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In this paper, we studied the stability of the Cauchy problem for a class of first-order linear quasi-periodically forced PDEs on the m-dimensional torus: {∂tu + (ξ + f(x, ωt, ξ)) · ∂xu = 0, u(x, 0) = u0(x), where ξ ∈ ℝm, x ∈ Tm, ω ∈ ℝd, in the case of multidimensional Liouvillean forced frequency. We proved that for each compact set O ∈ ℝm there exists a Cantor subset Oγ of O with positive Lebesgue measure such that if ξ ∈ Oγ, then for a perturbation f being small in some analytic Sobolev norm, there exists a bounded and invertible quasi-periodic family of linear operator Ψ(ωt), such that the above PDEs are reduced by the transformation v := Ψ(ωt)−1 [u] into the following PDE: ∂tv + (ξ + m∞(ωt)) · ∂xv = 0, provided that the forced frequency ω ∈ ℝd possesses finite uniform Diophantine exponent, which allows Liouvillean frequency. The reducibility can immediately cause the stability of the above Cauchy problem, that is, the analytic Sobolev norms of the Cauchy problem are controlled uniformly in time. The proof is based on a finite dimensional Kolmogorov-Arnold-Moser (KAM) theory for quasi-periodically forced linear vector fields with multidimensional Liouvillean forced frequency. As we know, the results on Liouvillean frequency existing in the literature deal with two-dimensional frequency and exploit the theory of continued fractions to control the small divisor problem. The results in this paper partially extend the analysis to higher-dimensional frequency and impose a weak nonresonance condition, i.e., the forced frequency ω possesses finite uniform Diophantine exponent. Our result can be regarded as a generalization of analytic cases in the work [R. Feola, F. Giuliani, R. Montalto and M, Procesi, Reducibility of first order linear operators on tori via Moser’s theorem, J. Funct. Anal., 2019] from Diophantine frequency to Liouvillean frequency. [ABSTRACT FROM AUTHOR]

This paper introduces hybrid models designed to analyze daily and weekly bitcoin return spanning the periods from 18 July 2010 to 28 December 2023 for daily data, and from 18 July 2010 to 24 December 2023 for weekly data. Firstly, the fractal and chaotic structure of the selected variables was explored. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z2 −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. The R/S and Mandelbrot–Wallis tests confirmed long-term dependence and fractionality. The largest Lyapunov test, the Rosenstein, Collins and DeLuca, and Kantz methods of Lyapunov exponents, and the HCT and Shannon entropy tests tracked by the Kolmogorov–Sinai (KS) complexity test determined the evidence of chaos, entropy, and complexity. The BDS test of independence test approved nonlinearity, and the TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, the LR test for threshold nonlinearity, and White's test and Engle test confirmed nonlinearity and heteroskedasticity, in addition to fractionality and chaos. In the second stage, the standard ARFIMA method was applied, and its results were compared to the LieNLS and LieOLS methods. The results showed that, under conditions of chaos, entropy, and complexity, the ARFIMA method did not yield successful results. Both baseline models, LieNLS and LieOLS, are enhanced by integrating them with deep learning methods. The models, LieLSTMOLS and LieLSTMNLS, leverage manifold-based approaches, opting for matrix representations over traditional differential operator representations of Lie algebras were employed. The parameters and coefficients obtained from LieNLS and LieOLS, and the LieLSTMOLS and LieLSTMNLS methods were compared. And the forecasting capabilities of these hybrid models, particularly LieLSTMOLS and LieLSTMNLS, were compared with those of the main models. The in-sample and out-of-sample analyses demonstrated that the LieLSTMOLS and LieLSTMNLS methods outperform the others in terms of MAE and RMSE, thereby offering a more reliable means of assessing the selected data. Our study underscores the importance of employing the LieLSTM method for analyzing the dynamics of bitcoin. Our findings have significant implications for investors, traders, and policymakers. [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]

Fractional calculus has many advantages. Under consideration of this paper is a (2+1)-dimensional non-linear local fractional heat conduction equation with arbitrary degree non-linearity. Backlund transformation of a reduced form of the local fractional heat conduction equation is constructed by Painleve analysis. Based on the Backlund transformation, some exact non-differentiable solutions of the local fractional heat conduction equation are obtained. To gain more insights of the obtained solutions, two solutions are constrained to a Cantor set and then two spatio-temporal fractal structures with profiles of these two solutions are shown. This paper further reveals by local fractional heat conduction equation that fractional calculus plays important role in dealing with non-differentiable problems. [ABSTRACT FROM AUTHOR]

Based on the local fractional derivative, a new local fractional Klein-Fock-Gordon equation is derived in this paper for the first time. A simple method namely Yang's special function method is used to seek for the non-differentiable exact solutions. The whole calculation process strongly shows that the proposed method is simple and effective, and can be applied to investigate the non-differentiable exact solutions of the other local fractional PDE. [ABSTRACT FROM AUTHOR]

In this paper, a new spatiotemporal chaotic Parameter Uncertainty Mixed Coupled Map Lattice (PUMCML) is proposed. The Cantor diagonal matrix is generated from the Cantor set, and the ordered rotation scrambling strategy for this matrix is used to generate the scrambled image. Cantor set is a fractal system, so the Cantor set has a good effect on chaotic image encryption. The dynamic behavior of the PUMCML system is analyzed. The system has good chaotic property, so it is very suitable for chaotic image encryption. Using the PUMCML system, a diffusion strategy based on the mixture of Arnold and Logistic is proposed. Compared with other algorithms, the encryption effect of the proposed method is better and more secure. [ABSTRACT FROM AUTHOR]

In this paper, we present a constructive description of the function space of all real-valued functions on R (F (R , R)) by presenting a partition of it into 28 distinct blocks and a closed-form formula for the representative function of each of them. Each block contains elements that share common features in terms of the cardinality of their sets of continuity and differentiability. Alongside this classification, we introduce the concept of the Connection, which reveals a special relationship structure between the well-known representatives of four of the blocks: the Cantor function, the Dirichlet function, the Thomae function, and the Weierstrass function. Despite the significance of this field, several perspectives remain unexplored. [ABSTRACT FROM AUTHOR]

In this paper, within the scope of the local fractional derivative theory, bidirectional propagation system local fractional equations are researched. Compared with the unidirectional propagation of nonlinear waves in a pipeline, the bidirectional propagation system equations studied in this paper can better describe the propagation of nonlinear waves in a channel. This study is groundbreaking and offers more possibilities for the bidirectional propagation of nonlinear waves in the simulation pipeline. The exact traveling wave solutions of the non-differentiable type defined on the Cantor sets are obtained. The characteristics of the particular solutions of a fixed fractal dimension are discussed. It is proven that the local fractional nonlinear bidirectional wave equations can describe the interaction of fractal waves. It is also shown that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics. [ABSTRACT FROM AUTHOR]

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 Fβbe the subset of F consisting of all x such that the dimension of the intersection of T with its translate by x is β times the dimension of T. We find conditions on the retained digits sets under which Fβ is dense in F for all 0 ≤ β ≤ 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. [ABSTRACT FROM AUTHOR]

In this paper, the coupled local fractional sine-Gordon equations are studied in the range of local fractional derivative theory. The study of exact solutions of nonlinear coupled systems is of great significance for understanding complex physical phenomena in reality. The main method used in this paper is the local fractional homotopy perturbation method, which is used to analyze the exact traveling wave solutions of generalized nonlinear systems defined on the Cantor set in the fractal domain. The fractal wave with fractal dimension ε = l n 2 / l n 3 is numerically simulated. Through numerical simulation, we find that the obtained solutions are of great significance to explain some practical physical problems. [ABSTRACT FROM AUTHOR]

In this paper, we construct new discrete chaotic dynamical systems on a 2D Cantor set by using the shift map, a 0-preadded map and a 2-preadded map. We also obtain a collection of chaotic dynamical systems using the elements of the symmetry group of the square. [ABSTRACT FROM AUTHOR]

Let C 1 and C 2 be two Cantor sets in R . Suppose that the size of the largest gap of C 1 is not greater than the diameter of C 2 , and vice versa. Newhouse (Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, volume 35 of Cambridge Studies in Advanced Mathematics. Fractal dimensions and infinitely many attractors. Cambridge University Press, Cambridge, 1993) proved if τ (C 1) · τ (C 2) ≥ 1 , then the arithmetic sum C 1 + C 2 is an interval, where τ (C i) , 1 ≤ i ≤ 2 denotes the thickness of C i . In this paper, we generalize this thickness theorem as follows. Let K i ⊂ R , i = 1 , ... , d , be some Cantor sets in R . Suppose f (x 1 , ... , x d - 1 , z) ∈ C 1 is defined on R d . Denote the continuous image of f by f K 1 , ... , K d = f x 1 , ... , x d - 1 , z : x i ∈ K i , z ∈ K d , 1 ≤ i ≤ d - 1. In this paper, we give a sufficient condition under which f (K 1 , ... , K d) is a closed interval. Our idea can reprove the Newhouse thickness theorem. Various applications are given. [ABSTRACT FROM AUTHOR]

In this paper, the periodic solution for the local fractional Boussinesq equation can be obtained in the sense of the local fractional derivative. It is given by applying direct integration with symmetry condition. Meanwhile, the periodic solution of the non-differentiable type with generalized functions defined on Cantor sets is analyzed. As a result, we have a new point to look the local fractional Boussinesq equation through the local fractional derivative theory. [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]

P-sets (P stands for Packet) is a set model with dynamic characteristics, which is obtained by introducing dynamic characteristics into Cantor set and improving Cantor set. According to the fact that the characteristics of class I big data are completely consistent with the basic characteristics of P-sets, this paper gives research on theory and application on class I big data from the view of mathematics. Here we introduce Class I big data which need some new definitions of data block, microdata and data link. Based on these concepts, decomposition theorem of data block and microdata relation theorem are given, and then attribute reasoning theorem and microdata intelligent discovery and the intelligent secure acquisition algorithm of microdata are also proposed. By using these theoretical results, the applications of secure acquisition of microdata are presented. In summary, P-sets mathematical model provides a new theory and method for studying class I big data. [ABSTRACT FROM AUTHOR]

The lines of curvature of a surface embedded in R³ comprise its principal foliations. Principal foliations of surfaces embedded in R³ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincaré-Bendixson Theorem precludes ows on the two sphere S² with re-current trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus T2. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set C of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has breaks; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeo-morphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces. [ABSTRACT FROM AUTHOR]

P-sets (P stands for Packet), a set pair with dynamic and law characteristics, are made up of an internal and an outer P-set, which is obtained by introducing dynamic characteristics into the Cantor set and improving the Cantor set. The concepts of α F -sub-information, α F ¯ -sub-information, and (α F , α F ¯ ) -sub-information are presented in this paper based on P-sets, and it is then suggested that the relationship between the generation of sub-information and its attribute, the process of attribute reasoning, reasoning structure, and sub-information intelligent separation-acquisition be explored. These findings were used to design a sub-information intelligent separation-identification algorithm. By using these results, the application of intelligent separation and the identification of case information are given. [ABSTRACT FROM AUTHOR]

Ablowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies. [ABSTRACT FROM AUTHOR]

In this paper, we use the Laplace transform series expansion method to find the analytical solution for the local fractional heat-transfer equation defined on Cantor sets via local fractional calculus. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of U-metric space of n-tuples which generalizes several known metric type spaces. Also we study the topological properties of such newly constructed spaces and prove Cantor's intersection like theorem therein. Banach contraction principle theorem has been proved in this space and finally we apply the theorem to obtain the stability of a fixed point equation. [ABSTRACT FROM AUTHOR]

In this paper, we present the fractal complex transform via a local fractional derivative. The traveling wave solutions for the fractal Korteweg-de Vries equations within local fractional derivative are obtained based on the special functions defined on Cantor sets. The technology is a powerful tool for solving the local fractional non-linear partial differential equations. [ABSTRACT FROM AUTHOR]

This paper addresses a new application of the local fractional variational iteration algorithm III to solve the local fractional diffusion equation defined on Cantor sets associated with non-differentiable heat transfer. [ABSTRACT FROM AUTHOR]

In this paper, a new application of the fractal complex transform via a local fractional derivative is presented. The solution for the fractal relaxation and time-fractal diffusion equations are obtained based on the sup-exponential functions defined on Cantor sets. [ABSTRACT FROM AUTHOR]

Given a non-negative, decreasing sequence a with sum 1, we consider all the closed subsets of [0, 1] such that the lengths of their complementary open intervals are given by the terms of a. These are the so-called complementary sets, or rearrangements of the Cantor set, constructed from a. In this paper we determine the almost sure value of the Φ -dimension of these sets given a natural model of randomness. The Φ -dimensions are intermediate Assouad-like dimensions which include the Assouad and quasi-Assouad dimensions as special cases. The answers depend on the size of Φ , with one size behaving almost surely like the Assouad dimensions of the associated Cantor set and the other, like the quasi-Assouad dimensions. These results are new even for the Assouad dimensions. [ABSTRACT FROM AUTHOR]

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials Pa(z) = zd-1(z - da/(d-1)), the family fλ = λtanpzq. These functions have a super-attractive fixed point, and, depending on p, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family fλ. Again there are similarities to and differences from the parameter plane of the family Pa and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points. [ABSTRACT FROM AUTHOR]

In this paper, the 2-D and 3-D diffusions defined on Cantor sets with local fractional differential operator were discussed in different co-ordinate systems. The 2-D diffusion in Cantorian co-ordinate system can be converted into the symmetric diffusion defined on Cantor sets. The 3-D diffusions in Cantorian co-ordinate system can be observed in the Cantor-type cylindrical and spherical co-ordinate methods. [ABSTRACT FROM AUTHOR]

In this paper we use the notion of pre-open sets and pre-closed sets in topological space to prove the Cantor's intersection theorem for pre-closed sets. [ABSTRACT FROM AUTHOR]

In this paper we establish a topological property of geometric objects (lines, surfaces and solids) called Euler-Poincaré characteristic. Since the paper is intended for a large profile of mathematics teachers, our approach is entirely intuitive and majority of readers can omit two addenda whose understanding requires a solid knowledge of topology. E-P characteristic is an integer which we calculate here decomposing lines into fibers being finite sets of points, surfaces into fibers being lines and solids into fibers being surfaces. When an object is subjected to a "plastic" deformation its shape and size changes as well as the alternating sums resulting from the method of calculation, but the value of these sums stays unchanged. This fact serves to convince the reader that E-P characteristic is a stable topological property. The same fact gives this approach an advantage over usually practiced ones which require a triangulation of geometric objects. [ABSTRACT FROM AUTHOR]

We study the topological μ-calculus, based on both Cantor derivative and closure modalities, proving completeness, decidability, and finite model property over general topological spaces, as well as over T0 and TD spaces. We also investigate the relational μ-calculus, providing general completeness results for all natural fragments of the μ-calculus over many different classes of relational frames. Unlike most other such proofs for μ-calculi, ours is model theoretic, making an innovative use of a known method from modal logic (the 'final' submodel of the canonical model), which has the twin advantages of great generality and essential simplicity. [ABSTRACT FROM AUTHOR]

Local fractional calculus has gained wide attention in the field of circuit design. In this paper, we propose the zero-input response(ZIR) of fractal RC circuit modeled by local fractional derivative(LFD) for the first time. With help of the law of switch and the Kirchhoff Voltage Laws, the transient local fractional ordinary differential equation is established, and the corresponding exact solution behavior defined on Cantor sets is presented. What we found especially interesting was that the fractal RC becomes the ordinary one in the particular case κ = 1. The results obtained in this paper reveal that the local fractional calculus is a powerful tool to analyze the fractal circuit systems. [ABSTRACT FROM AUTHOR]

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group G ∗ . These results can further be generalized to arbitrary locally compact abelian groups. [ABSTRACT FROM AUTHOR]

This paper addresses the systems of the incompressible Navier-Stokes equations on Cantor sets without the external force involving the fractal heat-conduction problem vial local fractional derivative. The spherical Cantor type co-ordinate method is used to transfer the incompressible Navier-Stokes equation from the Cantorian co-ordinate system into the spherical Cantor type co-ordinate system. [ABSTRACT FROM AUTHOR]

The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM). We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs). The efficiency of the scheme is examined by two illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we address the free (uncontrolled) dynamics of a snakeboard consisting of two wheel pairs fastened to a platform. The snakeboard is one of the well-known sports vehicles on which the sportsman executes necessary body movements. From the theoretical point of view, this system is a direct generalization of the classical nonholonomic system of the Chaplygin sleigh. We carry out a topological and qualitative analysis of trajectories of this dynamical system. An important feature of the problem is that the common level set of first integrals is a compact two-dimensional surface of genus 5. We specify conditions under which the reaction forces infinitely increase during motion and the so-called phenomenon of nonholonomic jamming is observed. In this case, the nonholonomic model ceases to work and it is necessary to use more complex mechanical models incorporating sliding, elasticity, etc. [ABSTRACT FROM AUTHOR]

In this paper, we use a local fractional Laplace decomposition method to solve the diffusion equation with the diffusive parameter variable in fractal media. The obtained result illustrates the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives. [ABSTRACT FROM AUTHOR]

The local fractional variational iteration method is applied to a modified Fisher's equation defined on Cantor sets with the fractal conditions. The solution process is simple, and the accuracy of the approximate solution is high. The method provides an unrivaled tool for local differential equations. [ABSTRACT FROM AUTHOR]

One of the most important results in basic set theory is without doubt Cantor's Theorem which states that the power set of any set X is strictly bigger than X itself. Specker once stated, without providing a proof, that a generalization is possible: for any natural exponent m, there is a natural number N for which if X has at least N distinct elements, then the power set of X is strictly bigger than Xm. The aim of this paper is to formalize and prove Specker's claim and to provide a way to compute the values of N for which the theorem holds. [ABSTRACT FROM AUTHOR]

This paper deals with an open problem posed by Jleli and Samet in [1, M. Jleli and B. Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl, 20(3) 2018]. In [1, Remark 5.1], they asked whether the Cantor's intersection theorem can be extended to F-metric spaces or not. In this manuscript, we give an affirmative answer to this open question. Additionally, keeping in mind the fact that totally boundedness is not a topological property, in the setting of Ƒ-metric spaces are equivalent to that of usual metric spaces. [ABSTRACT FROM AUTHOR]

Following the works of Rosendal and Mann and Rafi, we try to answer the following question: when is the mapping class group of an infinite-type surface quasi-isometric to a graph whose vertices are curves on that surface? With the assumption of tameness as defined by Mann and Rafi, we describe a necessary and sufficient condition, called translatability, for a geometrically non-trivial big mapping class group to admit such a quasi-isometry. In addition, we show that the mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard, which we believe represents the first known example of a hyperbolic mapping class group that is not virtually free. [ABSTRACT FROM AUTHOR]

We will consider a family of fractal measures on the real line R which are fixed, in the sense of Hutchinson, under a finite family of contractive affine mappings. The maps are chosen such as to leave gaps on R. Hence they have fractal dimension strictly less than 1. The middle-third Cantor construction is one example. Depending on the gaps and the scaling factor, it is known that the corresponding Hilbert space L²(μ) exhibits strikingly different properties. In this paper we show that when μ is fixed in a certain class, there are positive integers p such that multiplication by p modulo 1 induces an ergodic automorphism on the measure space (support(μ), μ). [ABSTRACT FROM AUTHOR]

In this paper we obtain precise estimates for the L² norm of the s-dimensional Riesz transforms on very general measures supported on Cantor sets in Rd, with d - 1 < s < d. From these estimates we infer that, for the so-called uniformly disconnected compact sets, the capacity γs associated with the Riesz kernel x/|x|s+1 is comparable to the capacity C2/3(d-s),3/2 from non-linear potential theory. [ABSTRACT FROM AUTHOR]

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials Pa(z) = zd-1(z - da/(d-1)), the family fλ = λtanpzq. These functions have a super-attractive fixed point, and, depending on p, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family fλ. Again there are similarities to and differences from the parameter plane of the family Pa and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points. [ABSTRACT FROM AUTHOR]

Research in mathematics education indicates that in the transition from given systems to wider ones prospective teachers tend to attribute all the properties that hold for the former also to the latter. In particular, it has been found that, in the context of Cantorian Set Theory, prospective teachers have been found to erroneously attribute properties of finite sets to infinite ones – using different methods to compare the number of elements in infinite sets. These methods which are acceptable for finite sets, lead to contradictions with infinite ones. This paper describes a course in Cantorian Set Theory that relates to prospective secondary mathematics teachers‘ tendencies to overgeneralize from finite to infinite sets. The findings clearly indicate that when comparing the number of elements in infinite sets the prospective teachers who took the course were more successful and were also more consistent in their use of a single method than those who studied the traditional, formally-based Cantorian Set Theory course. [ABSTRACT FROM AUTHOR]

The linear and non-linear Klein-Gordon equations are considered. The fractional complex transform is used to convert the equations on a continuous space/time to fractals ones on Cantor sets, the resultant equations are solved by local fractional reduced differential transform method. Three examples are given to show the effectiveness of the technology. [ABSTRACT FROM AUTHOR]

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group Diff+ω(S¹) of orientation-preserving real-analytic circle diffeomorphisms, which include all subgroups of Diff+ω(S¹) acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by Dippolito [Ann. of Math. (2) 107 (1978), 403-453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work (by a slightly different list of authors). [ABSTRACT FROM AUTHOR]

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives. [ABSTRACT FROM AUTHOR]

Theelectric field intensity in one-dimensional (1D) quasiperiodic and hybrid photonics band-gap structures is studied in the present paper. The photonic structures are ordered according to Fibonacci, Thue-Morse, Cantor, Rudin-Shapiro, Period-Doubling, Paper-Folding, and Baum-Sweet sequences. The study shows that the electric field intensity is higher for the Thue-Morse multilayer systems. After that the Thue-Morse structure will be combined with a periodic structure to form a hybrid photonic structure. It is shown that this hybrid system is the best for a strong localization of light. The proposed structures have been modeled using the Transfer Matrix Method. [ABSTRACT FROM AUTHOR]

Given a positive integer M , a number q > 1 is called a univoque base if there is exactly one sequence ( c i ) = c 1 c 2 ⋯ with integer digits c i belonging to the set { 0 , 1 , … , M } , such that 1 = ∑ i = 1 ∞ c i q − i . The topological and combinatorial properties of the set of univoque bases U and their corresponding sequences ( c i ) have been investigated in many papers since a pioneering work of Erdős, Horváth and Joó 25 years ago. While in most studies the attention was restricted to univoque bases belonging to ( M , M + 1 ] , a recent work of Kong and Li on the Hausdorff dimension of unique expansions demonstrated the necessity to extend the earlier results to all univoque bases. This is the object of this paper. Although the general research strategy remains the same, a number of new arguments are needed, several new properties are uncovered, and some formerly known results become simpler and more natural in the present framework. [ABSTRACT FROM AUTHOR]

In this paper, we investigate elliptic functions of the form , where is the Weierstrass elliptic function on a real rhombic lattice. We show that a typical function in this family has a superattracting fixed point at the origin and five other equivalence classes of critical points. We investigate conditions on the lattice which guarantee that has a double toral band, and we show that this family contains the first known examples of elliptic functions for which the Julia set is disconnected but not Cantor. [ABSTRACT FROM AUTHOR]