Stochastic Adsorption of Diluted Solute Molecules at Interfaces

<div> <p>Here an analytical solution of Fick’s 2^nd law is used to predict the diffusion and the stochastic adsorption of single diluted solute molecules on flat and patterned surfaces. The equations are then compared to the results of several numerical Monte Carlo simulations using a random walk model. The 1D diffusion simulations clarify that the dependence of the solute-surface collision rate on the observation-time (measurement time resolution) is because of the multiple collisions of the same molecules over different time regions. It also surprisingly suggests that due to the self-mimetic fractal function of diffusion, the equation should be corrected by a factor of two. The absorption rate of solute on an adsorptive surface is found to follow a power-law decay function due to an evolving concentration gradient near the surface along with the depletion of the bulk solute molecules on the surface, for example, in a self-assembled monolayer adsorption kinetics. Thus, the analytical equations developed to calculate the collision at a fixed measuring frequency can be extended to map the whole curve over time. In the last section of this work, 3D diffusion simulations suggest that the analytical solution is valid to predict the adsorption rate of the bulk solute to a small group of adsorptive target molecules/area on a bouncing surface, which is a critical process in analyzing the kinetics of many bio-sensing platforms.</p> </div>

Multivariate Fractal Functions in Some Complete Function Spaces and Fractional Integral of Continuous Fractal Functions

There has been a considerable evolution of the theory of fractal interpolation function (FIF) over the last three decades. Recently, we introduced a multivariate analogue of a special class of FIFs, which is referred to as α-fractal functions, from the viewpoint of approximation theory. In the current note, we continue our study on multivariate α-fractal functions, but in the context of a few complete function spaces. For a class of fractal functions defined on a hyperrectangle Ω in the Euclidean space Rn, we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces Lp(Ω), Sobolev spaces Wm,p(Ω), and Hölder spaces Cm,σ(Ω), which are Banach spaces. As a simple consequence, for some special choices of the parameters, we provide bounds for the Hausdorff dimension of the graph of the corresponding multivariate α-fractal function. We shall also hint at an associated notion of fractal operator that maps each multivariate function in one of these function spaces to its fractal counterpart. The latter part of this note establishes that the Riemann–Liouville fractional integral of a continuous multivariate α-fractal function is a fractal function of similar kind.

Graph-directed random fractal interpolation function

"Barnsley introduced in [1] the notion of fractal interpolation function (FIF). He said that a fractal function is a (FIF) if it possess some interpolation properties. It has the advantage that it can be also combined with the classical methods or real data interpolation. Hutchinson and Ruschendorf [7] gave the stochastic version of fractal interpolation function. In order to obtain fractal interpolation functions with more exibility, Wang and Yu [9] used instead of a constant scaling parameter a variable vertical scaling factor. Also the notion of fractal interpolation can be generalized to the graph-directed case introduced by Deniz and  Ozdemir in [5]. In this paper we study the case of a stochastic fractal interpolation function with graph-directed fractal function."

Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set

We present an analytic solvability of a class of Langevin differential equations (LDEs) in the asset of geometric function theory. The analytic solutions of the LDEs are presented by utilizing a special kind of fractal function in a complex domain, linked with the subordination theory. The fractal functions are suggested for the multi-parametric coefficients type motorboat fractal set. We obtain different formulas of fractal analytic solutions of LDEs. Moreover, we determine the maximum value of the fractal coefficients to obtain the optimal solution. Through the subordination inequality, we determined the upper boundary determination of a class of fractal functions holding multibrot function ϑ(z)=1+3κz+z3.

Fractal-multifractal ensembles of downscaled precipitation and temperature sets as implied by climate models

&lt;p&gt;Describing the specific details and textures implicit in real-world hydro-climatic data sets is paramount for the proper description and simulation of variables such as precipitation, streamflow, and temperature time series. To this aim, a couple of decades ago, a deterministic geometric approach, the so-called&amp;#160;fractal-multifractal&amp;#160;(FM)&amp;#160;method,&lt;sup&gt;1,2&lt;/sup&gt; was introduced. Such is a holistic approach capable of faithfully encoding (describing)&lt;sup&gt;3&lt;/sup&gt;, simulating&lt;sup&gt;4&lt;/sup&gt;, and downscaling&lt;sup&gt;5&lt;/sup&gt; hydrologic records in time, as the outcome of a fractal function illuminated by a multifractal measure. This study employs the FM method to generate ensembles of daily precipitation and temperature sets obtained from global circulation models (GCMs). Specifically, this study uses data obtained via ten GCM models, two sets of daily records, as implied from the past, over a year, and three sets projected for the future, as downscaled via localized constructed analogs (LOCA) for a couple of sites in California. The study demonstrates that faithful representations of all sets may be achieved via the FM approach, using encodings relying on 10 and 8 geometric (FM) parameters for rainfall and temperature, respectively. They result in close approximations of the data's histogram, entropy, and autocorrelation functions. By presenting a sensitivity study of FM parameters' for historical and projected data, this work concludes that the FM representations are useful for tracking and foreseeing the records' complexity&lt;sup&gt;6&lt;/sup&gt; in the past and the future and other applications in hydrology such as bias correction.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;

On a subclass of analytic functions of fractal power with negative coefficients

The purpose of this article is to introduce a new general family of normalized analytic fractal function in the open unit disk. We employ this class to define a fractional differential operator of two fractals. This operator, under some conditions involves the well known Salagean differential operator. Our method is based on the Hadamard product and its generalization of functions with negative coeffcients.

Preserving Information Security Using Fractal-Based Cryptosystem

The chapter intends to propose a hybrid cryptosystem based on a chaotic map and a fractal function. The sequential order of process execution provides a computationally less expensive and simple approach that still designed a secure cryptosystem. A one-dimensional Ricker map and its modified form are employed to initially shuffle the image pixels twice, and also a pseudo-random sequence is generated using both maps. The algorithm implemented a sequence of pixel confusion-diffusion steps using the image rotation and a transcendental anti-Mandelbrot fractal function (TAMFF) and its Mann-iterated fractal function (Sup-TAMFF). Finally, the pixel value of an image obtained in the last step and the recent two pixels of the encrypted image is XORed with the corresponding pseudo-random matrix value to get the cipher image. Subsequently, various performance tests are conducted to verify the suitability of the given method to be used in real-world information transmission.