dependence. Based on the concepts above, this article proposes the hidingdependence theorem of f-hiding laws, the hiding-dependence theorem of F-hiding laws, the hiding-dependence separation theorem, the hiding dependence-discovery principle of unknown laws. Finally, the application of the hiding dependence of hiding laws in the discovery of system laws is given.

Vibrations of FGM thin cylindrical shells with exponential volume fraction law

In this paper, the influence of an exponential volume fraction law on the vibration frequencies of thin functionally graded cylindrical shells is studied. Material properties in the shell thickness direction are graded in accordance with the exponential law. Expressions for the strain-displacement and curvature-displacement relationships are taken from Love＇s thin shell theory. The Rayleigh-Ritz approach is used to derive the shell eigenfrequency equation. Axial modal dependence is assumed in the characteristic beam functions. Natural frequencies of the shells are observed to be dependent on the constituent volume fractions. The results are compared with those available in the literature for the validity of the present methodology.

Closed-Form and Analytical BER Expression for OFDM System with μ-Law Companding Operation

The μ-law companding function has been applied widely in orthogonal frequency division multiplexing (OFDM) to reduce the peak-to-average power ratio (PAPR). However, nonlinear distortion caused by the μ-law companding function is considered a key impairment in OFDM communication systems. Few studies have addressed theoretical nonlinear distortion caused by μ-law companding function for OFDM systems. In this paper, we derive a closed-form expression of signal distortion as well as the closed-form bit error rate (BER) of OFDM system caused by the μ-law companding function. Based on the theoretical signal distortion and BER expression, the theoretical BER value and signal distortion value can also be calculated, which can guide us to choose appropriate μ value for different BER condition and bit-to-noise (Eb/N0) condition efficiently. Then the PAPR performance can also be predicted. The results show good agreement on the Monte-Carlo simulation results and the obtained theoretical BER results. Furthermore, based on theoretical signal distortion and theoretical BER expression, the figure of the relationship among BER value, Eb/N0 and μ is also given. Based on this figure, we can find the appropriate μ law for different BER and Eb/N0 condition. And then the PAPR performance can also be predicted.

FORMULA OF GLOBAL SMOOTH SOLUTION FOR NON-HOMOGENEOUS M-D CONSERVATION LAW WITH UNBOUNDED INITIAL VALUE

In this article, we prove the existence and obtain the expression of its solution formula of global smooth solution for non-homogeneous multi-dimensional（m-D） conservation law with unbounded initial value; our methods are new and essentially different with the situation of bounded initial value.

Function S-rough Sets and Their Law Characteristic

Function S-rough sets(Function Singular rough sets) are defined by R-function equivalence class which has dynamic characteristic, and a function is s law, function S-rough sets have law characteristic. Function S-rough sets has these forms: function one direction S-rough sets, function two direction S-rough sets and dual of function one direction S-rough sets. This paper presents the law characteristic of function one direction S-rough sets and puts forward the theorems of law-chain-attribute and law-belt. Function S-rough sets is s new research direction of the rough sets theory.

Pointwise Estimates on Travelling Wave Solution of the Scalar Viscous Conservation Laws

This paper research on the pointwise behavior of perturbations from a viscous shock solution to a scalar viscous conservation law by introducing an approximate Green’s function. The authors obtain not only the pointwise decay of the perturbation and but also the high derivative of it. Stability in anyL(p≥1) norm is a direct consequence.

Study on the Size Distribution and Similarity Law of Bubble Nuclei in Water

To study the distribution characteristics and similarity laws of nuclei under different pressures,based on the selfdesigned decompression chamber and the acoustic measuring system,the size distributions of nuclei in the degassed tap water under negative ambient pressures were measured.A number density distribution function of nuclei based on the modified Weibull distribution function was proposed and verified by the experimental measurement results and some published data of nuclei size distribution.Based on this nuclei number density distribution function,the similarity law of the nuclei size distribution was analyzed:in the scale experiment,the value of exponential in the similarity law of the nuclei number density should be determined by the nuclei size distribution of the water in the prototype experiment and the actual nuclei size distribution of the water in the model experiment.And a precondition is that the nuclei size distributions are similar.

Extended Conservation Laws of Some Nonlinear Partial Differential Equations

Self-adjoint theorem is introduced to match the corresponding functional of the given differential equations,and then Noether＇s theorem is used to determine the extended conservation laws of the original equations. Finally, as the application of the method, the conservation laws of Drinfel＇d-Sokolov-Wilson equation and Benjamin-Bona-Mahony equation are constructed.

A refinement of scaling laws in wall turbulence

As a universal conclusion of turbulent scale, scaling laws are important to the research on statistic turbulence. We measured two-dimensional instantaneous velocity field in turbulent boundary layers of flat plate with the momentum thickness Reynolds number Reθ=2 167. Scaling laws have different forms in different wall distance and scale. We proposed an expected scaling law and compared it with the She-Leveque (SL) scaling law based on the wavelet analysis and traditional statistical methods. Results show that the closer to the wall, the more the expected scaling law approached to the SL scaling law.