Fractal Density Research Articles

Enhanced catalytic activity of modified g-C3N4 heterojunction for photocatalytic degradation of methylene blue from wastewater

Rapid recombination of photoinduced charge carriers and poor photocatalytic degradation performance greatly hinder the large-scale application of g-C3N4 photocatalysis. Therefore, the g-C3N4 is modified by BiPO4 to overcome its poor photocatalytic performance, which is investigated by methylene blue (MB) removal. The S scheme heterojunction is constructed by modifying g-C3N4 with BiPO4, which can effectively preserve the redox activities of holes and electrons on VB and CB, exhibiting good photocatalytic activity. Therefore, the modified g-C3N4 exhibits excellent photocatalytic activity with efficient separation of the photoelectron-hole, compared to pure g-C3N4 and BiPO4. The modified g-C3N4 exhibits large MB degradation removal of 96.7 %, which is significantly larger than the pure g-C3N4 (46.3 %) and BiPO4 (3.3 %) owe to synergy, respectively. The photodegradation rate constant of the modified g-C3N4 is 4.8 and 43 times larger than that of the pure g-C3N4 and BiPO4, respectively. The modified g-C3N4 still has large MB removal of 92.9 % after 5 cycles, indicating excellent stability and recyclability. The fractal density calculation of the modified g-C3N4 is investigated combined with LUMO and HOMO analysis. The MB photocatalytic degradation process follows the S scheme charge transfer mechanism, based on degradation mechanism analysis combined with semiconductor energy band and density functional theory calculation analysis. The MB photocatalytic degradation path is systemically analyzed based on MS-UPLC result analysis, which indicates that MB is finally decomposed into CO2 and H2O, achieving the degradation of MB.

Assessment of reservoir heterogeneities and hydrocarbon potential zones using wavelet-based fractal and multifractal analysis of geophysical logs of Cambay basin, India

The scaling behaviour of physical properties of the earth's subsurface e.g. porosity, permeability, density, etc significantly influence reservoir characteristics. The characterization of the parameters, by analysing nonlinear geophysical logs using the application of linear and traditional empirical relations, is very tedious, complicated, and yields less accurate results. In this perspective, wavelet-based fractals and multifractals techniques provide effective insights into local and global characteristics of heterogeneity of reservoir parameters. The results from the numerical experiment on simulated synthetic data reveal that the wavelet-based fractal and multifractals are more useful to delineate the shorter fluctuations in signal and heterogeneities of the reservoir. The scaling exponent and fractal density, a component of fractal analysis, have been estimated for each sample from power law relationship that facilitates the understanding of the local characteristics of the subsurface formation. On the other hand, the multifractal spectrum, a generalization curve of the hurst exponents, provides the global nature of heterogeneities corresponding to reservoir parameters. We analyse gamma ray, density, and neutron logs of three wells (well-204, 203, and 172) of the Limbodara oil field of Cambay Basin, India. The fractal density curve of gamma ray log is found like the low-pass filtered version of the original signal, and map the lithologies by clustering their similar values while the scaling exponent curve on either side of 0.5 of scaling exponent demarcates the lithological boundaries at the intersection points. The scaling exponent curve highlights the finer variations embedded in the signal and is feasible to enhance the thin layers of the reservoir. The multifractal spectrum of gamma-ray logs across three wells encompasses three major formations emphasizing that the Kalol formation exhibits higher degree of heterogeneity than other formations (Tarapur and Olpad). The multifractal spectrum of neutron and gas corrected log for each well of Kalol formation shows the greater influence on well-203 and lowest influence on well-172, indicating the effective hydrocarbon bearing zones of the former well than the latter. This evidence that an increase in lithological heterogeneity enhances the quality of hydrocarbon reservoir as well as reduces the multifractality of porosity logs.

Build Network Intrusion Detection System based on combination of Fractal Density Peak Clustering and Artificial Neural Network

Imbalanced data poses a serious problem in intrusion detection systems. In this article, we propose a network intrusion detection system based on fractal density peak clustering and an artificial neural network (FD-ANN). The proposed detection system consists of three parts: data clustering based on the density-peak clustering (DPC) method, using the fractal concept as a membership weight of all data to the cluster, and a neural network to classify the data. The DPC method uses categorization of the tare data into subgroups with strongly correlated attributes to reduce the size of the training data and the imbalance of the sample. Each subgroup has its neural network to train the data. Based on fractal membership weights, the output of all classifiers of the sub-neural networks is combined using the aggregation function. The benchmarks of this model are based on the data sets NSL-KDD and UNSW-NB15. The proposed solution outperforms other known classification approaches in terms of overall accuracy, recall, precision, and F1 score.

(Invited) Long Range Coherent Transport in Proteins via Electron Pairing

Recently, Lindsay et al. showed in a series of papers (summarized in Lindsay, S. Life 2020, 10, 72.) that single proteins coupled to metal electrodes have unexpectedly good conductance properties. It became clear that much of the resistance drop happens at the metallic contacts, and the internal conductivity of the protein itself can be very close to the quantum limit 2e^2/h for short proteins. The localization length can be as large as ten nanometers for longer ones. Cahen et al. showed that the conductance is temperature independent in the range of 30-300 Kelvins in a series of measurements. All this indicates that proteins support fast and at least partially coherent electron transport at room temperature. According to electronic structure calculations, proteins have significant HOMO-LUMO gaps surrounded by localized orbitals, a typical arrangement for insulators ruling out efficient single-electron quantum transport even in the presence of photon mediation. However, due to the repetitive nature of the basic peptide backbone of the proteins, there are states deeper below the HOMO energy, which are intermediary between extended states typical in metallic conductance bands and the localized ones. These so-called critical states (Vattay G. et al. 2015 J. Phys.: Conf. Ser. 626 012023) have fractal density distributions and show large electron density correlations between other orbitals in the same energy region. It has been shown that in the presence of electron-electron attraction, these states are especially suited for pair formation of electrons, and an insulator-superconductor transition can occur even at high temperatures. Here we argue that polarizable sidechains of amino acids can create an attractive force between electrons propagating along the backbone. We show that the typical size and polarization of amino acids make the proper magnitude of pairing energy to maintain superconducting patches even at room temperature. We calculate and display the geometry of the superconducting gap over the protein structure and propose this new mechanism as a possible explanation for the observed coherent transport.

Identification of Multi-Element Geochemical Anomalies for Cu–Polymetallic Deposits Through Staged Factor Analysis, Improved Fractal Density and Expected Value Function

Identification of Multi-Element Geochemical Anomalies for Cu–Polymetallic Deposits Through Staged Factor Analysis, Improved Fractal Density and Expected Value Function

Singularity analysis of igneous zircon U-Pb age and Hf isotopic record in the Zhongdian arc, northwest Yunnan, China: Implications for Indosinian magmatic flare-up and the formation of porphyry copper deposits

Situated in the southern part of Yidun arc in the southeastern Tibetan Plateau, the Zhongdian arc records magmatic activity that ranges from approximately 270 to 190 Ma, which is likely attributable to subduction of the Garze-Litang slab and involved steady low-volume magmatic lulls punctuated by short-lived high-volume flare-ups. Many porphyry copper deposits have been found in these magmatic rocks, but the temporal-spatial relationships among magmatism, porphyry copper deposits, and subduction processes around the Zhongdian arc have long been debated. Frequency distributions calculated from igneous zircon U-Pb ages are commonly utilized to interpret the duration of magmatic events. Recently, a new concept of fractal density and a local singularity analysis (LSA) method have been applied to analyze the geometric properties of zircon U-Pb age peak anomalies, which are linked to deeply rooted avalanches associated with short spurts of convection during the formation of supercontinents and continental crust growth. In this paper, the igneous zircon U-Pb dating data collected from the Zhongdian arc are analyzed by the LSA method from the perspective of fractal density to interpret the causational relationship between age peak and subduction process of the Garze-Litang oceanic crust. The results show that the age density around the age peak of 215 Ma can be well fitted and described by power law functions, and the strong singularity (k > 0.8) indicates the occurrence of extreme geological events. Meanwhile, systematic changes in the hafnium isotopic composition of igneous zircons with time reflect the upwelling of asthenospheric mantle materials, characterized by a rapid increase in εHf(t) values that reach the highest around the 215 Ma peak. Moreover, the log–log plot that shows the cumulative number-age distribution reveals another transitional age of 235 Ma. By comparison of the magmatic intensity between the western and eastern belts of the Zhongdian arc, it is indicated that the eastward magmatic migration might have begun at ca. 235 Ma. Combined with the previous work, we propose that the Indosinian magmatic flare-up in the Zhongdian arc might have been caused by superimposition of slab subduction, steepening of slab dip, and slab breakoff. This long-term magmatic hydrothermal evolution system and extreme singularity events also provide indispensable conditions for the formation of large-scale and high-grade porphyry copper deposits in the region. Our findings provide new insights into the subduction process of the Garze-Litang oceanic crust and Indosinian magmatic framework around the Zhongdian arc. It is believed that the LSA method has potential applications in depicting the variation of magmatic activity and predicting mineral resources.

Sudden cessation of fluoxetine before alcohol drinking reinstatement alters microglial morphology and TLR4/inflammatory neuroadaptation in the rat brain

Preclinical studies on the effects of abrupt cessation of selective serotonin reuptake inhibitors (SSRIs), a medication often prescribed in alcohol use disorder (AUD) patients with depression, results in alcohol consumption escalation after resuming drinking. However, a potential neuroinflammatory component on this escalation remains unexplored despite the immunomodulatory role of serotonin. Here, we utilized a rat model of 14-daily administration of the SSRI fluoxetine (10 mg/kg/day) along alcohol self-administration deprivation to study the effects of fluoxetine cessation on neuroinflammation after resuming alcohol drinking. Microglial morphology and inflammatory gene expression were analyzed in prelimbic cortex, striatum, basolateral amygdala and dorsal hippocampus. Results indicated that alcohol drinking reinstatement increased microglial IBA1 immunoreactivity and altered morphometric features of activated microglia (fractal dimension, lacunarity, density, roughness, and cell area, perimeter and circularity). Despite alcohol reinstatement, fluoxetine cessation modified microglial morphology in a brain region-specific manner, resulting in hyper-ramified (spatial complexity of branching), reactive (lower heterogeneity and circularity)-like microglia. We also found that microglial cell area correlated with changes in mRNA expression of chemokines (Cx3cl1/fractalkine, Cxcl12/SDF1α, Ccl2/MCP1), cytokines (IL1β, IL6, IL10) and the innate immune toll-like receptor 4 (TLR4) in dorsal hippocampus. Specifically, TLR4 correlated with microglial spatial complexity assessed by fractal dimension in striatum, suggesting a role in process branching. These findings suggest that alcohol drinking reinstatement after fluoxetine treatment cessation disturbs microglial morphology and reactive phenotype associated with a TLR4/inflammatory response to alcohol in a brain region-specific manner, facts that might contribute to alcohol-induced damage through the promotion of escalation of alcohol drinking behavior.

Nonlocal quantum system with fractal distribution of states

In this paper, we generalize simple model of quantum mechanics and statistics to take into account power-law nonlocality by using fractional differential equations, and fractal distribution of states. These equations with derivatives of non-integer orders are widely used in describing such complex properties of systems as power-law spatial nonlocality and spatial dispersion, long-range interactions, as well as the openness of quantum systems (interactions with the environment). We suggest a generalization of quantum statistical model of an ideal gas of particles for the case of spatial nonlocality, long-range interactions and fractal distribution of permitted states. To take into account the power-law nonlocality and long-range interactions, we use the fractional Laplacian in the Riesz–Trujillo form in the Schrodinger equation. To take into account the fractal distribution of states, we use non-integer dimensional space approach. Expression for the fractal density of states for fractal distribution of permitted quantum states is proposed. The equation to calculate energy and total number of particles for quantum systems with power-law non-locality and fractal distribution of quantum states are derived. Examples of these calculations for degenerate quantum system are suggested in the framework of the proposed quantum statistical model.

Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes

Integral and differentiation are two mathematical operations in modern calculus and analysis which have been commonly applied in many fields of science. Integration and differentiation are associated and linked as inverse operation by the fundamental theorem of calculus. Both integral and differentiation are defined based on the concept of additive Lebesgue measure although various generations have been developed with different forms and notations. Fractals can be considered as geometry with fractal dimension (e.g., non-integer) which no longer possesses Lebesgue additive property. Accordingly, the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity. This paper introduces a recently developed concept of fractal differentiation and integral operations. These operations are expressed using the similar notations of the ordinary operations except the measures are defined in fractal space or measures with fractal dimension. The calculus operations can be used to describe the new concept of fractal density, the density with fractal dimension or density of matter with fractal dimension. The concept and methods are also applied to interpret the Bouguer anomaly over the mid-ocean ridges. The results show that the Bouguer gravity anomaly depicts singularity over the mid-ocean ridges. The development of new calculus operations can significantly improve the accuracy of geodynamic models.

Fractal Density Matrix for the Case of Time Dependent Radio Frequency Pulses Defined in the First Rotating Frame

We treat in this effort the problem of time evolution during a time dependent radio-frequency pulse in the First Rotating Frame (FRF) with Fractal Time Derivatives for the Representation of the Schroedinger Equation and Fractal Density Matrix for Like Spins ½. The resultant time evolutions during a Sin-Cos Pulse are compared with the Time Dependence during the Fractal Bloch equations without relaxation solved using Standard Runge Kutta methods.

Evaluation of the depressive state of mobile technical systems operator subjected to electromagnetic noise radiation

This paper presents the results of the work aimed at a study of the patterns of changes in the nonlinear electroencephalogram (EEG) parameters, including fractal dimension and self-similarity exponent, when the operator is irradiated with electromagnetic noise radiation. Together with the above-mentioned nonlinear parameters, a change in the spectral power density of the rhythmic components in EEG (delta-, theta-, alpha-, and beta-rhythms) has been studied. Investigation of the fractal dimension, self-similarity exponent, and spectral power density during irradiation was associated with possible changes of the parameters in the case of operator’s depression or minor depression. The radiation source was represented by a transistor electromagnetic-noise generator with the power of 30 mW operating over the frequency range of 5 GHz. The methods for calculation of the nonlinear parameters including fractal dimension and self-similarity exponent have been described. To realize the principal objectives of the work, the Java-based software was developed. The relevant literature demonstrating the changes in fractal dimension, self-similarity exponent, spectral power density of the delta-, theta-, alpha-, beta-rhythms in the case of depression and minor depression has been reviewed. Electroencephalograms were registered according to the “10/20” scheme using the MBN Neurocartograph electroencephalograph. The analyzed leads were Fp1, Fp2, T3, T4, P3, P4, O1, O2, F3, F4, C3, C4. As shown by the results of this work, there is no distinct depressive state of the operator exposed to electromagnetic noise radiation, judging by changes in the self-similarity exponent, fractal dimension, and spectral power density. However, when the operator was irradiated with electromagnetic noise radiation, the observed tendency in variation of the parameters was characteristic for minor depression.

On a formula finding fractal dimension

Purpose: The article is devoted to the determination of the fractal dimension of cellular concrete, in particular foam concrete, and the further clarification of the relationship between fractal dimension and porosity and average density of cellular concrete. Design/methodology/approach: In the theoretical description of disordered systems, the fundamentals of fractal theory are actively used, which allow obtaining statistical indicators of chaotic natural and artificially disordered systems, which include cellular concrete. The parameters of the pore structure are difficult to quantify by conventional methods because of the complexity and irregularity of the pore structure due to their random distribution. Findings: Formulas for calculating the fractal dimension and average density of highly porous material are calculated and proposed. The formula for calculating the average density takes into account the density of the material between the pore walls. Research limitations/implications: The calculation of the fractal dimension is one of the main factors affecting the practical application of the theory of fractals, a natural problem arises on a theoretical basis to justify these calculations. Practical implications: The formulas proposed in this work for calculating the fractal dimension and density of a highly porous structure improve research on methods for producing substances with a controlled fractal structure, which will help create materials with unusual mechanical properties, density, and porosity. Originality/value: The formula for calculating the fractal dimension obtained in the work improves the well-known Hausdorff-Bezikovich formula. On the other hand, it makes it possible to obtain a highly porous structure with a given density of the material under study.

KONTROL GEOLOGI TERHADAP PEMUNCULAN MANIFESTASI PANASBUMI DI KAWASAN GUNUNG LAWU

Gunung Lawu memiliki potensi panasbumi yang menjanjikan untuk dieksploitasi. Keberadaan berbagai jenis manifestasi dikawasan gunungapi tersebut mengindikasikan keberadaan sistem panas bumi di daerah ini yang perlu diteliti secara lebih rinci. Salah satu kajian yang perlu dilakukan adalah mengetahui faktor pengontrol pemunculan manifestasi panasbumi di kawasan gunung Lawu sehingga model sistem panas bumi bisa disusun dengan baik. Kami melakukan analisis peta geologi gunungapi, kerapatan kelurusan, dan geokimia air untuk mengetahui kaitan antara kondisi geologi, terutama struktur geologi, terhadap pemunculan air panas di kawasan Gunung Lawu. Hasil penelitian kami menunjukkan bahwa pemunculan manifestasi panasbumi di Gunung Lawu tidak berasosiasi secara langsung dengan jalur sesar-sesar turun utama yang berkembang di daerah penelitian. Analisa kerapatan kelurusan juga tidak menunjukkan korelasi yang signifikan antara nilai kerapatan tinggi dengan jalur sesar-sesar turun yang telah dipetakan sebelumnya. Jalur sesar-sesar tersebut berada pada nilai kerapatan kelurusan sedang. Hal ini menunjukkan bahwa nilai kerapatan kelurusan yang tinggi lebih banyak dipengaruhi oleh aspek morfologi yang didominasi oleh proses-proses di permukaan. Hasil penelitian kami menunjukkan bahwa kemunculan manisfetasi panasbumi di Gunung Lawu dikontrol terutama oleh keberadaan zona rekahan dan kekar yang diindikasikain oleh nilai kerapatan kelurusan sedang. Analisis geokimia air mendukung interpretasi tersebut dimana tipe air panas bumi yang muncul pada mata air panas Cumpleng dan Ngunut menunjukkan tipe air bikarbonat yang dicirikan oleh hadirnya endapan limonit dan travertin. Tipe fluida panasbumi dan endapan mineral ubahan tersebut biasanya berasosiasi dengan keberadaan rekahan dan kekar dibawah atau dekat permukaan.

Combined crystalline, lamellar and granular structural insights into in vitro digestion rate of native starches

In low moisture foods, such as biscuits, starch supramolecular and granular structures are largely retained, and these govern starch digestibility in such systems. Here, the supramolecular (crystalline, lamellar, and granular) structures of ten different starches were investigated to evaluate how such structures together determine the digestion rate of native starch. In particular, the Spearman correlation analysis revealed that starch digestion rate could be affected by the crystalline type and the granule pores or channels. However, these structural features did not govern the digestion rate in isolation. For instance, starch granules with the same crystalline type could show significantly different digestibility. Hence, the relationship between combinations of multiple structural features and the digestion rate was further investigated through the correlation analysis. Specifically, we found that 1) thinner lamellae caused a lower digestion rate for either A- or B/C type starch; 2) the degree of crystallinity tended to significantly affect the digestion rate of B/C-type starch rather than A-type starch; and 3) the compactness of starch fractals (power law exponent α) was negatively correlated with the digestion rate. The underlying mechanisms were discussed for these correlations. The results could facilitate the strategic selection of starch resources for developing low moisture foods with reduced starch digestion rate (glycaemic index), using combined structural features such as lamella thickness, fractal density and crystallinity.

Can spatial distribution of ungulates be predicted by modeling camera trap data related to landscape indices? A case study in a fragmented Mediterranean landscape

Camera trap applications range from studying wildlife habits to detecting rare species, which are difficult to capture by more traditional techniques. In this work, we aimed at finding the best model to predict the distribution pattern of wildlife and to explain the relationship between environmental conditions with the species detected by camera traps. We applied two types of statistical models in a specific Mediterranean landscape case. The results of both models shown adjustments over 80 %. First, we ran a Principal Components Analysis (PCA). Discriminant, and logistic analyses were performed for ungulates in general, and three species in particular: Barbary sheep, mouflon, and wild boar. The same environmental conditions explained the presence of these species in all the proposed models. Hence, we proved the generally positive influence of patch size on the presence of ungulates and negative influence of the fractal dimension and density edge. We quantified the relationships between a suite of landscape metrics measured in different grids to test whether spatial heterogeneity plays a major role in determining the distribution of ungulates. We explained much of the variation in distribution with metrics, specifically related to habitat heterogeneity. That outcome highlighted the potential importance of spatial heterogeneity in determining the distribution of large herbivores. We discussed our results in the forestry conservation practices context and discuss potential ways to integrate ungulate management and forestry practices better.

A tale of two habitats: preliminary comparison of fish abundance and diversity between saltmarsh- and mangrove-dominated creeks in the Nahoon Estuary, South Africa

A preliminary assessment of abundance and richness of fishes utilising saltmarsh and mangrove habitats in the Nahoon Estuary, situated on the eastern coast of South Africa, was undertaken during July 2017. The structural composition and complexity of the two habitats were assessed and underwater cameras were used to provide preliminary insight into the use of the different habitats by fish taxa. The saltmarsh habitat had a higher fractal dimension and density of stems than the mangrove habitat. The species richness of fishes was slightly higher in the mangroves than in the saltmarsh, with nine and seven taxa recorded in the two habitats, respectively. The mean relative abundance of fishes was higher in the saltmarsh than in the mangroves and this was mainly due to the dominance of shoaling estuarine zooplanktivores in the saltmarsh. The mean relative abundance of estuary- associated marine fish species was, however, higher in the mangroves, which may be indicative of greater nursery use of this habitat by marine fishes. Although this is the first study to simultaneously assess the role of both saltmarsh and mangroves as fish habitat in estuaries of temperate South Africa, the findings are preliminary, and further study on seasonal differences in species assemblages is recommended.

Fractal x-ray edge problem at the critical point of the Aubry-André model

We study the Anderson orthogonality catastrophe, and the corresponding x-ray edge problem, in systems that are at a localization transition driven by a deterministic quasiperiodic potential. Specifically, we address how the ground state of the Aubry-Andre model, at its critical point, responds to an instantaneous local quench. At this critical point, both the single-particle wavefunctions and the density of states are fractal. We find, numerically, that the overlap between post-quench and pre-quench wavefunctions, as well as the "core-hole" Green function, evolve in a complex, non-monotonic way with system size and time respectively. We interpret our results in terms of the fractal density of states at this critical point. In a given sample, as the post-quench time increases, the system resolves increasingly finely spaced minibands, leading to a series of alternating temporal regimes in which the response is flat or algebraically decaying. In addition, the fractal critical wavefunctions give rise to a quench response that varies strongly from site to site across the sample, which produces broad distributions of many-body observables. Upon averaging this broad distribution over samples, we recover coarse-grained power laws and dynamical exponents characterizing the x-ray edge singularity. We discuss how these features can be probed in ultra-cold atomic gases using radio-frequency spectroscopy and Ramsey interference.

Gravity modeling finds a large magma body in the deep crust below the Gulf of Naples, Italy

We analyze a wide gravity low in the Campania Active Volcanic Area and interpret it by a large and deep source distribution of partially molten, low-density material from about 8 to 30 km depth. Given the complex spatial-temporal distribution of explosive volcanism in the area, we model the gravity data consistently with several volcanological and petrological constraints. We propose two possible models: one accounts for the coexistence, within the lower/intermediate crust, of large amounts of melts and cumulates besides country rocks. It implies a layered distribution of densities and, thus, a variation with depth of percentages of silicate liquids, cumulates and country rocks. The other reflects a fractal density distribution, based on the scaling exponent estimated from the gravity data. According to this model, the gravity low would be related to a distribution of melt pockets within solid rocks. Both density distributions account for the available volcanological and seismic constraints and can be considered as end-members of possible models compatible with gravity data. Such results agree with the general views about the roots of large areas of ignimbritic volcanism worldwide. Given the prolonged history of magmatism in the Campania area since Pliocene times, we interpret the detected low-density body as a developing batholith.

New Insights into Element Distribution Patterns in Geochemistry: A Perspective from Fractal Density

Multifractal features of element concentrations in the Earth’s crust have demonstrated to be closely associated with multiple probability distributions such as normal, lognormal and power law. However, traditional understanding of geochemical distribution satisfying normal, lognormal or power-law models still faces a serious problem in adjusting theoretical statistics with the empirical distribution. Given that the differences among different geochemical distribution populations may have considerable effects on the target estimation, a new perspective from the singularity of fractal density is adopted to investigate mixed geochemical distribution patterns within frequency and space domains. In the framework of fractal geometry, ordinary density such as volume density (e.g., g/cm3 and kg/m3) described in Euclidean space can be considered as a special case of the fractal density (e.g., g/cmα and kg/mα). According to the nature of fractal density, geochemical information obtained from Euclidean geometry may not sufficiently reflect inherent geochemical features, because some information might be hidden within fractal geometry that can be only revealed by means of a set of fractional dimensions. In the present study, stream sediment geochemical data collected from west Tianshan region, Xinjiang (China), were used to explore element distribution patterns in the Earth’s crust based on a fractal density model. Four elements Cu, Zn, K and Na were selected to study the differences between minor and major elements in terms of their geochemical distribution patterns. The results strongly suggest that element distribution patterns can be well revealed and interpreted by means of a fractal density model and related statistical and multifractal parameters.

Singularity analysis of magmatic flare-ups caused by India – Asia collisions

Frequency distribution (histogram) calculated on the basis of igneous and detrital zircon U-Pb ages has been commonly utilized to interpret the age (range) of magmatic events. The temporal properties revealed by these types of data have also been integrated with other types of isotope data (e.g., neodymium, hafnium, and oxygen) to describe the high magmatic addition rate (MAR) and its association with the growth or reworking of the continental crust. Major peaks are picked to associate with pulses of high-volume magmatic flare-ups related to episodic evolution of continental crust. With the development of modern isotope identification technology which results both in reduction of analytical error of isotope data and rapid accumulation of massive high-quality data, the temporal and frequency variances of these types of data ought to be quantitatively analyzed to study magmatic evolution with respect to tectonic background. A new fractal density concept and a local singularity analysis (LSA) method have been recently and successfully applied to analyze the geometric property of the age peaks in global zircon U-Pb age database by power law models. The anomalies of age peaks identified are linked to deeply rooted avalanches associated with short spurts of convection during formation of supercontinents and continent crust growth. In this paper, the method is further used to analyze a small dataset of U-Pb ages from Gangdese arc to characterize the causational relationship of age peak and India-Asia collision. The results show that the age density around peak at 51Ma can be fitted by power law functions. Both the scaling range of the age distribution and the exponents of the power law functions observed from the data suggest that the age peak may reflect magmatic flare-ups which would have been caused by superimposing of subduction, exhumation and slab breakoffs. It has been demonstrated that the LSA can be used as a new way to quantitatively characterize magmatic flare-ups based on U-Pb age data from a fractal density point of view.