Search

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is "disjoint" in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake. [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]

In this paper, we analyze a certain family of holomorphic correspondences on ... x ... and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map F is such that for any a ∈ ..., we have that (Fn)*(δa) converges to a probability measure μF for which F*(μF) = μF d where d is the degree of F. This result is used to show that the minimal Hutchinson invariant set, introduced by P. Alexandersson, P. Brändén, and B. Shapiro [An inverse problem in Pólya--Schur theory. I. Non-degenerate and degenerate operators, preprint, 2024], of a large class of operators and for sufficiently large n exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, the minimal Hutchinson-invariant set is a Cantor set. [ABSTRACT FROM AUTHOR]

Given a metric continuum X, we consider the collection of all regular subcontinua of X and the collection of all meager subcontinua of X, these hyperspaces are denoted by D(X) and M(X), respectively. It is known that D(X) is compact if and only if D(X) is finite. In this way, we find some conditions related about the cardinality of D(X) and we reduce the fact to count the elements of D(X) to a Graph Theory problem, as an application of this, we prove in particular that |D(X)| G {2, 3,4, 5, 8, 9} for any continuum X. Also, we prove that D(X) is never homeomorphic to N. On the other hand, given a point p G X, we consider the meager composant and the filament composant of p in X, denoted by Mpf and FcsX (p), respectively, and we study some relations between Mpx and FcsX (p) such as the equality of them as a subset of X. Also, we construct examples showing that the collection Fcs(X) = {FcsX (p): p G X} can be homeomorphic to: any finite discrete space, the harmonic sequence, the closure of the harmonic sequence and the Cantor set. Finally, we study the contractibility of M(X); we prove the arc of pseudo-arcs, which is a no contractible continuum, satisfies that its hyperspace of meager subcontinua is contractible, given a solution to Problem 3 of [10]. Most of the results shown in this paper are focus to answer problems and questions posed in [6], [9] and [10]. Also, we rise open problems. [ABSTRACT FROM AUTHOR]

This paper investigates tangent measures in the sense of Preiss for non-homogeneous Cantor sets on ℝ d 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 ℝ d . [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]

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]

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]

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]

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]

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, 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]

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 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]

The fractional derivatives have been widely applied in many fields and has attracted widespread attention. This paper extracts a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKe) with the local fractional derivative (LFD) for the first time. Two special functions, namely, the LT δ (Ξ δ) and LC δ (Ξ δ) functions that are derived on the basis of the Mittag-Leffler function (MLF) defined on the Cantor set (CS), are employed to construct the auxiliary trial function to look into the exact solutions (ESs). Aided by Yang's non-differentiable (ND) transformation, six groups of the ND ESs are found. The ND ESs on the CS for δ = ln 2 / ln 3 are depicted graphically. Additionally, as a comparison, the ESs of the classic (3+1)-dimensional MZKe for δ = 1 are also illustrated. The outcomes reveal that the derived method is powerful and effective, and can be used to deal with the other local fractional PDEs. [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]

Samuel J. Wheeler defends Wittgenstein's criticism of Cantor's set theory against the objections raised by Hilary Putnam. Putnam claims that Wittgenstein's dismissal of the basic tenets of this set theory concerning the noncountability of the set of real numbers was unfounded and ill‐conceived. In Wheeler's view, Putnam's charges result from his failure to grasp Wittgenstein's intention and, in particular, to consider the difference between empirical and logical impossibility. In my paper, I argue that Wheeler's defence is unsuccessful and, at the same time, that Putnam's objections go too far. [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]

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]

In this paper, we study the generalized Cantor-integers { C n } n ≥ 1 with the base conversion function f : { 0 , ... , m } → [ 0 , p ] being strictly increasing and satisfying f (0) = 0 and f (m) = p. We show that the sequence { C n n α } n ≥ 1 with α = log m + 1 (p + 1) is dense in the closed interval with the endpoints being its inferior and superior, respectively. Moreover, every homogeneous Cantor set ℭ satisfying open set condition can be induced by some generalized Cantor-integers, we get the exact point which attains the maximal interval density of the form [ 0 , x ] with respect to the self-similar probability measure supported on ℭ. This result partially confirms a conjecture of E. Ayer and R. S. Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets, Trans. Am. Math. Soc. 351(9) (1999) 3725–3741]. [ABSTRACT FROM AUTHOR]

This paper investigates the spectral properties of Jacobi matrices with limit‐periodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit‐periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off‐diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off‐diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero‐dimensional spectrum. [ABSTRACT FROM AUTHOR]

In this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$. We also prove more general results where the distance map $|x-y|$ is replaced by a function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if $K_1,K_2, \phi$ are as above then there exists an open set S so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior. [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 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]

A fractal integral identity with the parameter τ related to twice-differentiable mappings is first proposed in this paper. Based on the identity, the parameterized inequalities over the fractal domains are then derived for the mappings whose second-order derivatives in absolute value at certain powers are generalized n -polynomial convex, which is the main purpose of this investigation. Moreover, a series of fractal findings of some applications, involving the special mean values, the midpoint formulas, the moments of random variable and the wave equations on Cantor sets, are acquired correspondingly. [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]

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]

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper by Jordan et al. (2007) [6]. For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure. [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]

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]

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]

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 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]

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]

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]

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive processes with zero mean. Such processes can be represented as X t = B V (t) where B is Brownian motion and V is a continuous increasing function. We show that these dimensions depend on the local uniform Hölder indices. In particular, if V is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if V admits at least one point with positive lower Hölder regularity. It is also possible to estimate the Hausdorff dimension of the graph through the L q spectrum of V. We will show that if V is generated by a self-similar measure on R 1 with convex open set condition, the Hausdorff dimension of the graph can be precisely computed by its L q spectrum. An illustrating example of the Cantor Devil Staircase function, the Hausdorff dimension of the graph is 1 + 1 2 ⋅ log 2 log 3 . Moreover, we will show that the graph of the Brownian staircase surprisingly has Fourier dimension zero almost surely. [ABSTRACT FROM AUTHOR]

Let C(a),C(b)⊂[0,1] be the central Cantor sets generated by sequences a,b(0,1)N. The first main result of the paper gives a necessary and a sufficient condition for sequences a and b which inform when C(a)-C(b) is equal to [-1,1] or is a finite union of closed intervals. One of the corollaries following from this results shows that the product of thicknesses of two central Cantor sets which algebraic difference is an interval may be arbitrarily small. We also show that there are sets C(a) and C(b) with the Hausdorff dimension equal to 0 such that their algebraic difference is an interval. Finally, we give a full characterization of the case, when C(a)-C(b) is equal to [-1,1] or is a finite union of closed intervals. [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]

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, we study the generalized Cantor sequence c , which is an ℓ -automatic sequence. We prove that the abelian complexity of the 2-block sequence of c is ℓ -regular if the factor set of the sequence c is mirror invariant. As a consequence, we show that the 2-abelian complexity of a generalized Cantor sequence satisfying certain conditions is ℓ -regular. • We prove that abelian complexity of 2-block coding of generalized Cantor sequences satisfying certain conditions is regular. • We build a bridge between 2-abelian complexity of generalized Cantor sequences and abelian complexity of its 2-block coding. • We prove that 2-abelian complexity of generalized Cantor sequences satisfying certain conditions is regular. [ABSTRACT FROM AUTHOR]

Based on inverse packet sets and fuzzy set, the concept of inverse packet fuzzy sets is proposed in this paper by means of introducing dynamic characteristics into a fuzzy set A and improving the fuzzy set A. The inverse packet fuzzy sets is composed of a pair of fuzzy sets generated by the fuzzy set A and has dynamic characteristic and fuzzy characteristic. Furthermore, on certain conditions, it can be restored to the fuzzy set A or the inverse packet sets generated by the Cantor set X = ker ⁡ (A) . As a result, an inverse packet fuzzy sets can be expressed as a series of inverse packet sets by defining its scalar multiplication and its cut sets. Meanwhile some attribute-dependent characteristics of inverse packet fuzzy sets are obtained. The new set model will both enrich the set theory and provide a new basic tool for further practical applications. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Riemann wave model (RWM) on Cantor sets by using the local fractional derivative (LFD). A novel computational approach is provided to seek the exact traveling-wave solution of the non-differential type for the local fractional Riemann wave model (LFRWM). The proposed scheme is called local fractional traveling-wave method (LFTWM). An example is given to illustrate that the LFTWM is simple and excellent. The properties of the obtained traveling-wave solutions are elaborated by some 3D graphs. The LFTWM sheds a new light on solving the local fractional wave equations (LFWE) in physics and engineering. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical computation of the harmonic-measure distribution function, or h -function for short, associated with a particular planar domain. This function describes the hitting probability of a Brownian walker released from some point with the boundary of the domain. We use a fast and accurate boundary integral method for the numerical calculation of the h -functions for symmetric multiply connected slit domains with high connectivity. In view of the fact that the middle-thirds Cantor set C is a special limiting case of these slit domains, the proposed method is used to approximate the h -function for the set C. We also numerically analyze some asymptotic features of the calculated h -functions. [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]