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In this work, we address the following question: Is it possible for a one‐dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by n$$ n $$, each one only able to bend on the set associated with the n$$ n $$‐th step in the construction of the Cantor set and compute the Γ$$ \Gamma $$‐limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension ln2/ln3$$ \ln 2/\ln 3 $$, which is the dimension of the Cantor set. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fractal nature of the local fractional Landau–Ginzburg–Higgs Equation (LFLGHE) describing nonlinear waves with weak scattering in a fractal medium. The main goal of the paper is to introduce and apply the Local Fractional Elzaki Variational Iteration Method (LFEVIM) for solution of LFLGHE. Convergence analysis of LFEVIM solution for general nonlinear local fractional partial differential equation is also provided. Two examples of the local fractional LFLGHE are considered to demonstrate the applicability of the proposed technique with numerical simulations on Cantor set. [ABSTRACT FROM AUTHOR]

In this work, we report the eigenenergies spectra for GA sequences present on human chromosome 7, obtained from an entirely atomistic Extended Huckel single-strand DNA model. For GA sequences ranging from 10 to 330 nucleotides (10,739 atoms) in steps of 10 nucleotides cumulatively, the density of states presented a beautiful and clear trend in the form of larger conductance in the region of [ - 20 , - 10 ] eV, like a digital signature, while the sequence length is increased. From a fractal analysis point of view, both level spacing and cumulative level spacing distribution of the energy spectra presented one kind of random Cantor set fractal behavior, including a devil’s staircase for their energy spectra. Besides this, a more accurate f (α) analysis has confirmed that these spectra are multifractal. [ABSTRACT FROM AUTHOR]

We prove that on every compact Riemann surface M, there is a Cantor set C ⊂ M such that M \ C admits a proper conformal constant mean curvature one (CMC-1 ) immersion into hyperbolic 3-space H 3 . Moreover, we obtain that every bordered Riemann surface admits an almost proper CMC-1 face into de Sitter 3-space S 1 3 , and we show that on every compact Riemann surface M, there is a Cantor set C ⊂ M such that M \ C admits an almost proper CMC-1 face into S 1 3 . These results follow from different uniform approximation theorems for holomorphic null curves in C 2 × C ∗ that we also establish in this paper. [ABSTRACT FROM AUTHOR]

In this paper, the solution and behaviour of local fractional Fokker–Planck equation (LFFPE) is investigated in fractal media. For this purpose, the local fractional homotopy perturbation Elzaki transform method (LFHPETM) is proposed and utilized to explore the solution of LFFPE. The proposed scheme is a merger of well known local fractional homotopy perturbation technique and recently introduced local fractional Elzaki transform (LFET). The convergence and uniqueness analyses for LFHPETM solution of the general partial differential equation is also presented along with the computational procedure of this new hybrid combination. Three examples of LFFPE are illustrated to depict the applicability of the employed method with graphical simulations on Cantor set. The copulation of LFET with local fractional homotopy perturbation method (LFHPM) efficiently provides the faster solution for LFFPE in a fractal domain as compared to the LFHPM. Furthermore, the achieved solutions are also in a good match with existing solutions. The 3D behavior of solutions of LFFPEs are presented for fractal order ln 2 / ln 3 . Figures illustrate the 3D surface graphics of solutions with respect to input variables. [ABSTRACT FROM AUTHOR]

We study the topology of all possible subsums of the generalized multigeometric series k 1 f (x) + k 2 f (x) + ⋯ + k m f (x) + ⋯ + k 1 f (x n) + ⋯ + k m f (x n) + ⋯ , where k 1 , k 2 , ⋯ , k m are fixed positive real numbers and f runs along a certain class of non-negative functions on the unit interval. We detect particular regions of this interval for which this achievement set is, respectively, a compact interval, a Cantor set and a Cantorval. [ABSTRACT FROM AUTHOR]

Given two coprime integers p ≥ 2 and q ≥ 3 , let D p ⊂ [ 0 , 1) consist of all rational numbers which have a finite p -ary expansion, and let K (q , A) = ∑ i = 1 ∞ d i q i : d i ∈ A ∀ i ∈ N , where A ⊂ 0 , 1 , ... , q − 1 with cardinality 1 < # A < q. In 2021 Schleischitz showed that # (D p ∩ K (q , A)) < + ∞. In this paper we show that for any r ∈ Q and for any α ∈ R , # ((r D p + α) ∩ K (q , A)) < + ∞. [ABSTRACT FROM AUTHOR]

We examine the structure of two possible candidates of isometry groups for the spectral triples on AF-algebras introduced by Christensen and Ivan. In particular, we completely determine the isometry group introduced by Park and observe that these groups coincide in the case of the Cantor set. We also show that the construction of spectral triples on crossed products given by Hawkins, Skalski, White, and Zacharias is suitable for the purpose of lifting isometries. [ABSTRACT FROM AUTHOR]

The Ruler function or the Gros sequence is a classical infinite integer sequence that underlies some interesting mathematical problems. In this paper, we provide four new problems containing this type of sequence: (i) demographic discrete dynamical automaton, (ii) the middle interval Cantor set, (iii) construction by duplication of polygons and (iv) the horizontal visibility sequence at the accumulation point of the Feigenbaum cascade. In all of them, the infinite sequence is obtained through a recursive procedure of duplication. The properties of the ruler sequence, in particular, those relating to recursiveness and self-containing, are used to achieve a deeper understanding of these four problems. These new representations of the ruler sequence could inspire new studies in the field of discrete mathematics. [ABSTRACT FROM AUTHOR]

Let C(Δ) denote the Banach space of all continuous real-valued functions on the Cantor set Δ and C0(Δ) = ff 2 C(Δ) : f(1) = 0g. From the 1966 theorem of Cambern, it is well-known that the Banach--Mazur distance d(C(Δ);C0(Δ)) ≥ 2. We prove that, in fact, d(C(Δ);C0(Δ)) = 2. As a consequence, we answer a question left open in the 2012 paper of Candido and Galego. [ABSTRACT FROM AUTHOR]

In this paper, we consider Mahler's question for intrinsic Diophantine approximation on the triadic Cantor set K , i.e., approximating the points in K by rational numbers inside K : W K (ψ) : = x ∈ K : | x - p / q | < ψ (q) , for infinitely many p / q ∈ K. By using the intrinsic denominator q int instead of the regular denominator q of a rational p / q ∈ K in ψ (·) , we present a complete metric theory for this variant of the set W K (ψ) , which yields a divergence theory of Mahler's question. [ABSTRACT FROM AUTHOR]

Let K = 2 N be the Cantor set, let M be the set of all metrics d on K that give its usual (product) topology, and equip M with the topology of uniform convergence, where the metrics are regarded as functions on K 2 . We prove that the set of metrics d ∈ M for which the Lipschitz-free space F (K , d) has the metric approximation property is a residual F σ δ set in M , and that the set of metrics d ∈ M for which F (K , d) fails the approximation property is a dense meager set in M . This answers a question posed by G. Godefroy. [ABSTRACT FROM AUTHOR]

We compare the regularity of the boundary of a convex set with the value of its Finslerian volume entropy. The main result states that the volume entropy of a two-dimensional domain whose associated curvature measure is Ahlfors α -regular is 2 α α + 1 . [ABSTRACT FROM AUTHOR]

For m ≥ 2 , we determine the Dirichlet spectrum in R m with respect to simultaneous approximation and the maximum norm as the entire interval [0, 1]. This complements previous work of several authors, especially Akhunzhanov and Moshchevitin, who considered m = 2 and Euclidean norm. We construct explicit examples of real Liouville vectors realizing any value in the unit interval. In particular, for positive values, they are neither badly approximable nor singular. Thereby we obtain a constructive proof of the main claim in a recent paper by Beresnevich, Guan, Marnat, Ramírez and Velani, who proved existence of such vectors but without being able to provide any concrete value in the Dirichlet spectrum. Our constructive proof is considerably shorter and less involved than previous work on the topic. Moreover, it is flexible enough to show that the according set of vectors with prescribed Dirichlet constant has large packing dimension and rather large Hausdorff dimension as well, thereby contributing to the metrical problem raised in the aforementioned paper by Beresnevich et alia. We further establish a more general result on exact uniform approximation, applicable to a wide class of approximating functions. Moreover, minor twists in the proof yield similar, slightly weaker results when restricting to a certain class of classical fractals or considering other norms on R m . In an Appendix we address the situation of a linear form. [ABSTRACT FROM AUTHOR]

We address a classical open question by H. Brezis and R. Ignat concerning the characterization of constant functions through double integrals that involve difference quotients. Our first result is a counterexample to the question in its full generality. This counterexample requires the construction of a function whose difference quotients avoid a sequence of intervals with endpoints that diverge to infinity. Our second result is a positive answer to the question when restricted either to functions that are bounded and approximately differentiable almost everywhere, or to functions with bounded variation. We also present some related open problems that are motivated by our positive and negative results. [ABSTRACT FROM AUTHOR]

This study proposes a new fractal modified equal width-Burgers equation (MEWBE) with the local fractional derivative (LFD) for the first time. By defining the Mittag-Leffler function (MLF) on the Cantor set (CS), two special functions, namely, the T H υ (μ υ) and C H υ (μ υ) functions, are derived for constructing the auxiliary function to seek the non-differentiable (ND) exact solutions. And 16 groups of the ND exact solutions are successfully established. The solutions on the CS are depicted graphically to interpret the nonlinear dynamic behaviors. Furthermore, the comparative results of the fractal MEWBE and the classical MEWBE are also discussed. The obtained results confirm that the proposed method is effective and powerful, and can provide a promising way to find the ND exact solutions of the local fractional PDEs. [ABSTRACT FROM AUTHOR]

The Ruler function or the Gros sequence is a classical infinite integer sequence that underlies some interesting mathematical problems. In this paper, we provide four new problems containing this type of sequence: (i) demographic discrete dynamical automaton, (ii) the middle interval Cantor set, (iii) construction by duplication of polygons and (iv) the horizontal visibility sequence at the accumulation point of the Feigenbaum cascade. In all of them, the infinite sequence is obtained through a recursive procedure of duplication. The properties of the ruler sequence, in particular, those relating to recursiveness and self-containing, are used to achieve a deeper understanding of these four problems. These new representations of the ruler sequence could inspire new studies in the field of discrete mathematics.

The paper considers the problems that arise when using the theory of fuzzy sets to solve applied problems. Unlike stochastic methods, which are based on statistical data, fuzzy set theory methods make sense to apply when statistical data are not available. In these cases, algorithms should be based on membership functions formed by experts who are specialists in this field of knowledge. Ideally, complete information about membership functions is required, but this is an impractical procedure. More often than not, even the most experienced expert can determine only their carriers or separate sets of the α -cuts for unknown fuzzy parameters of the system. Building complete membership functions of unknown fuzzy parameters on this basis is risky and unreliable. Therefore, the paper proposes an extension of the fuzzy sets theory axiomatics in order to introduce non-traditional (less demanding on the completeness of data on membership functions) extension principles and operations on fuzzy sets. The so-called α -weak operations on fuzzy sets are proposed, which are based on the use of separate sets of the α -cuts. It is also shown that all classical theorems of Cantor sets theory apply in the extended axiomatic theory. New extension principles of generalization have been introduced, which allow solving problems in conditions of significant uncertainty of information.

In order to explore the fractal characteristic in Dempster–Shafer evidence theory, a fractal dimension of mass function is proposed recently, to reveal the invariance of scale of belief entropy. When mass function degenerates to probability, the fractal dimension is equivalent to classical Renyi information dimension only with α = 1 , which can measure the change rate of Shannon entropy with the size of framework. For Renyi dimension, different parameters α represent the relationship between different entropies and framework size. However, this compatibility is not shown in existing fractal dimension. Thus, in this paper, we introduce parameter α to generalize the existing dimension. Due to the diversity of the value of α , we name the new dimension: multifractal dimension of mass function. In addition, inspired by multifractal spectrum of Cantor set, we explore the relation between the belief degree of focal element and the number of focal element with same belief degree for some special assignments. Relevant results are also presented by spectrum. We provide a static discounting coefficient generating method to modify mass function to improve the accuracy of classify result. The experiment is conducted in three datasets, and the result shows the effectiveness of our method. [ABSTRACT FROM AUTHOR]

We consider the space Ψ(C) of equivalence classes of continuous open maps defined on the Cantor set C and endowed with the Vietoris topology. It is shown that Ψ(C) is homeomorphic to the space of irrational numbers. [ABSTRACT FROM AUTHOR]

We apply the charge simulation method (CSM) in order to compute the logarithmic capacity of compact sets consisting of (infinitely) many "small" components. This application allows to use just a single charge point for each component. The resulting method therefore is significantly more efficient than methods based on discretizations of the boundaries (for example, our own method presented in Liesen et al. (Comput. Methods Funct. Theory17, 689–713, 2017)), while maintaining a very high level of accuracy. We study properties of the linear algebraic systems that arise in the CSM, and show how these systems can be solved efficiently using preconditioned iterative methods, where the matrix-vector products are computed using the fast multipole method. We illustrate the use of the method on generalized Cantor sets and the Cantor dust. [ABSTRACT FROM AUTHOR]

Given a positive integer M and a real number x ∈ (0 , 1 ] , we call q ∈ (1 , M + 1 ] a univoque simply normal base of x if there exists a unique simply normal sequence (d i) ∈ { 0 , 1 , ... , M } N such that x = ∑ i = 1 ∞ d i q − i. Similarly, a base q ∈ (1 , M + 1 ] is called a univoque irregular base of x if there exists a unique sequence (d i) ∈ { 0 , 1 , ... , M } N such that x = ∑ i = 1 ∞ d i q − i and the sequence (d i) has no digit frequency. Let U S N (x) and U I r (x) be the sets of univoque simply normal bases and univoque irregular bases of x , respectively. In this paper we show that for any x ∈ (0 , 1 ] both U S N (x) and U I r (x) have full Hausdorff dimension. Furthermore, given finitely many rationals x 1 , x 2 , ... , x ℓ ∈ (0 , 1 ] so that each x i has a finite expansion in base M + 1 , we show that there exists a full Hausdorff dimensional set of q ∈ (1 , M + 1 ] such that each x i has a unique expansion in base q. [ABSTRACT FROM AUTHOR]

We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and Miska (Results Math, 2021. ) and demonstrate that the original construction of a special series with unique subsums remains valid when using a weaker estimate that we prove to be true. Additionally, we present a weaker version—without the uniqueness of subsums—of the Thm. 2.1 from Marchwicki and Miska (2021), but with a very simple proof based on the concept of semi-fast convergent series. [ABSTRACT FROM AUTHOR]

For λ ∈ (0 , 1 / 3 ] let C λ be the middle- (1 − 2 λ) Cantor set in R . Given t ∈ [ − 1 , 1 ] , excluding the trivial case we show that Λ (t) := λ ∈ (0 , 1 / 3 ] : C λ ∩ (C λ + t) ≠ ∅ is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of Λ (t) , which reveals a dimensional variation principle. Furthermore, for any β ∈ [ 0 , 1 ] we show that the level set Λ β (t) : = { λ ∈ Λ (t) : dim H (C λ ∩ (C λ + t)) = dim P (C λ ∩ (C λ + t)) = β log 2 − log λ } has equal Hausdorff and packing dimension (− β log β − (1 − β) log 1 − β 2) / log 3. We also show that the set of λ ∈ Λ (t) for which dim H (C λ ∩ (C λ + t)) ≠ dim P (C λ ∩ (C λ + t)) has full Hausdorff dimension. [ABSTRACT FROM AUTHOR]

The main result is that the celebrated Guthrie–Nymann’s Cantorval has comeager set of uniqueness. On the other hand many other Cantorvals have meager set of uniqueness. [ABSTRACT FROM AUTHOR]

We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire functions. Specifically for signals in L 2 (R d) , we obtain norm estimates for Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, which we also use to derive explicit upper bound asymptotes for the multidimensional Cantor iterates, in particular. Finally, we translate the fractal uncertainty principle to discrete Gaussian Gabor multipliers. [ABSTRACT FROM AUTHOR]

We give a new way to obtain the standard isomorphisms of complex Clifford algebras, known as the tensor product of Pauli matrices, by representing the complex Clifford algebras on the space of complex valued functions defined over a finite subset of the Cantor set. [ABSTRACT FROM AUTHOR]

We consider the convergence theory for dyadic approximation in the middle-third Cantor set, K , for approximation functions of the form ψ τ (n) = n − τ (τ ⩾ 0). In particular, we show that for values of τ beyond a certain threshold we have that almost no point in K is dyadically ψ τ -well approximable with respect to the natural probability measure on K. This refines a previous result in this direction obtained by the first, third, and fourth named authors. [ABSTRACT FROM AUTHOR]

We study direct limits of embedded Cantor sets and embedded Sierpinski curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called ω-Cantor spaces, and, similarly, all limits of Sierpi'nski curves give homeomorphic spaces, called ω-Sierpi'nski curves. We then show that the former occur naturally as Morse boundaries of right-angled Artin groups and fundamental groups of non-geometric graph manifolds, while the latter occur as Morse boundaries of fundamental groups of finite-volume, cusped hyperbolic 3-manifolds. [ABSTRACT FROM AUTHOR]

Let { S j : 1 ≤ j ≤ 3 } be a set of three contractive similarity mappings such that S j (x) = r x + j - 1 2 (1 - r) for all x ∈ R , and 1 ≤ j ≤ 3 , where 0 < r < 1 3 . Let P = ∑ j = 1 3 1 3 P ∘ S j - 1 . Then, P is a unique Borel probability measure on R such that P has support the Cantor set generated by the similarity mappings S j for 1 ≤ j ≤ 3 . Let r 0 = 0.1622776602 , and r 1 = 0.2317626315 (which are ten digit rational approximations of two real numbers). In this paper, for 0 < r ≤ r 0 , we give a general formula to determine the optimal sets of n-means and the nth quantization errors for the triadic uniform Cantor distribution P for all positive integers n ≥ 2 . Previously, Roychowdhury gave an exact formula to determine the optimal sets of n-means and the nth quantization errors for the standard triadic Cantor distribution, i.e., when r = 1 5 . In this paper, we further show that r = r 0 is the greatest lower bound, and r = r 1 is the least upper bound of the range of r-values to which Roychowdhury formula extends. In addition, we show that for 0 < r ≤ r 1 the quantization coefficient does not exist though the quantization dimension exists. [ABSTRACT FROM AUTHOR]

We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes, which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations.

For x > 0 , let Υ (x) = (a , b) : x ∈ E a , b , a > 0 , b > 0 , a + b ≤ 1 , where E a , b is the unique nonempty compact invariant set generated by the inhomogeneous IFS Ψ a , b = f 0 (x) = a x , f 1 (x) = b (x + 1). We show that the set Υ (x) is a Lebesgue null set with full Hausdorff dimension and the intersection of the sets Υ (x 1) , ... , Υ (x ℓ) still has full Hausdorff dimension for any finite number of positive numbers x 1 , ... , x ℓ . [ABSTRACT FROM AUTHOR]

We construct a function that lies in L^p(\mathbb {R}^d) for every p \in (1,\infty ] and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač's maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove that there are no previously unknown relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported in the set. [ABSTRACT FROM AUTHOR]

We study stochastic wave equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we give an improved estimate on the uniform norm of eigenfunctions and approximate the wave propagator using the resolvent density. Afterwards, we establish existence and uniqueness of mild solutions to stochastic wave equations provided some Lipschitz and linear growth conditions. We prove H¨older continuity in space and time and compute the H¨older exponents. Moreover, we are concerned with the phenomenon of weak intermittency. [ABSTRACT FROM AUTHOR]

We exploit the mirror and complementary symmetries of elementary cellular automata (ECAs) to rewrite their rules in terms of logical operators. The operator representation based on these fundamental symmetries enables us to construct a periodic table of ECAs that maps all unique rules in clusters of similar asymptotic behavior. We also expand the elementary cellular automaton (ECA) dynamics by introducing a parameter that scales the pace with which operators iterate the system. While tuning this parameter continuously, further emergent behavior in ECAs is unveiled as several rules undergo multiple phase transitions between periodic, chaotic and complex (class 4) behavior. This extension provides an environment for studying class transitions and complex behavior in ECAs. Moreover, the emergence of class 4 structures can potentially enlarge the capacity of many ECA rules for universal computation. [ABSTRACT FROM AUTHOR]

The various aspects of differential calculus are always on the path to progress and excellence, and these trends have been more highlighted in recent decades. More specifically, tremendous advances have been made in the field of fractional calculus. One of the main branches in this field is the local fractional derivative, which has been used successfully to describe many real‐world phenomena in science, and engineering. The present contribution aims to adopt a newly proposed analytical technique to construct exact solutions to local fractional Schamel's equation defined on Cantor sets. To this end, a set of elementary functions are constructed on Cantor sets is defined. Furthermore, the combination of these modified functions is utilized to constitute a formal representation of the searched exact solution for the equation. Numerical simulations related to some of the obtained solutions are also included. The acquired results confirm that the method used is not only very straightforward but also efficient in terms of application. In order to perform complicated and tedious calculations while solving this problem, the use of a symbolic computational packages in Maple or Mathematica is inevitable. The work emphasizes the power of the employed method in providing various exact solutions to different physical problems involving local fractional derivatives. [ABSTRACT FROM AUTHOR]

This paper derives a new fractional Fokas system with the aid of the local fractional derivative. A novel technology named the extended rational fractal sine–cosine method is presented for the first time ever to develop the abundant traveling wave solutions. Two families (six sets) of the traveling wave solutions on Cantor set are successfully constructed. The dynamic behaviors of the solutions on Cantor sets are provided via numerical simulations. The obtained results in this work strongly prove that the proposed approach is simple but effective, which is expected to shed a new light on the study of the traveling wave theory of the local fractional equations. [ABSTRACT FROM AUTHOR]

In this work, we derive a new fractional (3 + 1) -dimensional Jimbo–Miwa equation via the local fractional derivative. With the help of the traveling wave transform of the non-differentiable type, the local fractional (3 + 1) -dimensional Jimbo–Miwa equation is converted into a nonlinear local fractional ordinary differential equation. Then, a new method named as Mittag-Leffler function-based method is used to construct the exact traveling wave solutions for the first time. Four families (eight sets) of the traveling wave solutions are obtained and the behaviors of the solutions on Cantor sets are presented via the 3D plot. The results show that the proposed method is a powerful tool to study the traveling wave theory of the local fractional equations. [ABSTRACT FROM AUTHOR]

The Julia set of any quadratic rational map is either connected or a Cantor set. In this paper, we extend this dichotomy to any cubic rational map with all critical points escaping to an attracting fixed point. [ABSTRACT FROM AUTHOR]

To each colored graph one can associate its closure in the universal space of isomorphism classes of pointed colored graphs, and this subspace can be regarded as a generalized subshift. Based on this correspondence, we introduce two definitions for chaotic (colored) graphs, one of them analogous to Devaney’s. We show the equivalence of our two novel definitions of chaos, proving their topological genericity in various subsets of the universal space.

The basic idea we use throughout this paper is to generalize the construction of the Cantor set as the attractor of an iterated function system $\big(f_0,\ f_1\big)$, replacing the classical functions acting on $ [0,1]$ via $f_0(x) =\frac{x}{3}$ and $f_1(x)= \frac{2}{3} + \frac{x}{3}$ with the new functions acting via $f_0(x)= \theta x$ and $f_1(x) = 1 - \theta + \theta x$, where $\theta \in \Big[0,\ \frac{1}{2} \Big]$ is an arbitrary number. We add to the schema two positive numbers $p_0, \ p_1$ such that $p_0 + p_1 = 1$, obtaining the iterated function system with probabilities $\big(f_0,\; f_1 ; \; p_0,\; p_1\big)$ which generates the invariant set (attractor) and the invariant measure, according to J. Hutchinson. The structure of the invariant set (genera-lized Cantor set) being known, we focus on the detailed computation of the invariant measure. At the end of the paper, we study the dependence upon the parameter $\theta$ of the invariant set and of the invariant measure. [ABSTRACT FROM AUTHOR]

Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

In this paper we consider a retained digits Cantor set \(T\) based on digit expansions with Gaussian integer base. Let \(F\) be the set all \(x\) such that the intersection of \(T\) with its translate by \(x\) is non-empty and let \(F_{\beta}\) be the subset of \(F\) consisting of all \(x\) such that the dimension of the intersection of \(T\) with its translate by \(x\) is \(\beta\) times the dimension of \(T\). We find conditions on the retained digits sets under which \(F_{\beta}\) is dense in \(F\) for all \(0\leq\beta\leq 1\). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.

Fractal square is a concept evolved from mathematics, which subsequently achieved its importance in the area of computer graphics, pattern recognition, and machine vision to analyze or generate complex objects. The self-similarity and self-affine nature of fractals was introduced by Mandelbrot to create different patterns. Furthermore, the multiple reduction copy machine (MRCM) concept has been utilized for a long time in the design of fractals. Apart from this, some new programs have recently been employed in this area. However, all the existing models were limited in terms of their topological properties and not even concerned with the connected and disconnected set of fractals. Even the fractal square that is used to distinguish different distinct patterns is not being studied deeply. In this paper, we have focused on fractal square to generate cantor set by using its different topological properties in terms of disconnected fractal sets, designing the process of various patterns with the help of fractal square and to make a topological classification among them. [ABSTRACT FROM AUTHOR]

In this article, we derive a new local fractional Gardner equation based on the local fractional derivative. A novel traveling wave transform of the non‐differentiable type is utilized to convert the local fractional Gardner equation into a nonlinear local fractional ordinary differential equation. Then a new method called the Mittag–Leffler function based method is proposed for the first time ever to construct the abundant traveling wave solutions. By this method, three families (six sets) of the traveling wave solutions on Cantor set are obtained. Finally, the performances of the various solutions on Cantor set are presented through the three‐dimensional contours. It shows that the proposed method is straightforward, powerful and can provide more forms of the traveling wave solutions for the local fractional equations, which is expected to be helpful for the study of the traveling wave theory for local fractional equations. [ABSTRACT FROM AUTHOR]