THE FUNCTIONAL DIMENSION OF SOME CLASSES OF SPACES

The functional dimension of countable Hilbert spaces has been discussed by some authors. They showed that every countable Hilbert space with finite functional dimension is nuclear. In this paper the authors do further research on the functional dimension, and obtain the following results: (1) They construct a countable Hilbert space, which is nuclear, but its functional dimension is infinite. (2) The functional dimension of a Banach space is finite if and only if this space is finite dimensional. (3)Let B be a Banach space, B* be its dual, and denote the weak * topology of B* by σ(B*, B). Then the functional dimension of (B*, σ(B*, B)) is 1. By the third result, a class of topological linear spaces with finite functional dimension is presented.

A Study on Radar Cross Section of the Two-Dimensionally Fractal Rough Surface

The analysis of the RCS from the rough sea and ground surface is made. The two dimensionally band limited fractal function is used to model the sea and ground surface, the scattered electromagnetic field is calculated by using Kirchhoff approximation. The validity of this result is assured by some references, which indicates that the methods are reliable.

A Novel Variant of Moth Flame Optimizer for Higher Dimensional Optimization Problems

Moth Flame Optimization(MFO)is a nature-inspired optimization algorithm,based on the principle of navigation technique of moth toward moon.Due to less parameter and easy implementation,MFO is used in various field to solve optimization problems.Further,for the complex higher dimensional problems,MFO is unable to make a good trade-off between global and local search.To overcome these drawbacks of MFO,in this work,an enhanced MFO,namely WF-MFO,is introduced to solve higher dimensional optimization problems.For a more optimal balance between global and local search,the original MFO’s exploration ability is improved by an exploration operator,namely,Weibull flight distribution.In addition,the local optimal solutions have been avoided and the convergence speed has been increased using a Fibonacci search process-based technique that improves the quality of the solutions found.Twenty-nine benchmark functions of varying complexity with 1000 and 2000 dimensions have been utilized to verify the projected WF-MFO.Numerous popular algorithms and MFO versions have been compared to the achieved results.In addition,the robustness of the proposed WF-MFO method has been evaluated using the Friedman rank test,the Wilcoxon rank test,and convergence analysis.Compared to other methods,the proposed WF-MFO algorithm provides higher quality solutions and converges more quickly,as shown by the experiments.Furthermore,the proposed WF-MFO has been used to the solution of two engineering design issues,with striking success.The improved performance of the proposed WF-MFO algorithm for addressing larger dimensional optimization problems is guaranteed by analyses of numerical data,statistical tests,and convergence performance.

LEVEL SETS AND EQUIVALENCES OF MORAN-TYPE SETS

In the paper, we consider Moran-type sets E;given by sequences {a;};and{n;};. we prove that E;may be decompose into the disjoint union of level sets. Moreover,we define three type of equivalence between two dimension functions associated to two Morantype sets, respectively, and we classify Moran-type sets by these equivalent relations.

STRESS INTENSITY FACTORS FOR CORNER CRACKS AT A HOLE UNDER REMOTE TENSION

The three-dimensional weight function method recently developed by the authors is used to determine stress intensity factors for two symmetric quarter-elliptical corner cracks at a hole in a wide finite-thickness plate subjected to remote tensile loading. The geometry parameters considered are r / t = 0.5, 1, 2; a / c= 0.2, 0.5, 1, 2; a / t = 0.2, 0.5 within c/r= 2. The results are compared, where possible, with other solutions available in the literature. Generally good agreement is observed. The effect of an approximation of the two-dimensional unflawed stress distribution on the accuracy of stress intensity factors by the weight function method is discussed.

Enhancing PIV image and fractal descriptor for velocity and shear stresses propagation around a circular pier

In this study,the fractal dimensions of velocity fluctuations and the Reynolds shear stresses propagation for flow around a circular bridge pier are presented.In the study reported herein,the fractal dimension of velocity fluctuations（u′,v′,w′） and the Reynolds shear stresses（u′v′ and u′w′） of flow around a bridge pier were computed using a Fractal Interpolation Function（FIF） algorithm.The velocity fluctuations of flow along a horizontal plane above the bed were measured using Acoustic Doppler Velocity meter（ADV）and Particle Image Velocimetry（P1V）.The PIV is a powerful technique which enables us to attain high resolution spatial and temporal information of turbulent flow using instantaneous time snapshots.In this study,PIV was used for detection of high resolution fractal scaling around a bridge pier.The results showed that the fractal dimension of flow fluctuated significantly in the longitudinal and transverse directions in the vicinity of the pier.It was also found that the fractal dimension of velocity fluctuations and shear stresses increased rapidly at vicinity of pier at downstream whereas it remained approximately unchanged far downstream of the pier.The higher value of fractal dimension was found at a distance equal to one times of the pier diameter in the back of the pier.Furthermore,the average fractal dimension for the streamwise and transverse velocity fluctuations decreased from the centreline to the side wall of the flume.Finally,the results from ADV measurement were consistent with the result from PIV,therefore,the ADV enables to detect turbulent characteristics of flow around a circular bridge pier.

Investigation of Rotor Radial Rotating Error upon AMB System

This paper first suggests the use of the Fourier frequency transmission method of two dimensions function （ 2D FFT） to analyze radial rotating errors that occurred in a rotor. Based on this method a magnetic rotor is measured. The authors point out that the main cause to affect radial rotating accuracy of the rotating shaft at a high speed is the dynamic imbalance of the shaft itself. Finally the feedforward control scheme is suggested to improve the accuracy of the shaft in an active magnetic bearing （ AMB ） system.

Finite dimensional of Brownian rough characteristic functions path

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2. Equivalently, it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough path in IRd and obtain an explicit formula for the case when d = 2.

Accuracy of three-dimensional quantification of left ventricular function using magnetic sensor acquisition: a dynamic in vitro model

Objective To evaluate the accuracy of a three-dimensional (3D) magnetic position sensor system in the quantification of ventricular stroke volumes in a dynamic model.Methods A latex balloon model of the left ventricle was suspended in a water bath connected to a pump producing 10 different pulsatile stroke volumes (15-65mi/beat). Scanning was performed using a 5.0 mHz transducer mounted with a Flock of Birds (FOB) magnetic receiver (GE System Five). The probe was scanned to sweep continuously across and over the balloon volume over 3 - 7 seconds. Digital loops were stored on magneto-optical disks and reviewed retrospectively using 3D Echopac software (GE)based on Simpson's method and compared with a two-dimensional (2D) biplane area-length method (1/2L x R) measurements at end systole and end diastole. Both 3D and 2D derived stroke volumes were compared with the reference stroke volume calculated by direct measurement of balloon capacity.Results There was an improved correlation between 3D stroke volume and reference stroke volume ( y = 0.91 x + 0.41, r = 0.97, SEE = 2.83 ml, P = 0.0001 ) compared to 2D stroke volume and reference stroke volume (y=0.49x+8.68, r=0.87, SEE=3.87 ml, P=0.0011, difference between 2D and 3D P＜0.003).Conclusion 3D magnetic FOB scanning is practical, accurate and should facilitate assessment of left ventricular function.

The Fractional Derivatives of a Fractal Function

The present paper investigates the fractional derivatives of Weierstrass function, proves that there exists some linear connection between the order of the fractional derivatives and the dimension of the graphs of Weierstrass function.