• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.229-241
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• 1999

The fractal space is often associated with natural phenomena with many length scales and the functions defined on this space are usually not differentiable. First we define a $\sigma$-multifractal from $\sigma$-iterated function systems with probability. We introduce the measure derivative through the invariant measure of the $\sigma$-multifractal. We show that the non-differentiable function on the $\sigma$-multifractal can be differentiable with respect to this measure derivative. We apply this result to some examples of ordinary differential equations and diffusion processes on $\sigma$-multifractal spaces.

• Transactions of the Korean Society of Mechanical Engineers
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• v.19 no.9
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• pp.2342-2352
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• 1995

Deposition of flame-synthesized silica particles onto a target is utilized in optical fiber preform fabrication processes. The particles are convected and deposited onto the target. Falkner-Skan wedge flow was chosen as the particle laden flow. Typically the particles are polydisperse in size and follow a lognormal size distribution. Brownian diffusion, thermophoresis, and coagulation of the particles were considered and effects of these phenomena on particle deposition were studied. A moment model was developed in order to predict the particle number density and the particle size distribution simultaneously. Particle deposition with various wedge configurations was examined for conditions selected for a typical VAD process. When coagulation was considered, mean particle size and its standard deviation increased and particle number density decreased, compared to the case without coagulation. These results proved the fact that coagulation effect expands particle size distribution. The results were discussed with characteristics of thermal and diffusion boundary layers. As the boundary layers grow in thickness, overall temperature and concentration gradients decrease, resulting in decrease of deposition rate and increase of particle residence time in the flow and thus coagulation effect.

• Particle and aerosol research
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• v.9 no.3
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• pp.163-171
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• 2013

Soot particles emitted from combustion processes are often coated by non-absorbing organic materials, which enhance the global warming effect of soot particles. It is of importance to study the condensation characteristics of soot particles experimentally and theoretically to reduce the uncertainty of the climate impact of soot particles. In this study, the condensational growth of soot particles in a tubular coater was modeled by a one-dimensional (1D) plug flow model and a two-dimensional (2D) laminar flow model. The effects of 2D heat and mass transports on the predicted particle growth were investigated. The temperature and coating material vapor concentration distributions in radial direction, which the 1D model could not accounted for, affected substantially the particle growth in the coater. Under the simulated conditions, the differences between the temperatures and vapor concentrations near the wall and at the tube center were large. The neglect of these variations by the 1D model resulted in a large error in modeling the mass transfer and aerosol dynamics occurring in the coater. The 1D model predicted the average temperature and vapor concentration quite accurately but overestimated the average diameter of the growing particles considerably. At the outermost grid, at which condensation begins earliest due to the lowest temperature and saturation vapor concentration, condensing vapor was exhausted rapidly because of the competition between condensations on the wall and on the particle surface, decreasing the growth rate. At the center of the tube, on the other hand, the growth rate was low due to high temperature and saturation vapor concentration. The effects of Brownian diffusion and thermophoresis were not high enough to transport the coating material vapor quickly from the tube center to the wall. The 1D model based on perfect radial mixing could not take into account this phenomenon, resulting in a much higher growth rate than what the 2D model predicted. The result of this study indicates that contrary to a previous report for a thermodenuder, 2D heat and mass transports must be taken into account to model accurately the condensational particle growth in a coater.

• The Korean Journal of Financial Management
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• v.20 no.2
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• pp.95-126
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• 2003

This paper introduces multifractal processes and presents the empirical investigation of the multifractal asset pricing. The multifractal stock price process contains long-tails which focus on Levy-Stable distributions. The process also contains long-dependence, which is the characteristic feature of fractional Brownian motion. Multifractality introduces a new source of heterogeneity through time-varying local reqularity in the price path. This paper investigates multifractality in stock prices. After finding evidence of multifractal scaling, the multifractal spectrum is estimated via the Legendre transform. The distinguishing feature of the multifractal process is multiscaling of the return distribution's moments under time-resealing. More intensive study is required of estimation techniques and inference procedures.