R￣d中Hausdorff维数和packing维数的确定

本文目的在于建立确定Ｒ￣ｄ中Ｈａｕｓｄｏｒｆｆ维数ｄｉｍ和ｐａｃｋｉｎｇ维数Ｄｉｍ的两个命题（定理１和定理２），进而寻求Ｒ￣ｄ中Ｈａｕｓｄｏｒｆｆ维数ｄｉｍ与ｐａｃｋｉｎｇ维数Ｄｉｍ相等的条件；这使得我们能够引入分形测度的测度论定义。     

The fifth dimension: Theodor Kaluza's ground‐breaking idea

Theodor Kaluza (1885–1954) attracted the attention of the physical community since 1921 with his unified field theory of gravitation and electromagnetism in five dimensions. Despite Einstein's great interest in Kaluza's theory, 50 years elapsed before it contributed toward a paradigm shift in modern theoretical physics. The biography of this still unknown scientist is briefly presented along with an outline of his work in physics. A short history of the theories of unification and the dimensionality of space‐time is followed by a discussion of the significance of Kaluza's five‐dimensional unified theory in modern physics from the point of view of superstring and M‐theory.     

ANTI FUZZY VECTOR SPACES（Ⅱ）

ＡＮＴＩＦＵＺＺＹＶＥＣＴＯＲＳＰＡＣＥＳ（Ⅱ）ＣｈｅｎＹｏｎｇｎｉａｎ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＸｉｎｊｉａｎｇＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，８３００５４，Ｘｉｎｊｉａｎｇ，Ｐ．Ｒ．Ｃｈｉｎａ）ＡＮＴＩＦＵＺＺＹＶＥＣＴ...     

Logarithmic corrections for dilute uniaxial ferromagnets at the critical dimension

The leading logarithmic corrections to the critical behavior of a dilute uniaxial (Ising) ferromagnet in the disordered phase are derived using renormalization group methods. The values of the exponents in the logarithmic terms differ from those given by previous authors.     

Picture Fuzzy Threshold Graphs with Application in Medicine Replenishment

In this study, a novel concept of picture fuzzy threshold graph (PFTG) is introduced. It has been shown that PFTGs are free from alternating 4-cycle and it can be constructed by repeatedly adding a dominating or an isolated node. Several properties about PFTGs are discussed and obtained the results that every picture fuzzy graph (PFG) is equivalent to a PFTG under certain conditions. Also, the underlying crisp graph (UCG) of PFTG is a split graph (SG), and conversely, a given SG can be applied to constitute a PFTG. A PFTG can be decomposed in a unique way and it generates three distinct fuzzy threshold graphs (FTGs). Furthermore, two important parameters i.e., picture fuzzy (PF) threshold dimension (TD) and PF partition number (PN) of PFGs are defined. Several properties on TD and PN have also been discussed. Lastly, an application of these developed results are presented in controlling medicine resources.     

An Upper Bound for the w-Weak Global Dimension of Pullbacks

Let R be a commutative ring with identity and I0 an ideal of R.We introduce and study the c-weak global dimension c-w.gl.dim(R/I0) of the factor ring R/I0.Let T be a w-linked extension of R,and we also introduce the wR-weak global dimension wR-w.gl.dim(T) of T.We show that the ring T with wR-w.gl.dim(T) =0 is exactly a field and the ring T with wR-w.gl.dim(T) ≤ 1 is exactly a PwRMD.As an application,we give an upper bound for the w-weak global dimension of a Cartesian square (RDTF,M).More precisely,if T is w-linked over R,then w-w.gl.dim(R) ≤ max{wR-w.gl.dim(T) + w-fdR T,c-w.gl.dim(D) + w-fdn D}.Furthermore,for a Milnor square (RDTF,M),we obtain w-w.gl.dim(R) ≤ max{wR-w.gl.dim(T) + w-fdR T,w-w.gl.dim(D) + w-fdR D}.     

Application of Two Novel Acoustic Emission Parameters on Identifying the Instability of Granite

The issue of monitoring and early warning of rock instability has received increasing critical attention in the study of rock engineering. To investigate the damage evolution process of granite under triaxial compression tests, acoustic emission (AE) tests were performed simultaneously. This study firstly introduced two novel parameters, i.e., the coefficient of variation (CoV) of the information entropy and correlation dimension of the amplitude data from the AE tests, to identify the precursor of the failure of granite. Then the relationship between the changes in these parameters and the stress-time curve was compared and analyzed. The results of this study show that: (1) There is a strong correlation between the CoV of the information entropy and the failure process of granite. The granite failed when the CoV curve raised to a plateau, which could be used as an indicator of rock instability. (2) The fluctuation of the correlation dimension indicates the different stages during the loading process, i.e., the initial compaction stage, the linear elastic stage, the yield stage, and the failure stage. Each stage contains a descending and a rising process in the correlation dimension curve, and the exhibited starting point or the bottom point at the correlation dimension curve could be selected as the indicator point for the rock instability. (3) The combined analysis of the Information entropy and Correlation dimension can improve the accuracy of rock instability prediction. This study provides new insights into the prediction of rock instability, which has theoretical implications for the stability of subsurface engineering rock masses.     

有雨概率与分维数关系的初步研究

对武汉地区１９８２～１９９４年有雨分维数与有雨概率之间的关系进行了初步的分析和研究，并给出了研究结果．为今后修改降雨衰减预报模型作准备．     

DIMENSIONS FORRANDOM SELF——CONFORMAL SETS

A set is called regular if its Hausdorff dimension and upper box-counting dimension coincide. In this paper, we prove that the random self-con formal set is regular almost surely. Also we determine the dimen-sions for a class of random self-con formal sets.     

The Coordinate of the Singular Point of Generating Functions of Clusters in the High-Temperature Dynamics of Spin Lattice Systems with Axially Symmetric Interaction

We investigate generating functions for equipped trees composed of double bonds of two sorts on a hypercubic lattice of dimension d with built-in fragments. Rules for constructing these clusters are chosen to ensure the estimate for coefficients of power series in time for the longitudinal and transverse autocorrelation functions of the spin system with axially symmetric interaction. We derive a system of two equations for the tree-generating functions and an equation for the generating functions of chains leading from the root to a fragment in a tree using the Bethe approximation and under the condition that mainly bonds of one sort are taken into account. For the face-centered hypercubic lattice, we find the first terms of the 1/d expansion for the coordinate of the singular point of the generating function in both the anisotropic and the isotropic cases taking fragments in the forms of a triangle from four bonds and a four-fold bound pair into account. The obtained result is written in terms of ratios of lattice sums and is generalized to nuclear spin systems with dipole–dipole interaction. The theoretical value of the singular-point coordinate agrees well with the experimental value calculated from the tail of the absorption spectrum of the nuclear magnetic resonance in a barium fluoride monocrystal.