Abstract:
Natural radioactivity is very important for the assessment of the marine sand property and usability. By using gamma spectrometry, the concentration of the natural radionuclides ^{226}Ra, ^{232}Th and ^{40}K have been measured in marine sand deposits from Liaodong Bay (LDB), North Yellow Sea (NYS), Zhoushan area (ZS), Taiwan Shoal (TS) and Pearl River Mouth (PR), offshore China, which are potential marine sand mining areas. The radiation activity equivalent (Raeq), indoor gamma absorbed dose rate (DR), annual effective dose (HR), alpha index (Ia), gamma index (Ig), external radiation hazard index (Hex), internal radiation hazard index (Hin), representative level index (RLI), excess lifetime cancer risk (ELCR) and annual gonadal dose equivalent (AGDE) associated with the natural radionuclides are calculated to assess the radiation hazard of the natural radioactivity in the marine sands offshore China. From the analysis, it is found that these marine sands are safe for the constructions. The Pearson correlation coefficient reveals that the ^{226}Ra distribution in the marine sands offshore China is controlled by the variation of the ^{40}K concentration. Principal component analysis (PCA) yields a two-component representation of the entire data from the marine sands, wherein 98.22% of the total variance is explained. Our results provide good baseline data to expand the database of radioactivity of building materials in China and all over the world.

Abstract:
The East Asian monsoon system influences the sedimentation and transport of organic matter in East Asian marginal seas that is derived from both terrestrial and marine sources. In this study, we determined organic carbon(OC) isotope values, concentrations of marine biomarkers, and levels of OC and total nitrogen(TN) in core YSC-1 from the central South Yellow Sea(SYS). Our objectives were to trace the sources of OC and variations in palaeoproductivity since the middle Holocene, and their relationships with the East Asian monsoon system. The relative contributions of terrestrial versus marine organic matter in core sediments were estimated using a two-end-member mixing model of OC isotopes. Results show that marine organic matter has been the main sediment constituent since the middle Holocene. The variation of terrestrial organic carbon concentration(OCter) is similar to the EASM history. However, the variation of marine organic carbon concentration(OCmar) is opposite to that of the EASM curve, suggesting OCmar is distinctly influenced by terrestrial material input. Inputs of terrestrial nutrients into the SYS occur in the form of fluvial and aeolian dust, while concentrations of nutrients in surface water are derived mainly from bottom water via the Yellow Sea circulation system, which is controlled by the East Asian winter monsoon(EAWM). Variations in palaeoproductivity represented by marine organic matter and biomarker records are, in general, consistent with the recent EAWM intensity studies, thus, compared with EASM, EAWM may play the main role to control the marine productivity variations in the SYS

Abstract:
The Fourier transform and spectral analysis are employed to estimate the fractal dimension and explore the fractal parameter relations of urban growth and form using mathematical experiments and empirical analyses. Based on the models of urban density, two kinds of fractal dimensions of urban form can be evaluated with the scaling relations between the wave number and the spectral density. One is the radial dimension of self-similar distribution indicating the macro-urban patterns, and the other, the profile dimension of self-affine tracks indicating the micro-urban evolution. If a city's growth follows the power law, the summation of the two dimension values may be a constant under certain condition. The estimated results of the radial dimension suggest a new fractal dimension, which can be termed “image dimension”. A dual-structure model named particle-ripple model (PRM) is proposed to explain the connections and differences between the macro and micro levels of urban form. 1. Introduction Measurement is the basic link between mathematics and empirical research in any factual science [1]. However, for urban studies, the conventional measures based on Euclidean geometry, such as length, area, and density, are sometimes of no effect due to the scale-free property of urban form and growth. Fortunately, fractal geometry provides us with effective measurements based on fractal dimensions for spatial analysis. Since the concepts of fractals were introduced into urban studies by pioneers, such as Arlinghaus [2], Batty and Longley [3], Benguigui and Daoud [4], Frankhauser and Sadler [5], Goodchild and Mark [6], and Fotheringham et al. [7], many of our theories of urban geography have been reinterpreted using ideas from scaling invariance. Batty and Longley [8] and Frankhauser [9] once summarized the models and theories of fractal cities systematically. From then on, research on fractal cities has progressed in various aspects, including urban forms, structures, transportation, and dynamics of urban evolution (e.g., [10–20]). Because of the development of the cellular automata (CA) theory, fractal geometry and computer-simulated experiment of cities became two principal approaches to researching complex urban systems (e.g., [21–25]). Despite all the above-mentioned achievements, however, we often run into some difficult problems in urban analysis. The theory on the fractal dimensions of urban space is less developed. We have varied fractal parameters on cities, but we seldom relate them with each other to form a systematic framework. Moreover, the estimation

Abstract:
Fractal growth is a kind of allometric growth, and the allometric scaling exponents can be employed to describe growing fractal phenomena such as cities. The spatial features of the regular fractals can be characterized by fractal dimension. However, for the real systems with statistical fractality, it is incomplete to measure the structure of scaling invariance only by fractal dimension. Sometimes, we need to know the ratio of different dimensions rather than the fractal dimensions themselves. A fractal-dimension ratio can make an allometric scaling exponent (ASE). As compared with fractal dimension, ASEs have three advantages. First, the values of ASEs are easy to be estimated in practice; second, ASEs can reflect the dynamical characters of system's evolution; third, the analysis of ASEs can be made through prefractal structure with limited scale. Therefore, the ASEs based on fractal dimensions are more functional than fractal dimensions for real fractal systems. In this paper, the definition and calculation method of ASEs are illustrated by starting from mathematical fractals, and, then, China's cities are taken as examples to show how to apply ASEs to depiction of growth and form of fractal cities. 1. Introduction Dimension is a measurement of space, and measurement is the basic link between mathematical models and empirical research. So, dimension is a necessary measurement for spatial analysis. Studying geographical spatial phenomena of scaling invariance such as cities and systems of cities has highlighted the value of fractal dimension [1–6]. However, there are two problems in practical work. On the one hand, sometimes it is difficult for us to determine the numerical value of fractal dimension for some realistic systems, but it is fairly easy to calculate the ratio of different fractal parameters. On the other hand, in many cases, it is enough to reveal the system’s information by the fractal-dimension ratios and it is unnecessary to compute fractal dimension further [7]. The ratio of different dimensions of a fractal can constitute an allometric coefficient under certain conditions. As a parameter of scale-free systems, the allometric coefficient is in fact a scaling exponent, and the fractal-dimension ratio can be called allometric scaling exponent (ASE). The use of ASEs for the simple regular fractals in the mathematical world is not very noticeable. But for the quasifractals or random fractals in the real world, the function of ASEs should be viewed with special respect. A city can be regarded as an evolutive fractal. The land-use

Abstract:
Hierarchy of cities reflects the ubiquitous structure frequently observed in the natural world and social institutions. Where there is a hierarchy with cascade structure, there is a Zipf's rank-size distribution, and vice versa. However, we have no theory to explain the spatial dynamics associated with Zipf's law of cities. In this paper, a new angle of view is proposed to find the simple rules dominating complex systems and regular patterns behind random distribution of cities. The hierarchical structure can be described with a set of exponential functions that are identical in form to Horton-Strahler's laws on rivers and Gutenberg-Richter's laws on earthquake energy. From the exponential models, we can derive four power laws including Zipf's law indicative of fractals and scaling symmetry. A card-shuffling model is built to interpret the relation between Zipf's law and hierarchy of cities. This model can be expanded to illuminate the general empirical power-law distributions across the individual physical and social sciences, which are hard to be comprehended within the specific scientific domains. This research is useful for us to understand how complex systems such as networks of cities are self-organized.

Abstract:
The method of spectral analysis is employed to research the spatial dynamics of urban population distribution. First of all, the negative exponential model is derived in a new way by using an entropy-maximizing idea. Then an approximate scaling relation between wave number and spectral density is derived by Fourier transform of the negative exponential model. The theoretical results suggest the locality of urban population activities. So the principle of entropy maximization can be utilized to interpret the locality and localization of urban morphology. The wave-spectrum model is applied to the city in the real world, Hangzhou, China, and spectral exponents can give the dimension values of the fractal lines of urban population profiles. The changing trend of the fractal dimension does reflect the localization of urban population growth and diffusion. This research on spatial dynamics of urban evolvement is significant for modeling spatial complexity and simulating spatial complication of city systems by cellular automata.

Abstract:
Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Using the idea from general fractals and scaling, I propose a dual competition hypothesis of city development to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another, but represent different processes of urban evolution, respectively. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, the parameter interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf distribution, a size correlation function can be built, and it is opposite to the first one. By the second correlation function and multifractals notion, the Pareto exponent interval is derived as [1, 2). Thus the process of urban evolution falls into two effects: one is the Pareto effect indicating city number increase (external complexity), and the other the Zipf effect indicating city size growth (internal complexity). Because of struggle of the two effects, the scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium with each other, the scaling exponent approaches 1. A series of mathematical experiments on hierarchical correlation are employed to verify the models and a conclusion can be drawn that if cities in a given region follow Zipf's law, the frequency and size correlations will follow the scaling law. This theory can be generalized to interpret the inverse power-law distributions in various fields of physical and social sciences.

Abstract:
To remove the confusion of concepts about different sorts of geographical space and dimension, a new framework of space theory is proposed in this paper. Based on three sets of fractal dimensions, the geographical space is divided into three types: real space (R-space), phase space (P-space), and order space (O-space). The real space is concrete or visual space, the fractal dimension of which can be evaluated through digital maps or remotely sensed images. The phase space and order space are both abstract space, the fractal dimension values of which cannot be estimated with one or two maps or images. The dimension of phase space can be computed by using time series, and that of order space can be determined with cross-sectional data in certain time. Three examples are offered to illustrate the three types of spaces and fractal dimension of geographical systems. The new space theory can be employed to explain the parameters of geographical scaling laws, such as the scaling exponent of the allometric growth law of cities and the fractal dimension based on Horton’s laws of rivers.

Abstract:
Spatial autocorrelation plays an important role in geographical analysis; however, there is still room for improvement of this method. The formula for Moran’s index is complicated, and several basic problems remain to be solved. Therefore, I will reconstruct its mathematical framework using mathematical derivation based on linear algebra and present four simple approaches to calculating Moran’s index. Moran’s scatterplot will be ameliorated, and new test methods will be proposed. The relationship between the global Moran’s index and Geary’s coefficient will be discussed from two different vantage points: spatial population and spatial sample. The sphere of applications for both Moran’s index and Geary’s coefficient will be clarified and defined. One of theoretical findings is that Moran’s index is a characteristic parameter of spatial weight matrices, so the selection of weight functions is very significant for autocorrelation analysis of geographical systems. A case study of 29 Chinese cities in 2000 will be employed to validate the innovatory models and methods. This work is a methodological study, which will simplify the process of autocorrelation analysis. The results of this study will lay the foundation for the scaling analysis of spatial autocorrelation.

Abstract:
This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann's equation. For the normalized data, Boltzmann's equation is equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is made that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise, and the city system will fall into disorder. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos seem to be related with some kind of replacement process.