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We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in ℝ⁴ and its projection in ℝ³ is taken. The extra degree of freedom coming from ℝ⁴ provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both self-similar and non-self-similar FIS simultaneously and show that the hidden variable fractal surface may be self-similar under certain conditions.

Two methods for representing discrete image data on rectangular lattices using fractal surfaces are proposed. They offer the advantage of a more general fractal modeling compared to previous one-dimensional fractal interpolation techniques resulting in higher compression ratios. Theory, implementation and analytical study of the proposed methods are also presented.

We present a method of construction of vector valued bivariate fractal interpolation functions on random grids in ℝ². Examples and applications are also included.

No abstract received.

A new method for constructing recurrent bivariate fractal interpolation surfaces through points sampled on rectangular lattices is proposed. This offers the advantage of a more flexible fractal modeling compared to previous fractal techniques that used affine transformations. The compression ratio for the above mentioned fractal scheme as applied to real images is higher than other fractal methods or JPEG, though not as high as JPEG2000. Theory, implementation and analytical study are also presented.

A new fractal interpolation method called PPA (Pointed Point Algorithm) based on IFS is proposed to interpolate the self-affine signals with the expected interpolation error, solving the problem that the ordinary fractal interpolation can't get the value of any arbitrary point directly, which has not been found in the existing literatures. At the same time, a new method to calculate the vertical scaling factors is proposed based on the genetic algorithm, which works together with the PPA algorithm to get the better interpolation performance. Experiments on the theoretical data and real field seismic data show that the proposed interpolation schemes can not only get the expected point's value, but also get a great accuracy in reconstruction of the seismic profile, leading to a significant improvement over other trace interpolation methods.

In the present paper, the bounds on fractal dimension of Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS) in ℝ³ on a equispaced mesh are found. These bounds determine the conditions on the free parameters for fractal dimension of the constructed CHFIS to become close to 3. The results derived here are tested on a tsunami wave surface by computing the lower and upper bounds of the fractal dimension of its CHFIS simulation.

Quantum logic (QL) and fuzzy logic (FL) have been gaining attention nowadays due to its potential to be used in quantum computing and information theory. This paper aims to provide a comprehensive overview of QL models of fuzzy representations viz. fuzzy sets (FSs), interval valued fuzzy sets (IVFSs), intuitionistic fuzzy sets (IFSs) and interval valued intuitionistic fuzzy sets (IVIFSs). These QL models can be used to analyze the behavior of quantum logical systems in a fuzzy environment, which is particularly useful for dealing with uncertainty and imprecision in quantum environment. Furthermore, this paper explores the concept of effect algebras (EAs), which are algebraic structures that provide a natural framework for studying FL. Specifically, it is shown that the family of FS, IVFS, IFS and IVIFS can be organized into an EA if the Lukasiewicz operations are considered. This result is significant because EAs can be used to model a wide range of physical and mathematical systems, including quantum systems, and provide a useful tool for analyzing the properties of FL.

IFS fractals — the attractors of Iterated Function Systems — have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design of fractal antennas. The statement and resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of certain applications. This paper intends to take further steps towards this resolution, building on the literature. For the broad class of hyperdense fractals, a verifiable condition guaranteeing intersection with any line passing through the convex hull of a planar IFS fractal is shown, in general ℝ^d for hyperplanes. The condition also implies a constructive algorithm for finding the points of intersection. Under certain conditions, an infinite number of approximate intersections are guaranteed, if there is at least one. Quantification of the intersection is done via an explicit formula for the invariant measure of IFS.

This paper focuses on two very important questions: “what is the future of a hybrid mathematical structure of soft set in science and social science?” and “why should we take care to use hybrid structures of soft set?”. At present, these are the most fundamental questions; which encircle a few prominent areas of mathematics of uncertainties viz. fuzzy set theory, rough set theory, vague set theory, hesitant fuzzy set theory, IVFS theory, IT2FS theory, etc. In this paper, we review connections of soft set theory and hybrid structures in a non-technical manner; so that it may be helpful for a non-mathematician to think carefully to apply hybrid structures in his research areas. Moreover, we must express that we do not have any intention to nullify contributions of fuzzy set theory or rough set theory, etc. to mankind; but our main intention is to show that we must be careful to develop any new hybrid structure with soft set. Here, we have a short discussion on needs of artificial psychology and artificial philosophy to enrich artificial intelligence.

In this paper, we propose a new fractal deformation technique. An “extended unit Iterated Shuffle Transformation (ext-unit-IST)” is a mapping that changes the order of the places of a code on a code space. When it is applied on a geometric space, it constructs a fractal-like repeated structure, named “local resemblance”. In our previously proposed fractal deformation technique, a geometric shape was deformed by applying an ext-unit-IST to displacement vectors (d-vectors) given on the shape. In the new technique proposed in this paper, the ext-unit-IST is applied to the increasing rates of the d-vectors. This allows the d-vectors to change widely without disturbing the shape and improves the deformation quality. Several examples demonstrate the performance of the newly proposed technique.