Decidability Problems for Self-induced Systems Generated by a Substitution

Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity. It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy’s program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization. Nevertheless, recently Berthe, Steiner, and Thuswaldner (2016) made some further progress on Rauzy’s program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982). Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincare section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin’s idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well. It is the aim of my series of lectures to review the above results.

The properties of a family of tiles with a parameter

On synchronized multi-tape and multi-head automata

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems.

Fractal (or transfractal) features are common in real-life networks and are known to influence the dynamic processes taking place in the network itself. Here, we consider a class of scale-free deterministic networks, called (u, v)-flowers, whose topological properties can be controlled by tuning the parameters u and v; in particular, for u > 1, they are fractals endowed with a fractal dimension df, while for u = 1, they are transfractal endowed with a transfractal dimension d̃f. In this work, we investigate dynamic processes (i.e., random walks) and topological properties (i.e., the Laplacian spectrum) and we show that, under proper conditions, the same scalings (ruled by the related dimensions) emerge for both fractal and transfractal dimensions.

Determination of the initial saturation of a hydrocarbon reservoir requires resistivity log data and saturation exponent. A number of experimental investigations has shown that reservoir wettability affect the exponent value. A remarkable divergence conclusions, however, still appear in the literature. The objective of the present study was therefore to investigate the effect of rock wettability on saturation exponent. Fractal concepts have been applied to the image of thin-sections of a core sample and used to derive the resistivity equation. The advantage of this approach is that it is independent of rock-fluids equilibrium problems that commonly influence laboratory measurements. A general equation of electrical resistivity has been developed in this study. Electrical tortuosity, clay content, and rock wettability are incorporated in the equation. The present work employed twenty thin-sections of limestone and sandstones. It was found that the lowest exponent of 1.61 was obtained for strongly water-wet shaly sandstone and the highest value of 4.99 is for strongly oil-wet. The exponent consistently increases as the wetting condition is shifted from strongly water-wet toward oil-wet. It is close to 2.0 for clean sandstones at strongly water-wet, supporting the empirical formula of Archie. Introduction Estimation of a hydrocarbon reserve is strongly dependent of electric log data and the value of saturation exponent used. This exponent is usually either assumed to be 2.0 regardless of the reservoir wettability or is derived from laboratory measurements of electrical properties of the cores. Saturation exponent has been the subject to many laboratory investigations. The researchers arrived at considerably different quantitative results, particularly for wetting conditions at oil-wet side. Detailed review of this matter has been made by Anderson and Sondenaa et al. A lack of analytical investigation has been conducted to study the effect of wettability on electrical properties of rocks containing fluids. Fractal concepts have been applied to a wide range of research area related to porous media at various scales, from microscale (pore level) to megascale (field scale). A detailed review of fractal concepts applications has been presented by Sahimi and Yortsos. Briefly, a fractal object is an object that has a geometric dimension greater than its topological (Euclidian) dimension. Fractal geometry may describe the irregurality and fragmentation of natural objects and patterns. To characterize a fractal object or set, the fractal dimension D must be determined. A simple method is by "box counting". This is simply done by drawing a mesh of squares of side dimension that covers the set. The fractal dimension is determined from the slope of the relation between count and square side dimension. which is the fractal dimension that can yield any real value. To handle the complexity of the problems, the present study was directed to applying concepts of fractal analysis to determination of the electrical properties and thus saturation exponent. Pore structures and geometries can be quite complex in nature. The complexity stems from the irregularity of pore wall surfaces. The presence of clays may increase the complexity of pore structures. As a matter of fact fluids distribution within the pores is influenced by the wetting condition. This in turn would affect the fluids connectivity. Therefore, electric tortuosity depends not only on pores and fluids connectivity but also on the irregurality of main grain surfaces and volume and distribution of clays within the pores. As pores of a porous medium can be magnified by employing microscope, identification of fine details of the pores can be performed through the use of thin sections made. Ultimately, mathematical analysis coupled with fractal analysis is thus possible to study electrical properties of porous media. P. 143

The plasma of the auroral region, where energetic particles precipitate from the magnetosphere into the ionosphere, is highly inhomogeneous and nonstationary. In this case, traditional methods of classical plasma physics turn out to be inapplicable. In order to correctly describe the dynamic regimes, transition processes, fluctuations, and self-similar scalings in this region, nonlinear dynamics methods based of the concepts of fractal geometry and percolation theory can be used. In this work, the fractal geometry and percolation theory are used to describe the spatial structure of the ionospheric conductivity. The topological properties, fractal dimensions, and connective indices characterizing the structure of the Pedersen and Hall conductivities on the nightside auroral zone are investigated theoretically. The restrictions imposed on the fractal estimates by the condition of ionospheric current percolation are analyzed. It is shown that the fluctuation scalings of the electric fields and auroral glow observed in the auroral zone fit well the restrictions imposed by the critical condition on the percolation of the Pedersen current. Thus, it is demonstrated that the fractal approach is a promising and convenient method for studying the properties of the ionosphere.

We explore the fundamental principles underlying the architecture of the human brain's structural connectome through the lens of spectral analysis of Laplacian and adjacency matrices. Building on the idea that the brain balances efficient information processing with minimizing wiring costs, our goal is to understand how the metric properties of the connectome relate to the presence of an inherent scale. We demonstrate that a simple generative model combining nonlinear preferential attachment with an exponential penalty for spatial distance between nodes can effectively reproduce several key features of the human connectome. These include spectral density, edge length distribution, eigenmode localization, local clustering, and topological properties. Additionally, we examine the finer spectral characteristics of human structural connectomes by evaluating the inverse participation ratios (IPR q ) across various parts of the spectrum. Our analysis shows that the level statistics in the soft cluster region of the Laplacian spectrum (where eigenvalues are small) deviate from a purely Poisson distribution due to interactions between clusters. Furthermore, we identify localized modes with large IPR values in the continuous spectrum. Multiple fractal eigenmodes are found across different parts of the spectrum, and we evaluate their fractal dimensions. We also find a power-law behavior in the return probability-a hallmark of critical behavior-and conclude by discussing how our findings are related to previous conjectures that the brain operates in an extended critical phase that supports multifractality.

A QCA in 2 + 1 and 3 + 1 dimensions

The author proposes that the following relations hold exactly for the lattice animal model of branched polymers. The resistivity exponent zeta a is exactly equal to da, the fractal dimensionality of the backbone of the anomals. The random walk fractal dimensionality dw is given by, dw=da+da, where da is the fractal dimensionality of the animals. The spectral dimension ds of the backbone of the animals is given by, ds=1 at all dimensions, i.e. the backbone is a highly decorated but essentially chain-like object. These are in contrast with earlier suggestions that one needs two (three) topological properties to obtain zeta a(dw). The author proposes that these relations may hold for all clusters for which loops are irrelevant and have a finite upper critical dimensionality. The implications of these results for diffusion-limited aggregates and percolation clusters are also discussed.

This paper covers a novel micro-level application of image processing in understanding the topological and petrophysical properties of Indian Gondwana shale using X-ray computed microtomography images. The complexity and randomness in the pore system are explained through the concept of fractal dimension (FD). In this paper, a quantitative analysis of 2D and 3D fractal dimensions of pores, grains and interfaces was performed for Indian Gondwana shale, using the box-counting method. A pore network is formed by the connection of many subpore clusters, each with a different volume. Hence, an image segmentation algorithm was applied to label different subclusters, and subsequently an analysis of FD was carried out on such subclusters of pores and grains. We implemented a novel application of Betti numbers (B0, B1 and B2) and Euler characteristics on our sample and calculated the possible flow channels of the sample. The FD of grains was found to be greater than the FD of the pore–grain interfaces, while the FD of pores was found to have the smallest value. Consequently, we also observed how the FD of both pores and grains was majorly controlled by the largest subcluster, and during fluid intrusion we observed a significant decrease in the FD of pores. Finally, the pore network with a larger B0 and larger difference of B1 was proved to be best for the storage of hydrocarbons and for fluid movement along more flow channels.

We study the solvability of the equation a(x) = f(x) on a sphere in a Banach space, where a is a closed surjective linear operator and f is an odd a-compact map. We also estimate the topological dimension of the solution set and give applications of the corresponding theorem to some problems in differential equations and other fields of mathematics.

Similarity search is fundamental to store and retrieve large volumes of complex data required by many real world applications. A useful mechanism for such concept is the query-by-similarity. Based on their topological properties, metric similarity functions can be used to index sets of data which can be queried effectively and efficiently by the so-called metric access methods. However, data produced by various application domains and the varying data types handled often lead to missing data, hence, they do not follow the metric similarity requirements. As a consequence, missing data cause distortions in the index structure and yield bias in the query answer. In this paper, we propose the Hollow-tree, a novel access method aimed at successfully retrieving data with missing attribute values. It employs new strategies for indexing and searching data elements, capable of handling the missing data issues when the cause of missingness is ignorable. The indexing strategy is based on a family of distance functions that allow measuring the distance between elements with missing values, along with a set of policies able to organize the elements in the index without causing distortions to its internal structure. The searching strategy employs fractal dimension property of the data to achieve accurate query answer while considering data with missing values part of the response. Results from experiments performed on a variety of real and synthetic data sets showed that, while other metric access methods deteriorate with small amounts of missing values, the Hollow-tree maintains a remarkable performance with almost 100% of precision and recall for range queries and more than 90% for k-nearest neighbor queries, for up to 40% of missing values.

For objects with topological and fractal dimensions (using the example of a “Coulomb explosion”), the physics of modification of electron-stimulated processes in porous media under irradiation with multiply charged ions is considered. A quasi-one-dimensional model has been constructed, which is a convenient methodological approach that describes characteristic phenomena in various media. The results obtained are assessed within the framework of the “complexity” concept.

Specimens from 1⅜-in.-thick plate of 7075-T651 and 7075-T7351 alloys were examined by electron transmission, light microscopy, and electron fractography to relate microstructure, fracture topology, and mechanical and fracture properties. Material in the T7351 temper had a lower yield strength, smaller and more numerous fracture dimples, and an entirely transgranular fracture path. The T651 temper had a higher yield strength, fewer and larger dimples, and a partly intergranular fracture path. It was concluded that the beneficial effect of the lower yield strength more than offset the disadvantage of more numerous fracture dimple sites, and led to greater fracture toughness in the T7351 temper. The partially intergranular fracture was not significant in relation to fracture.