FRACTAL ANALYSIS OF LIPASE–CATALYSED SYNTHESIS OF BUTYL BUTYRATE IN A MICROBIOREACTOR UNDER THE INFLUENCE OF NOISE

Fractals ◽  
2013 ◽  
Vol 21 (01) ◽  
pp. 1350007
Author(s):  
PRATAP R. PATNAIK
Keyword(s):  
Fractal Dimension ◽  
Process Model ◽  
Immobilized Lipase ◽  
Deterministic Chaos ◽  
Fractal Dimensions ◽  
External Noise ◽  
Butyl Butyrate ◽  
Stochastic Chaos ◽  
Neural Filter ◽  
Influence Of Noise

Microbioreactors operated in real environments are often subject to noise from the environment. This is commonly manifested as fluctuations in the flow rates of the feed streams. Previous studies with larger bioreactors have shown that noise can seriously impair the performance. Given this possibility, the effects of noise on the performance of a microbioreactor have been analyzed for the trans-esterification of vinyl butyrate by 1-butanol by immobilized lipase B to produce butyl butyrate. As in previous work for macrobioreactors, the analysis was done with (i) no noise, (ii) unfiltered noise, and (iii) noise filtered by four different methods, and the fractal dimension of the product was used as an index of the performance. All fractal dimensions decreased with increasing dilution rates, and significant stochastic chaos was likely at low dilution rates. Of the four types of filters, the auto-associative neural filter (ANF) was the most effective in reducing chaos and restoring of smooth, nearly noise-free performance. The ANF also does not require a process model, which is a significant advantage for real systems. Simulations also revealed that even in the absence of noise, deterministic chaos is possible at low dilution rates; this underscores the importance of efficient filtering under such conditions when external noise too is present. The results thus establish the importance of noise in microbioreactor behavior and the usefulness of the fractal dimension in characterizing the effects.

Restoration of Coherent Population Movement from Noise-Induced Chaos in the Chemotaxis of E. Coli

2013 ◽  
Vol 3 (2) ◽  
pp. 125-144
Author(s):  
Pratap R. Patnaik
Keyword(s):  
Binding Sites ◽  
Fractal Dimensions ◽  
Environmental Noise ◽  
Model System ◽  
Coherent Motion ◽  
Population Movement ◽  
E Coli ◽  
Stochastic Chaos ◽  
Neural Filter ◽  
The Impact

Bacteria navigating in a chemically guided manner are under the impact of noise from at least three sources – inside the cells, at the binding sites between chemoattractants in the environment and corresponding receptors of the cells, and in the environment itself. For Escherichia coli as model system, compounded effects of these sources of noise were investigated recently by using the fractal dimensions of the trajectories of the cells as an index of the nature of population motility. It was observed that environmental noise can drive synchronized movement of a population toward a chemoattractant into stochastic chaos. Those results have been used here to explore the effectiveness of different kinds of noise filters in restoring coherent motion of the cells. An auto-associative neural filter was the best, followed by the extended Kalman filter. The performance of either filter depended on the relative rates of motion of the bacteria and the chemoattractant, and on whether the responses of the cells to fluctuations in the external chemoattractant was non-adaptive or adaptive. The results establish: (a) the validity and usefulness of fractal indexes to characterize noise-affected chemotaxis, (b) the significance of the effect of environmental noise on chemotactic motility, and (c) the effectiveness of a neural filter in rescuing coherent population movement from noise-induced chaos.

Fractal Analysis of the Attenuation of Noise Inflow to Bioreactors under Monotonic, Oscillating and Chaotic Conditions

2007 ◽  
Vol 5 (1) ◽  
Author(s):  
Pratap R Patnaik
Keyword(s):  
Large Scale ◽  
Continuous Fermentation ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Undesirable State ◽  
Bioreactor Operation ◽  
Neural Filter ◽  
And Control ◽  
Microbial Systems ◽  
Influence Of Noise

The influx of noise through inlet streams is often a problem in the operation of large-scale microbial fermentations. It can distort the otherwise smooth performance and, more seriously, displace the fermentation to an undesirable state. Therefore, removal or reduction of the noise content of measured data is important for retrieving the true process variables for bioreactor operation and control. This is done by noise filters, which are soft devices that process noisy data and generate less noisy values with identifiable features. Three types of filters have been compared here by applying them to a continuous fermentation by Saccharomyces cerevisiae under (a) monotonic, (b) oscillating and (c) chaotic operation. Recognising self-similarity as a characteristic feature under the influence of noise, fractal dimensions of the output concentrations are suggested as effective indexes of both noise-affected and noise-filtered performance. On this basis, a hybrid neural filter (HNF) was the best, an auto-associative neural filter (ANF) was somewhat inferior and an extended Kalman filter (EKF) the poorest. While these results and similar observations for other microbial systems favour the use of both fractal dimensions and the HNF, the EKF and other algorithmic filters have some merits, which are discussed.

FRACTAL DIMENSION, PRIMES, AND THE PERSISTENCE OF MEMORY

2003 ◽  
Vol 06 (02) ◽  
pp. 241-249
Author(s):  
JOSEPH L. PE
Keyword(s):  
Number Theory ◽  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Distinguishing Characteristic ◽  
Local Behavior ◽  
Remarkable Similarity ◽  
Recursive Procedures

Many sequences from number theory, such as the primes, are defined by recursive procedures, often leading to complex local behavior, but also to graphical similarity on different scales — a property that can be analyzed by fractal dimension. This paper computes sample fractal dimensions from the graphs of some number-theoretic functions. It argues for the usefulness of empirical fractal dimension as a distinguishing characteristic of the graph. Also, it notes a remarkable similarity between two apparently unrelated sequences: the persistence of a number, and the memory of a prime. This similarity is quantified using fractal dimension.

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

scholarly journals Dynamic characteristics and fractal representations of crack propagation of rock with different fissures under multiple impact loadings

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bing Sun ◽  
Shun Liu ◽  
Sheng Zeng ◽  
Shanyong Wang ◽  
Shaoping Wang
Keyword(s):  
Fractal Dimension ◽  
Crack Propagation ◽  
Crack Growth ◽  
Impact Test ◽  
Fractal Theory ◽  
Dynamic Change ◽  
Fractal Dimensions ◽  
Peak Stress ◽  
Dynamic Mechanical Properties ◽  
Gravitational Potential Energy

AbstractTo investigate the influence of the fissure morphology on the dynamic mechanical properties of the rock and the crack propagation, a drop hammer impact test device was used to conduct impact failure tests on sandstones with different fissure numbers and fissure dips, simultaneously recorded the crack growth after each impact. The box fractal dimension is used to quantitatively analyze the dynamic change in the sandstone cracks and a fractal model of crack growth over time is established based on fractal theory. The results demonstrate that under impact test conditions of the same mass and different heights, the energy absorbed by sandstone accounts for about 26.7% of the gravitational potential energy. But at the same height and different mass, the energy absorbed by the sandstone accounts for about 68.6% of the total energy. As the fissure dip increases and the number of fissures increases, the dynamic peak stress and dynamic elastic modulus of the fractured sandstone gradually decrease. The fractal dimensions of crack evolution tend to increase with time as a whole and assume as a parabolic. Except for one fissure, 60° and 90° specimens, with the extension of time, the increase rate of fractal dimension is decreasing correspondingly.

LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS I: FRACTAL DIMENSION OF PHONON SPECTRA

1995 ◽  
Vol 09 (12) ◽  
pp. 1429-1451 ◽  
Author(s):  
WŁODZIMIERZ SALEJDA
Keyword(s):  
Fractal Dimension ◽  
Lattice Dynamics ◽  
Nearest Neighbor ◽  
Average Distance ◽  
Local Maximum ◽  
Fractal Dimensions ◽  
Dimension Formula ◽  
Phonon Spectra ◽  
Wide Range ◽  
Silver Mean

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

Fractal dimension based features are useful descriptors of leaf complexity and shape

1999 ◽  
Vol 29 (9) ◽  
pp. 1301-1310 ◽  
Author(s):  
Wojciech Borkowski
Keyword(s):  
Fractal Dimension ◽  
Tree Species ◽  
Fractal Dimensions ◽  
Plant Identification ◽  
Simple Method ◽  
Mass Dimension ◽  
Scale Range ◽  
Computer Identification

An application of fractal dimensions as measures of leaf complexity to morphometric studies and automated plant identification is presented. Detailed algorithms for the calculation of compass dimension and averaged mass dimension together with a simple method of grasping the scale range related variability are given. An analysis of complexity of more than 300 leaves from 10 tree species is reported. Several classical biometric descriptors as well as 16 fractal dimension features were computed on digitized leaf silhouettes. It is demonstrated that properly defined fractal dimension based features may be used to discriminate between species with more than 90% accuracy, especially when used together with other measures. It seems, therefore, that they can be utilized in computer identification systems and for purely taxonomical purposes.

scholarly journals Development of Back-calculation Model for Aggregate Gradation Determination Using Fractal Theory

2018 ◽  
Vol 159 ◽  
pp. 01006
Author(s):  
Bagus Hario Setiadji ◽  
Supriyono ◽  
Djoko Purwanto
Keyword(s):  
Fractal Dimension ◽  
Fractal Theory ◽  
Asphalt Mixture ◽  
Fractal Dimensions ◽  
Calculation Model ◽  
Careful Consideration ◽  
Upper And Lower Bounds ◽  
Aggregate Gradation ◽  
Fractal Characteristic ◽  
Fractal Characteristics

Several studies have shown that fractal theory can be used to analyze the morphology of aggregate materials in designing the gradation. However, the question arises whether a fractal dimension can actually represent a single aggregate gradation. This study, which is a part of a grand research to determine aggregate gradation based on known asphalt mixture specifications, is performed to clarify the aforementioned question. To do so, two steps of methodology were proposed in this study, that is, step 1 is to determine the fractal characteristics using 3 aggregate gradations (i.e. gradations near upper and lower bounds, and middle gradation); and step 2 is to back-calculate aggregate gradation based on fractal characteristics obtained using 2 scenarios, one-and multi-fractal dimension scenarios. The results of this study indicate that the multi-fractal dimension scenario provides a better prediction of aggregate gradation due to the ability of this scenario to better represent the shape of the original aggregate gradation. However, careful consideration must be observed when using more than two fractal dimensions in predicting aggregate gradation as it will increase the difficulty in developing the fractal characteristic equations.

scholarly journals Spatial Measures of Urban Systems: from Entropy to Fractal Dimension

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 991 ◽  
Author(s):  
Yanguang Chen ◽  
Linshan Huang
Keyword(s):  
Fractal Dimension ◽  
Spatial Analysis ◽  
Fractal Dimensions ◽  
Scale Dependence ◽  
Urban Systems ◽  
Scale Free ◽  
Spatial Entropy ◽  
Spatial Systems ◽  
Urban Patterns ◽  
Generalized Entropies

One type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimensions can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connection between entropy and fractal dimensions, they have different application scopes and directions in urban studies. This paper focuses on exploring how to convert entropy measurements into fractal dimensions for the spatial analysis of scale-free urban phenomena using the ideas from scaling. Urban systems proved to be random prefractal and multifractal systems. The spatial entropy of fractal cities bears two properties. One is the scale dependence: the entropy values of urban systems always depend on the linear scales of spatial measurement. The other is entropy conservation: different fractal parts bear the same entropy value. Thus, entropy cannot reflect the simple rules of urban processes and the spatial heterogeneity of urban patterns. If we convert the generalized entropies into multifractal spectrums, the problems of scale dependence and entropy homogeneity can be solved to a degree for urban spatial analysis. Especially, the geographical analyses of urban evolution can be simplified. This study may be helpful for students in describing and explaining the spatial complexity of urban evolution.

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