Bounds on Fractional-Based Metric Dimension of Petersen Networks

The problem of investigating the minimum set of landmarks consisting of auto-machines(Robots)in a connected network is studied with the concept of location number ormetric dimension of this network.In this paper,we study the latest type of metric dimension called as local fractional metric dimension(LFMD)and find its upper bounds for generalized Petersen networks GP(n,3),where n≥7.For n≥9.The limiting values of LFMD for GP(n,3)are also obtained as 1(bounded)if n approaches to infinity.

Application of the Algorithm of Fractional Dimension to Extraction Image Edge

The idea of fractional dimension was stated in brief firstly.Then,adopting the fractional statistical similar principle, the method of the least square minimum error was applied to evaluate the fractional dimension of per image pixel depending on the fractional property of image.And the image edge is extracted by magnitude of fractional dimension of image pixel.We presented the algorithm of the local fractional dimension,which made the rule of window size and sentencing the fractional dimension of edge.Although this algorithm was waste time,it is better than the classical ones in extraction edge and anti-jamming.

THE FRACTIONAL DIMENSION IDENTIFICATION METHOD OF CRITICAL BIFURCATED PARAMETERS OF BEARING-ROTOR SYSTEM

The stable problem of rotor system, seen in many fields, has been cared for more. Nowadays the reasons of most losing stability are caused by nonlinear behaviors This presents higher requirements to the designing of motor system : considering nonlinear elements, avoiding the unstable parameter points or regions where nonlinear phenomena will be presented. If a family of time series of the unknown nonlinear dynamical system cart only be got ( may be polluted by noise), how to identify the change of motive properties at different parameters? In this paper, through the study of Jeffcott rotor system, the result that using the figures between the fractional dimension of rime-serial and parameter can be gained, and the critical bifurcated parameters of bearing-rotor dynamical system can be identified.

Raman spectra and structure study of silicate glasses and their liquids

Stress index of tetrahedron （SIT） was defined to describe the topological connectivities among various sili- con-oxygen tetrahedra （SiOT） in anionic clusters of binary silicate crystals, glasses, and melts. It was found that the value of SIT was well correlated with the wavenumber of Raman active symmetric stretching vibration of non-bridging oxygen of SiOT. The spatial fractional dimension of hyperfine structure was introduced while comparative analysis was made with the value of SIT. It can be concluded that the concepts of SIT, vibrational wavenumber, and spatial fractional dimension were inherently and holographically correlated and exhibit isomorphic representations of complex structure of binary silicates. Experimental Raman spectra of binary silicates with different alkali cations were investigated. It was demonstrated that alkali cations have little effect on the vibrational wavenumber of symmetric stretching of non-bridging oxygen （NBO） of SiOT, but remarkably affect its Raman active optical cross section, as was consensus resulted from ab initio calculation. It can also be concluded that the spatial fractional dimension of binary silicate is predominantly determined by the hyperfine structure of the anionic clusters and little affected by alkali cations, although the species of anionic clusters and their distributions were originally assigned by the content of alkali oxides. And Raman optical activity extinct effect of isolated SiOT at high basicity should be considered while being applied to quantitatively analysis.

Fractal and its application to sedimentology

In the paper,the foundation,development,basic conception and general characteristics of fractal and the calculating method of the fractional dimension are expounded briefly, and the current situation and prospect of the fractal application in sedimentology are discussed stressly. Both sedimentary process and sedimentary record behave the fractal feature of the self similarity structure. External form and internal texture of the sediments and the distribution of the grain size of the sediments are of fractal feature very well, and the size of the fractional dimension is the quantitative index of the complexity of the background when they are formed. The further analysis on the multi fractal feature of the sedimentary body is the base of the fractal simulation and forecast, and it is the key of the application of the fractal theory to sedimentology.

Study of the Stacking Sequences of an Irregular Mixed-Layer Illite/Smectite (I/S) Clay Mineral—with a Discussion on the Existence of Minerals with Two-Dimentional Lattice and One-Dimentional Quasi-Lattice

There are two aspects in the study of irregular mixed-layer clay minerals: one is the kinds and ratios of their basic structural unit layers and the other is the junction probabilities of the unit layers. Irregular mixed-layer illite/smectite clay minerals (I/S) are widespread in nature. While studying the clay minerals from the Permian-Triassic (P/T) boundary bed, the authors found that I/S clay minerals are developed in all P/T boundary clay layers in areas from the northwest to southeast of China. Systematic mineralogical studies of the I/S clay minerals from Hunan, Hubei, Sichuan and Zhejiang were made by means of X-ray, infrared spectroscopic, electron microscopic and chemical analyses and a deepened study of the stacking sequences of their structural unit layers was conducted by the MacEwan one—dimentional direct Fourier transform. It was found that the stacking of the illite and smectite crystal layers along the c axis can be derived from Fibonacci sequences. Hence, the authors propose that such I/S clay minerals are possessed of two—dimentional crystal lattice and one—dimentional quasicrystal lattice.

The Simultaneous Fractional Dimension of Graph Families

For a connected graph G with vertex set V,let RG{x,y}={z∈V:dG(x,z)≠dG(y,z)}for any distinct x,y∈V,where dG(u,w)denotes the length of a shortest uw-path in G.For a real-valued function g defined on V,let g(V)=∑s∈V g(s).Let C={G_(1),G_(2),...,G_(k)}be a family of connected graphs having a common vertex set V,where k≥2 and|V|≥3.A real-valued function h:V→[0,1]is a simultaneous resolving function of C if h(RG{x,y})≥1 for any distinct vertices x,y∈V and for every graph G∈C.The simultaneous fractional dimension,Sdf(C),of C is min{h(V):h is a simultaneous resolving function of C}.In this paper,we initiate the study of the simultaneous fractional dimension of a graph family.We obtain max1≤i≤k{dimf(Gi)}≤Sd_(f)(C)≤min{∑k i=1 dimf(Gi),|V|/2},where both bounds are sharp.We characterize C satisfying Sdf(C)=1,examine C satisfying Sdf(C)=|V|/2,and determine Sdf(C)when C is a family of vertex-transitive graphs.We also obtain some results on the simultaneous fractional dimension of a graph and its complement.

The Fractional Metric Dimension of Permutation Graphs

Let G =（V（G）, E（G）） be a graph with vertex set V（G） and edge set E（G）. For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equa l to the distance from y to z in G. For a function g defined on V（G） and for U V（G）, let g（U） =∑s∈Ug（s）. A real-valued function g ： V（G） → [0, 1] is a resolving function of G if g（RG{x, y}） ≥ 1 for any two distinct vertices x, y ∈ V（G）. The fractional metric dimension dimf（G）of a graph G is min{g（V（G）） ： g is a resolving function of G}. Let G1 and G2 be disjoint copies of a graph G, and let σ ： V（G1） → V（G2） be a bijection. Then, a permutation graph Gσ =（V, E） has the vertex set V = V（G1） ∪ V（G2） and the edge set E = E（G1） ∪ E（G2） ∪ {uv ｜ v = σ（u）}. First,we determine dimf（T） for any tree T. We show that 1 〈 dimf（Gσ） ≤1/2（｜V（G）｜ ＋ ｜S（G）｜） for any connected graph G of order at least 3, where S（G） denotes the set of support vertices of G. We also show that, for any ε 〉 0, there exists a permutation graph Gσ such that dimf（Gσ）- 1 〈 ε. We give examples showing that neither is there a function h1 such that dimf（G） 〈 h1（dimf（Gσ）） for all pairs（G, σ）, nor is there a function h2 such that h2（dimf（G）） 〉 dimf（Gσ） for all pairs（G, σ）. Furthermore,we investigate dimf（Gσ） when G is a complete k-partite graph or a cycle.