基于分形维数的爆炸分散过程中液体破碎特征提取

应用高速摄像仪对液体的爆炸分散实验过程进行了拍摄,得到了液体爆炸分散过程的时间序列.通过对灰度图像分形维数的分析,揭示了液体爆炸分散过程的机理.     

粗糙面分形计算理论研究进展

为提出一种工程上适用可靠的粗糙面分形维数计算方法,在分形曲线的维数计算方法(码尺法,盒维法)基础上,先后提出了星积分形曲面的维数计算方法、三角形棱柱表面积法、投影覆盖法、立方体覆盖法、改进的立方体覆盖法、分形的增变量描述法等曲面分形维数理论.鉴于上述方法的共有缺陷——获取三维坐标的激光表面仪器的扫描尺度限制,研究者提出了粗糙面图像维数计算理论,包括二值化图像维数、灰度图像维数、RGB图像维数计算理论.最后,本文展望了分形维数计算理论领域内亟待解决的三大问题.     

星积分形曲面及其维数

通过分形曲线定义了一类分形曲面(被称为星积分形曲面),讨论了这类分形曲面的分形维数,得出了分形曲线的维数与它们所构造出的分形曲面维数之间的关系。     

岩石裂隙网络的分形特征

分形特征与分形维数广泛应用于岩石裂隙网络的量化,及与工程参数的关系模型建立.然而,严格的分形维数的极限定义形式难以直接应用,工程应用中多用近似分形维数值代替,近似的结果在建立量化关系模型时会产生蝴蝶效应,在量化及预测过程中产生巨大偏差.本文回顾了分形研究一系列的发展过程,并基于最新的分形定义提出了一种新的分形维数计算方法.通过对于十个岩石裂隙网络分形维数的计算,证明该方法能够准确有效的计算出图形的复杂度,避免了以往计算分形维数所产生的问题.     

铜基复合材料组织形态分形特征的统计分析与研究

通过对铜基复合材料显微组织结构相图的分析和研究,根据分形理论,计算了不同实验条件下铜基复合材料横截面和平行压制力面的显微组织结构相图的分形维数,同时结合统计方法分析了铜基复合材料分形维数的一些统计特性,结果表明,分形维数反映了石墨在样品中的分布规律,分形维数越大,组织结构相图越复杂,石墨分布越不规则,故石墨分布的不规则性可用分形维数来刻画,分形维数可作为材料组织形态分析的一个表征参数,通过统计分析可知,铜基复合材料横截面和平行压制力面的组织结构相图的分形维数服从正态分布,且横截面和平行压制力面的分形维数随石墨含量变化的情况互不影响。     

Koch曲线及其分数阶微积分

给出了Koch曲线的一个复值表达式,并且估计了该表达式的分数阶微积分的分形维数,同时给出了此表达式的Weyl-Marchaud分数阶导数的图像.进一步讨论了Koch曲线的图像与某类自仿分形函数图像的联系.最后证明了这类自仿分形函数的分形维数与其分数阶微积分的分形维数成立着线性关系,一个特殊例子的图像和数值结果在文中给出.     

一类随机Moran集的分形维数

本文研究了随机压缩向量满足一定条件下的随机Moran集的分形维数.利用计算上盒维数的上界和分形维数之间的性质,得到Moran集各种分形维数. 并在一般情形下,给出随机Moran集的上盒维数的上界.     

Katugampola分数阶积分的阶与Weierstrass函数的分形维数之间的关系（英文）

计算Weierstrass函数的Katugampola分数阶积分的分形维数,如盒维数、K-维数和P-维数.证明了Weierstrass函数的Katugampola分数阶积分的阶与Weierstrass函数的分形维数之间存在线性关系.     

基于分形维数的Ni-Co电刷镀层表面形貌分析研究

采用电刷镀技术获得了Ni-Co合金电刷镀层,借助于扫描电子显微镱观察得到了镀层的表面形貌图.通过改变镀液中Co离子含量的比例,使镀层中Co元素含量发生变化,从而使镀层表面形貌发生变化.利用盒维数的计算方法,计算出了不同Co元素含量的表面形貌分形维数.以分形维数作为衡量指标,研究了镀液中Co元素比例对镀层表面形貌的影响.     

中药色谱指纹图谱的小波变换与分形

为了提取中药指纹图谱共性的特征,将小波变换与分形维数相结合,对同一种中药指纹图谱进行小波变换,并求其分形维数,利用相关系数法,考察了分形维数对温度和测试条件的抗干扰能力,结果表明小波变换的分形维数对温度变化具有较好的抗干扰能力,可以作为描述中药指纹图谱共性的特征.     

Global solvability of the problem on the motion of two fluids without surface tension

Unsteady motion of viscous incompressible fluids is considered in a bounded domain. The liquids are separated by an unknown interface on which the surface tension is neglected. This motion is governed by an interface problem for the Navier-Stokes system. First, a local existence theorem is established for the problem in Hölder classes of functions. The proof is based on the solvability of a model problem for the Stokes system with a plane interface, which was obtained earlier. Next, for a small initial velocity vector field and small mass forces, we prove the existence of a unique smooth solution to the problem on an infinite time interval. Bibliography: 7 titles.     

Solvability in weighted Hölder spaces for a problem governing the evolution of two compressible fluids

Local (in time) unique solvability of a problem on the motion of two compressible fluids, one of which has finite volume, is obtained in Hölder spaces of functions with a power-like decay at infinity. After passage to Lagrangian coordinates, we arrive at a nonlinear initial boundary value problem with a given closed interface between the liquids. We establish an existence theorem for this problem on the basis of the solvability of a linearized problem by means of the fixed-point theorem. To obtain estimates and to prove the solvability for the linearized problem, we use the Schauder method and an explicit solution of a model linear problem with a plane interface between the liquids. The results are obtained under some restrictions on the fluid density and viscosities, which mean that the fluids are not much different from each other. Bibliography: 8 titles.To Olga Aleksandrovna Ladyzhenskaya on the occasion of her jubilee__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 57–89.     

The Gompertzian curve reveals fractal properties of tumor growth

The normalized Gompertzian curve reflecting growth of experimental malignant tumors in time can be fitted by the power function y(t)=at^b with the coefficient of nonlinear regression r0.95, in which the exponent b is a temporal fractal dimension, (i.e., a real number), and time t is a scalar. This curve is a fractal, (i.e., fractal dimension b exists, it changes along the time scale, the Gompertzian function is a contractable mapping of the Banach space R of the real numbers, holds the Banach theorem about the fix point, and its derivative is 1). This denotes that not only space occupied by the interacting cancer cells, but also local, intrasystemic time, in which tumor growth occurs, possesses fractal structure. The value of the mean temporal fractal dimension decreases along the curve approaching eventually integer values; a fact consistent with our hypothesis that the fractal structure is lost during tumor progression.     

Problem of the motion of two compressible fluids separated by a closed free interface

We consider the problem of the simultaneous evolution for two barotropic capillary viscous compressible fluids occupying the space ℝ³ and separated by a closed free interface. Under some restrictions on the viscosities of the liquids, the local (in time) unique solvability of this problem is obtained in the Sobolev-Slobodetskii spaces. After the passage to Lagrangian coordinates it is possible to exclude the fluid density from the system of equations. The proof of the existence theorem for a nonlinear, noncoercive initial boundary-value problem is based on the method of successive approximations and on an explicit solution of a model linear problem with a plane interface between the liquids. The restrictions on the viscosities mentioned above appear in the intermediate estimation of this explicit solution in the Sobolev spaces with an exponential weight. Bibliography: 8 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 000, 1997, pp. 61–86. Translated by I. V. Denisova.     

Explosive-crystallization waves in thin amorphous layers on heat conducting substrates with thermal contact resistance

A model for explosive crystallization in a thin amorphous layer on a heat conducting substrate is presented. Rate equations are used to describe the kinetics of the homogeneous amorphous-crystalline transition. Heat conduction into the substrate and thermal contact resistance at the interface between layer and substrate are taken into account. The whole process is examined as a wave of invariant shape in a moving frame of reference. A coupled system of an integro-differential equation and ordinary differential equations is obtained and solved numerically. The propagation velocity of the wave is obtained as an eigenvalue of the system of equations. Some representative solutions are shown. Crystallization-wave velocities are compared with experimental values for explosive crystallization in germanium. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quarks confinement via Kaluza–Klein theory as a topological property of quantum classical spacetime phase transition

By introducing a curled fifth dimension, Kaluza–Klein theory predicted for the first time a connection between gravity and electromagnetism. An exacting look at this result shows that for a radius R of the fifth dimension equal to the Planck length, the coupling is exactly unity. The result is utilized to show that by introducing correction terms to the one loop renormalization equation of unification it can be made exact and subsequently quark confinement can be proven non-perturbatively as a property of the topology of quantum spacetime at the classical-quantum interface and the Planck phase transition.     

A Note on Semilocal Reciprocity Laws on Curves

The aim of this work is to offer a definition of the Contou-Carrère symbol associated with a closed point of an algebraic curve and with a local ring of dimension zero, first, and then with a semilocal ring of dimension zero, from the commutator of a certain central extension. When the curve is complete, we deduce the reciprocity law in both cases. Moreover, we give some applications to the residues, and obtain explicit relations between the classic residue and the Witt residue.

Communicated by C. Pedrini.     

A Jordan surface theorem for three-dimensional digital spaces

Many applications of digital image processing now deal with three-dimensional images (the third dimension can be time or a spatial dimension). In this paper we develop a topological model for digital three space which can be useful in this context. In particular, we prove a digital, three-dimensional, analogue of the Jordan curve theorem. (The Jordan curve theorem states that a simple closed curve separates the real plane into two connected components.) Our theorem here is a digital topological formulation of the Jordan-Brouwer theorem about surfaces that separate three-dimensional space into two connected components.     

Spatially explicit ecological models: a spatial convolution approach

Spatial structure tends to have a stabilizing influence on predator–prey interactions in which the local model predicts extinction of the system. This result is well supported by laboratory observations of simple systems. Here, we use a spatially explicit version of the Nicholson–Bailey model having Moran–Ricker host reproduction to repeat and extend some of these results. Our model is a discrete spatial convolution model analogous to the integrodifference equations (IDEs) used by other authors. We show a spatial rescue effect which prevents extinction of the system by reducing the size (standard deviation) of the dispersal pdf. We also show that very favorable habitat (K=∞) and marginal habitat (K=1.0), when mixed randomly together in an explicit map, are highly stabilizing whereas either kind of habitat alone will cause extinction. The marginal habitat in this situation has host densities below parasite replacement level and thus constitutes a host refuge (although not a complete one) from the parasite. When a host–parasitoid model having spiral wave dynamics in two-dimensional space was extended to one- and three-dimensional space, we observed analogous dynamics, i.e., traveling waves of evasion and pursuit in one dimension and ‘spiral-like’ structures in a three-dimensional spatial volume. We illustrate an approach to analysis of spatial convolution models via the frequency response of the system transfer function. In spatial convolution format, local interaction and dispersal are conveniently isolated from one another, and this allows us to vary these components independently and thus to study their effects on the dynamics of the total system. We show two examples of nonrandom dispersal pdf’s – a bimodal form representing two dispersal types in the population and a ‘ripple’ pdf representing a repulsive process.