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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]

In this paper, we discuss the thickness and quasisymmetric equivalence of a class of fractal sets based on the binary expansion of numbers. We give the sufficient and necessary conditions for the thickness of this kind of sets to be positive, and the necessary and sufficient condition for their quasisymmetric equivalence with the standard Cantor ternary set. This paper also studies the relationship between the thickness and the uniform completeness about the product sets of two sets defined by digit restrictions. [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]

But to what extent did Skolem have a complete and coherent view of the foundation of mathematics? Skolem 1920 One hundred years ago, the Norwegian mathematician Thoralf Albert Skolem published the first of a series of remarkable papers on logic and the foundations of mathematics [[33]]. Davies is in no doubt about what mathematics is: "Mathematics is an aspect of human culture, just as are language, law, music, and architecture" [[6], p. 101]. But we note that the "grand view" of mathematics toward which these developments aim stands in stark contrast to Skolem's skepticism and his insistence on the primacy of specific structures and doable algorithms. [Extracted from the article]

In this paper we explore the extent to which the algebraic structure of a monoid M determines the topologies on M that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or T_1 topologies; Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids and inverse monoids. If M is a topological monoid such that every homomorphism from M to a second countable topological monoid N is continuous, then we say that M has automatic continuity. We show that many well-known, and extensively studied, monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid \mathbb {N} ^\mathbb {N}; the full binary relation monoid B_{\mathbb {N}}; the partial transformation monoid P_{\mathbb {N}}; the symmetric inverse monoid I_{\mathbb {N}}; the monoid \operatorname {Inj}(\mathbb {N}) consisting of the injective transformations of \mathbb {N}; and the monoid C(2^{\mathbb {N}}) of continuous functions on the Cantor set 2^{\mathbb {N}}. The monoid \mathbb {N} ^\mathbb {N} can be equipped with the product topology, where the natural numbers \mathbb {N} have the discrete topology; this topology is referred to as the pointwise topology. We show that the pointwise topology on \mathbb {N} ^\mathbb {N}, and its analogue on P_{\mathbb {N}}, is the unique Polish semigroup topology on these monoids. The compact-open topology is the unique Polish semigroup topology on C(2 ^\mathbb {N}), and on the monoid C([0, 1] ^\mathbb {N}) of continuous functions on the Hilbert cube [0, 1] ^\mathbb {N}. The symmetric inverse monoid I_{\mathbb {N}} has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology. The full binary relation monoid B_{\mathbb {N}} has no Polish semigroup topologies, nor do the partition monoids. At the other extreme, \operatorname {Inj}(\mathbb {N}) and the monoid \operatorname {Surj}(\mathbb {N}) of all surjective transformations of \mathbb {N} each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on \mathbb {N} ^\mathbb {N}, P_{\mathbb {N}}, and \operatorname {Inj}(\mathbb {N}) coincide with the pointwise topology; and we characterise the Zariski topology on B_{\mathbb {N}}. Along the way we provide many additional results relating to the Markov topology, the small index property for monoids, and topological embeddings of semigroups in \mathbb {N}^{\mathbb {N}} and inverse monoids in I_{\mathbb {N}}. Finally, the techniques developed in this paper to prove the results about monoids are applied to function clones. In particular, we show that: the full function clone has a unique Polish topology; the Horn clone, the polymorphism clones of the Cantor set and the countably infinite atomless Boolean algebra all have automatic continuity with respect to second countable function clone topologies. [ABSTRACT FROM AUTHOR]

In this paper, within the scope of the local fractional derivative theory, the (4+1)-dimensional local fractional Fokas equation is researched. The study of exact solutions of high-dimensional nonlinear partial differential equations plays an important role in understanding complex physical phenomena in reality. In this paper, the exact traveling wave solution of generalized functions is analyzed defined on Cantor sets in high-dimensional integrable systems. The results of non-differentiable solutions in different cases are numerically simulated when the fractal dimension is equal to ρ = ln 2/ln 3. The results show that the exact solution of the local fractional Fokas equation represents the fractal waves on the shallow water surface. Through numerical simulation, we find that the exact solution of the local fractional Fokas equation can describe the fractal waves and waves characteristics of shallow water surface. It also shows that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics. [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]

A pattern is called universal in another collection of sets, when every set in the collection contains some linear and translated copy of the original pattern. Paul Erdős proposed a conjecture that no infinite set is universal in the collection of sets with positive measure. This paper explores an analogous problem in the topological setting. Instead of sets with positive measure, we investigate the collection of dense Gδ$G_{\delta }$ sets and the collection of generic sets (dense Gδ$G_{\delta }$ and complement has Lebesgue measure zero). We refer to such pattern as topologically universal and generically universal, respectively. It is easy to show that any countable set is topologically universal, while any set containing an interior cannot be topologically universal. In this paper, we will show that Cantor sets on Rd$\mathbb {R}^d$ are not topologically universal and Cantor sets with positive Newhouse thickness on R1$\mathbb {R}^1$ are not generically universal. This gives a positive partial answer to a question by Svetic concerning the Erdős similarity problem on Cantor sets. Moreover, we also obtain a higher dimensional generalization of the generic universality problem. [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 consider a class of normally degenerate quasi-periodically forced reversible systems, obtained as perturbations of a set of harmonic oscillators, { x ˙ = y + ϵ f 1 (ω t , x , y) , y ˙ = λ x l + ϵ f 2 (ω t , x , y) , where 0 ≠ λ ∈ ℝ, l > 1 is an integer and the corresponding involution G is (−θ, x, −y) → (θ, x, y). The existence of response solutions of the above reversible systems has already been proved in [22] if [f2(ωt, 0, 0)] satisfies some non-zero average conditions (See the condition (H) in [22]), here [ · ] denotes the average of a continuous function on T d . However, discussing the existence of response solutions for the above systems encounters difficulties when [f2(ωt, 0, 0)] = 0, due to a degenerate implicit function must be solved. This article will be doing work in this direction. The purpose of this paper is to consider the case where [f2(ωt, 0, 0)] = 0. More precisely, with 2p < l, if f2 satisfies [ f 2 (ω t , 0 , 0) ] = [ ∂ f 2 (ω t , 0 , 0) ∂ x ] = [ ∂ 2 f 2 (ω t , 0 , 0) ∂ x 2 ] = ⋯ = [ ∂ p − 1 f 2 (ω t , 0 , 0) ∂ x p − 1 ] = 0 , either λ − 1 [ ∂ p f 2 (ω t , 0 , 0) ∂ x p ] < 0 as l − p is even or λ − 1 [ ∂ p f 2 (ω t , 0 , 0) ∂ x p ] ≠ 0 as l − p is odd, we obtain the following results: (1) For λ ˜ > 0 (see λ ˜ in (2.2)) and ϵ sufficiently small, response solutions exist for each ω satisfying a weak non-resonant condition; (2) For λ ˜ < 0 and ϵ* sufficiently small, there exists a Cantor set ℰ ∈ (0 , ϵ ∗) with almost full Lebesgue measure such that response solutions exist for each ϵ ∈ ℰ if ω satisfies a Diophantine condition. In the remaining case where λ − 1 [ ∂ p f 2 (ω t , 0 , 0) ∂ x p ] > 0 and l − p is even, we prove the system admits no response solutions in most regions. [ABSTRACT FROM AUTHOR]

Let \Gamma be a countable discrete group and let H be a lcsc totally disconnected group, L a compact open subgroup of H, and \rho : \Gamma \rightarrow H a homomorphism with dense image. In this paper we construct, for every bi-L-invariant probability measure \theta on H, an explicit Furstenberg discretization \tau of \theta such that the Poisson boundary (B_\theta,\nu _\theta) of (H,\theta) is a \tau-boundary, where \Gamma acts on B_\theta via the homomorphism \rho. We also provide several criteria for when this \tau-boundary is maximal. Our technique can for instance be used to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but not L^p-irreducible for any p \geq 1, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we provide the first example of a countable discrete group \Gamma and two spread-out probability measures \tau _1 and \tau _2 on \Gamma such that the boundary entropy spectrum of (\Gamma,\tau _1) is an interval, while the boundary entropy spectrum of (\Gamma,\tau _2) is a Cantor set. [ABSTRACT FROM AUTHOR]

We prove a reducibility result for a linear wave equation with a time quasi-periodic driving on the one dimensional torus. The driving is assumed to be fast oscillating, but not necessarily of small size. Provided that the external frequency vector is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time-independent, block-diagonal one. With the present paper, we extend the previous work [26] to more general assumptions: we replace the analytic regularity in time with Sobolev one; the potential in the Schrödinger operator is a non-trivial smooth function instead of the constant one. The key tool to achieve the result is a localization property of each eigenfunction of the Schrödinger operator close to a subspace of exponentials, with a polynomial decay away from the latter. [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, we consider the one-dimensional Kirchhoff equation with x-dependent coefficients under Dirichlet boundary conditions, which models the forced vibrations of a clamped inhomogeneous string in the presence of a time periodic external forcing with period 2π/ω and amplitude ε. By using the Nash-Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period 2π/ω and order ε. Such results hold for parameters (ω, ε) in a positive measure Cantor set that has asymptotically full measure as the amplitude ε goes to zero. [ABSTRACT FROM AUTHOR]

Consider a normally degenerate Hamiltonian system with the following Hamiltonian H (θ , I , x , y , ϵ) = ω , I + λ x n + 1 n + 1 + y 2 2 + ϵ P (θ , x , y) , (θ , I , x , y) ∈ T d × R d + 2 , which is associated with the standard symplectic form dθ ∧ dI ∧ dx ∧ dy, where 0 ≠ λ ∈ R and n > 1 is an integer. The existence of response tori for the degenerate Hamiltonian system has already been proved by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] if [ ∂ P (θ , 0 , 0) ∂ x ] satisfies some non-zero conditions, see condition (H) in the work of Si and Yi [Nonlinearity 33, 6072–6098 (2020)], where [·] denotes the average value of a continuous function on T d . However, when [ ∂ P (θ , 0 , 0) ∂ x ] = 0 , no results were given by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] for response tori of the above system. This paper attempts at carrying out this work in this direction. More precisely, with 2p < n, if P satisfies [ ∂ j P (θ , 0 , 0) ∂ x j ] = 0 for j = 1, 2, ..., p and either λ − 1 [ ∂ p + 1 P (θ , 0 , 0) ∂ x p + 1 ] < 0 as n − p is even or λ − 1 [ ∂ p + 1 P (θ , 0 , 0) ∂ x p + 1 ] ≠ 0 as n − p is odd, we obtain the following results: (1) For λ ̃ < 0 [see λ ̃ in (2.1)] and ϵ sufficiently small, response tori exist for each ω satisfying a Brjuno-type non-resonant condition. (2) For λ ̃ > 0 and ϵ * sufficiently small, there exists a Cantor set E ∈ (0 , ϵ * ) with almost full Lebesgue measure such that response tori exist for each ϵ ∈ E if ω satisfies a Diophantine condition. In the case where λ − 1 [ ∂ p + 1 P (θ , 0 , 0) ∂ x p + 1 ] > 0 and n − p is even, we prove that the system admits no response tori in most regions. The present paper is regarded as a continuation of work by Si and Yi [Nonlinearity 33, 6072–6098 (2020)]. [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]

The paper is motivated by E. Akin's book about dynamical systems and closed relations [ 1 ], and by J. Kennedy's and G. Erceg's recent paper about the entropy of closed relations on closed intervals [ 9 ]. In present paper, we introduce the entropy of a closed relation G on any compact metric space X and show its basic properties. We also introduce when such a relation G generates a periodic point or finitely generates a Cantor set. Then we show that periodic points, finitely generated Cantor sets, Mahavier products and the entropy of closed relations are preserved by topological conjugations. Among other things, this generalizes the well-known results about the topological conjugacy of continuous mappings. Finally, we prove a theorem, giving sufficient conditions for a closed relation G on [ 0 , 1 ] to have a non-zero entropy. Then we present various examples of closed relations G on [ 0 , 1 ] such that (1) the entropy of G is non-zero, (2) no periodic point or exactly one periodic point is generated by G , and (3) no Cantor set is finitely generated by G. [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 paper, the bidirectional wave model is described by using the local fractional derivative (LFD) on Cantor sets for the first time. A novel algorithm is established to obtain the exact traveling-wave solution of the non-differential type for the local fractional bidirectional wave model (LFBWM), which is called the local fractional wave method (LFWM). The advantages of the LFWM are simple, efficient and accurate. The LFWM sheds a new light on solving the local fractional wave equations (LFWE) in physics and engineering. Finally, the physical properties of the obtained exact traveling-wave solution of the non-differential type are elaborated by some simulation figures. [ABSTRACT FROM AUTHOR]

This paper investigates tangent measures in the sense of Preiss for non-homogeneous Cantor sets on ℝd generated by specific iterated function systems that satisfy the strong separation condition. Through the dynamics of “zooming in” on any typical point, we derive an explicit and uniform formula for the tangent measures associated with this category of non-homogeneous Cantor sets on ℝd. [ABSTRACT FROM AUTHOR]

In this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let α:G↷X$\alpha:G\curvearrowright X$ be a minimal action of a locally finite‐by‐virtually Z${\mathbb {Z}}$ group G$G$ on the Cantor set X$X$. We prove that under certain assumptions, the action α$\alpha$ is almost finite in measure if and only if α$\alpha$ is essentially free. As an application, we obtain that any minimal topologically free action of a virtually Z${\mathbb {Z}}$ group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property Γ$\Gamma$ and Z$\mathcal {Z}$‐stability for their crossed product C∗$C^*$‐algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided. [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]

By means of He's fractal derivative, a new fractal (2 + 1)-dimensional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation is extracted in this paper. The semi-inverse method is employed to establish the generalized fractal variational principle. The generalized fractal variational principle can show the conservation laws through the energy form in the fractal space. Moreover, some semi-domain solutions are also explored by applying the variational approach and the one-step method namely Wang's direct mapping method-II. The dynamics of the extracted solutions on the Cantor set are unveiled graphically. The findings of this study are expected to provide some new insights into the exploration of the fractal PDEs. [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]

We study the recently suggested effective Wadge hierarchy in effective spaces, concentrating on the non-collapse property. Along with hierarchies of sets, we study hierarchies of k-partitions which are interesting on their own. In particular, we establish sufficient conditions for the non-collapse of the effective Wadge hierarchy and apply them to some concrete spaces, including the discrete space of natural numbers, the Baire space, and the Cantor space. While the latter two cases are obtained by easy effectivization of the corresponding topological result by T. Kihara and A. Montalban, the first case needs a different technique and is certainly the main technical result of this paper. [ABSTRACT FROM AUTHOR]

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]

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, we present a generalized form of the Cantor ternary set by studying the cantor 1/2λ+1 middle set where 1 ≤ λ < ∞ and its fractal dimension. The paper also introduces the Heine-Borel set and shows that the cantor set and its generalised 1/2λ+1 middle set where 1 ≤ λ < ∞ are Heine-Borel sets. [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]

For a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi-periodically forced oscillators. These solutions are well-understood in quasi-periodically perturbed nonlinear oscillators either in the presence of large damping or in the non-degenerate cases with small or free damping. In this paper, we consider the existence of response solutions in quasi-periodically perturbed, second order differential equations, including nonlinear oscillators, of the form x ¨ + λ x l = ϵ f (ω t , x , x ˙) , x ∈ R , where λ is a constant, 0 < ϵ ≪ 1 is a small parameter, l > 1 is an integer, ω ∈ R d is a frequency vector, and f : T d × R 2 → R 1 is real analytic and non-degenerate in x up to a given order p ≥ 0 , i.e., [ f (· , 0 , 0) ] = [ ∂ f (· , 0 , 0) ∂ x ] = [ ∂ 2 f (· , 0 , 0) ∂ x 2 ] = ⋯ = [ ∂ p - 1 f (· , 0 , 0) ∂ x p - 1 ] = 0 and [ ∂ p f (· , 0 , 0) ∂ x p ] ≠ 0 , where [ ] denotes the average value of a continuous function on T d . In the case that λ = 0 and f is independent of x ˙ , the existence of response solutions was first shown by Gentile (Ergod Theory Dyn Syst 27:427–457, 2007) when p = 1 . This result was later generalized by Corsi and Gentile (Commun Math Phys 316:489–529, 2012; Ergod Theory Dyn Syst 35:1079–1140, 2015; Nonlinear Differ Equ Appl 24(1):article 3, 2017) to the case that p > 1 is odd. In the case λ ≠ 0 , the existence of response solutions is studied by the authors Si and Yi (Nonlinearity 33(11):6072–6099, 2020) when p = 0 . The present paper is devoted to the study of response solutions of the above quasi-periodically perturbed differential equations for the case λ ≠ 0 by allowing p > 0 . Under the conditions that 0 ≤ p < l / 2 and λ [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 when l - p is even, we obtain a general result which particularly implies the following: (1) If either l is odd and λ < 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 , then as ϵ sufficiently small response solutions exist for each ω satisfying a Brjuno-like non-resonant condition; (2) If either l is odd and λ > 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] < 0 , then there exists an ϵ ∗ > 0 sufficiently small and a Cantor set E ∈ (0 , ϵ ∗) with almost full Lebesgue measure such that response solutions exist for each ϵ ∈ E and ω satisfying a Diophantine condition. Similar results are also obtained in the case λ = ± ϵ which particularly concern the existence of large amplitude response solutions. [ABSTRACT FROM AUTHOR]

Optical soliton is a physical phenomenon in which the waveforms and energy of optical fibers remain unchanged during propagation, which has important application value in information transmission. In this paper, the coupled nonlinear Schrödinger equations describe the propagation of optical solitons with different frequencies in sense of local fractional derivative is analyzed. The exact traveling wave solutions of the non-differentiable type defined on the Cantor sets are obtained. The characteristics of the particular solutions of fixed fractal dimension are discussed. It is proved that the local fractional coupled nonlinear Schrödinger equations can describe the interaction of fractal waves in optical fiber transmission. [ABSTRACT FROM AUTHOR]

Let 0 < ρ < 1 and let { a j , b j , n j } j = 1 ∞ be a sequence of positive integers with an upper bound. Associated with them, there exists a unique Borel probability measure μ ρ , { 0 , a j , b j } , { n j } generated by the following infinite convolution of discrete measures: μ ρ , { 0 , a j , b j } , { n j } = δ ρ n 1 ⁢ { 0 , a 1 , b 1 } ∗ δ ρ n 1 + n 2 ⁢ { 0 , a 2 , b 2 } ∗ δ ρ n 1 + n 2 + n 3 ⁢ { 0 , a 3 , b 3 } ∗ ⋯ , where gcd ⁡ (a j , b j) = 1 for all j ∈ ℕ . In this paper, we show that L 2 ⁢ (μ ρ , { 0 , a j , b j } , { n j } ) admits an exponential orthonormal basis if and only if the following two conditions are satisfied: (i) { a j , b j } ≡ { ± 1 } ⁢ (mod ⁢ 3) for all j ≥ 1 ; (ii) there exists a natural number r such that ρ - r ∈ 3 ⁢ ℕ and n j ∈ r ⁢ ℕ for all j ≥ 2 . [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]

Summary: Over the last few years, fractal structure has been used in several engineering fields such as surface physics, telecommunication, fluid mechanics, and so forth. In telecommunication, often there is a demand for miniaturized patch antennas with fractal structures because it is capable of handling very high frequencies with its diminutive structure. The space filling capability of the fractal structures is used to effectively distribute the surface current all over the radiating material. Fractal structures have a better size reduction capability which aids a miniaturized antenna design. This article provides a generalized and detailed survey on fractal structures in a Euclidean space using iterated function system (IFS) in patch antennas. There are various techniques to design a fractal structure, namely, IFSs, escape‐time fractals, strange attractors, L‐systems, random fractals, and so forth. IFSs are widely used technique to construct a finite set of construction mapping on a complete metric space. The fractal geometries designed using the technique are Sierpinski carpet and gasket, Koch snowflake, deterministic tree/binary tree, Gosper island, Minkowski island, Cantor set, T‐square, and some space filling fractals. This paper deals with the above given fractal structures with their design expressions and shapes in recent developments and applications mentioned. The fractal geometries are used in MIMO antennas to abridge the mutually coupled radiators and improve the gain and efficiency of the antenna. This paper also analyzes the effects of fractal structures on MIMO antennas and discusses the ascendancy fractal structures by cross‐referencing the significant antenna parameters. [ABSTRACT FROM AUTHOR]

The article presents the table of contents and editorial staff.

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]

In this paper, we study the metric theory of dyadic approximation in the middle-third Cantor set. This theory complements earlier work of Levesley et al. (Math Ann 338(1):97–118, 2007), who investigated the problem of approximation in the Cantor set by triadic rationals. We find that the behaviour when we consider dyadic approximation in the Cantor set is substantially different to considering triadic approximation in the Cantor set. In some sense, this difference in behaviour is a manifestation of Furstenberg's times 2 times 3 phenomenon from dynamical systems, which asserts that the base 2 and base 3 expansions of a number are not both structured. [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]

The Boussinesq–Kadomtsev–Petviashvili-like model is a famous wave equation which is used to describe the shallow water waves in ocean beaches and lakes. When shallow water waves propagate in microgravity or with unsmooth boundaries, the Boussinesq–Kadomtsev–Petviashvili-like model is modified into its fractal model by the local fractional derivative (LFD). In this paper, we mainly study the fractal Boussinesq–Kadomtsev–Petviashvili-like model (FBKPLM) based on the LFD on Cantor sets. Two efficient and reliable mathematical approaches are successfully implemented to obtain the different types of fractal traveling wave solutions of the FBKPLM, which are fractal variational method (FVM) and fractal Yang wave method (FYWM). Finally, some three-dimensional (3D) simulation graphs are employed to elaborate the properties of the fractal traveling wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, the fractal Zakharov–Kuznetsov–Benjamin–Bona–Mahony model (FZKBBM) is studied based on the local fractional derivative sense on Cantor sets for the first time. The different types of traveling wave solutions of the FZKBBM are successfully obtained by using two reliable and efficient approaches, which are fractal Yang wave method (FYWM) and fractal variational method (FVM). The properties of the obtained traveling wave solutions of non-differential type are elaborated by using some three-dimensional simulation graphs. [ABSTRACT FROM AUTHOR]