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This paper introduces self-similarity inherent in planar Milich-Jennings centered flip graphs derived from the Narayana sequence. We show that self-similarity found in a Narayana sequence yields a connected spanning subgraph with a centered flip. This paper has several main results (1) Every Narayana sequence constructs a flip graph, (2) Every Narayana sequence is self-similar and (3) Every Pascal 3-triangle has a free group presentation.

In this paper, we investigate the factor properties and gap sequence of the Tribonacci sequence, the fixed point of the substitution σ( a , b , c ) = ( ab , ac , a ). Let ω p be the p-th occurrence of ω and G p ( ω ) be the gap between ω p and ω p +1 . We introduce a notion of kernel for each factor ω , and then give the decomposition of the factor ω with respect to its kernel. Using the kernel and the decomposition, we prove the main result of this paper: for each factor ω , the gap sequence { G p ( ω )} p ≤1 is the Tribonacci sequence over the alphabet { G 1 ( ω ), G 2 ( ω ), G 4 ( ω )}, and the expressions of gaps are determined completely. As an application, for each factor ω and p ∈ ℕ , we determine the position of ω p . Finally we introduce a notion of spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors.

Complexity may be one of the most important measurements for analysing time series data; it covers or is at least closely related to different data characteristics within nonlinear system theory. This paper provides a comprehensive literature review examining the complexity testing techniques for time series data. According to different features, the complexity measurements for time series data can be divided into three primary groups, i.e., fractality (mono- or multi-fractality) for self-similarity (or system memorability or long-term persistence), methods derived from nonlinear dynamics (via attractor invariants or diagram descriptions) for attractor properties in phase-space, and entropy (structural or dynamical entropy) for the disorder state of a nonlinear system. These estimations analyse time series dynamics from different perspectives but are closely related to or even dependent on each other at the same time. In particular, a weaker self-similarity, a more complex structure of attractor, and a higher-level disorder state of a system consistently indicate that the observed time series data are at a higher level of complexity. Accordingly, this paper presents a historical tour of the important measures and works for each group, as well as ground-breaking and recent applications and future research directions.

This paper studies the problem of finding optimal parameters for a Poincare section used for determining the type of behavior of a time series: a deterministic or stochastic one. To reach that goal optimization algorithms are coupled with the Poincare & Higuchi (P&H) method, which calculates the Higuchi dimension using points obtained by performing a Poincare section of a certain attractor. The P&H method generates distinctive patterns that can be used for determining if a given attractor is produced by a deterministic or a stochastic system, but this method is sensitive to the parameters of the Poincare section. Patterns generated by the P&H method can be characterized using numerical measures which in turn can be used for finding such parameters for the Poincare section for which the patterns produced by the P&H method are the most prominent. This paper studies several approaches to parameterization of the Poincare section. Proposed approaches are tested on twelve time series, six produced by deterministic chaotic systems and six generated randomly. The obtained results show, that finding good parameters of the Poincare section is important for determining the type of behavior of a time series. Among the tested methods the evolutionary algorithm was able to find the best Poincare sections for use with the P&H method.

The subject of this paper is to give the description, up to topological conjugacy, of possible minimal sets of triangular maps of the square of type 2 ∞ . In [4] , we give a general method allowing to embed any zero-dimensional almost 1–1 extension of the dyadic odometer (in particular any dyadic Toeplitz system) as a minimal set of a triangular map of this type. In this paper we present a method (a combination of that described in [4] with one introduced in [1] ) of similarly embedding a special class of zero-dimensional almost 2–1 extensions of the odometer. We conjecture that these two embedding theorems exhaust all possibilities for nonperiodic minimal sets. The paper was inspired by the last unsolved problem in the Sharkovski classification program of triangular maps: does there exist a triangular map with positive entropy attained on the set of uniformly recurrent points but with entropy zero on the set of regularly recurrent points. The paper answers this question positively, concluding the program.

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup{\dim_{H}f^{-1}(y) : y \in \mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, provided that $K$ is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if $K$ is not totally disconnected and sufficiently homogeneous then $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in \inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. Finally, we prove that the graph of the generic $f\in C(K)$ has the same Hausdorff and topological Hausdorff dimension as $K$., 20 pages

As known, the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of convergence field of an infinite matrix in the space of almost convergent sequences is so new (see [15]). The purpose of this paper is to introduce the new spaces and consisting of all sequences whose Cesaro transforms of order one are in the spaces f and f0, respectively. Also, in this paper, we show that and are linearly isomorphic to the spaces f and f0, respectively. The β- and γ-duals of the spaces and are computed. Furthermore, the classes ( : μ) and (μ: ) of infinite matrices are characterized for any given sequence space μ, and determined the necessary and sufficient conditions on a matrix A to satisfy BC-core(Ax) ⊆ K-core(x), K-core(Ax) ⊆ BC-core(x), BC-core(Ax) ⊆ BC-core(x), BC-core(Ax) ⊆ st-core(x) for all x ∈ l∈.

Remarkably benign looking logistic transformations x n +1 = r x n (1 − x n ) for choosing x 0 between 0 and 1 and 0 r ⩽ 4 have found a celebrated place in chaos, fractals and discrete dynamics. The strong physical meaning of Mandelbrot and Julia sets is broadly accepted and nicely connected by Christian Beck [Beck C. Physical meaning for Mandelbrot and Julia sets. Physica D 1999;125(3–4):171–182. Zbl0988.37060] to the complex logistic maps, in the former case, and to the inverse complex logistic map, in the latter case. The purpose of this paper is to study the bounded behavior of the complex logistic map using superior iterates and generate fractals from the same. The analysis in this paper shows that many beautiful properties of the logistic map are extendable for a larger value of r .

In this paper, we study the following Neumann–Robin boundary value problem - ( ϕ p ( u ′ ( x ) ) ) ′ = λ f ( u ( x ) ) , x ∈ ( 0 , 1 ) , u ′ ( 0 ) = 0 , u ′ ( 1 ) + α u ( 1 ) = 0 , whereα ∈ R, λ > 0 are parameters and p > 1, and p ′ = p p - 1 is the conjugate exponent of p and ϕp(x): = ∣x∣p−2x for all x ∈ R where (ϕp(u′))′ is the one dimensional p-Laplacian and f ∈ C2[0, ∞) such that f(0) 0, and also f is increasing and concave up. We shall investigate the existence and multiplicity of nonnegative solutions. Note that in this paper, we shall establish our existence results by using the quadrature method.

Motivated by Mandelbrot's [The Fractal Geometry of Nature, Freeman, San Francisco, 1983] idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, we study the properties of these translation sets and show how they can be used for a classification purpose. This first paper of a series of two will be devoted to set up the fundamental properties of Hausdorff measures of those intersection sets. Using the triadic expansion of the shifting number, we determine the fractal structure of intersection of triadic Cantor sets with their translates. We found that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from those shifting numbers with a finite triadic expansion. We characterize this set of shifting numbers by giving a partition expression of it and the steps of its construction from a fundamental root set. Finally, we prove that intersection of Cantor sets with their translates verify a measure-conservation law with scales. The second paper will take advantage of the properties exposed here in order to utilize them in a classification context. Mainly, it will deal with the use of the discrete spectrum of measures to distinguish two Cantor-like sets of the same fractal dimension.

Some definitions such as m-chain recurrent set, weakly gradient flow and generalized Morse decomposition for a flow defined on a topological space are introduced in this paper. Some conclusions, include the chain recurrent set contain the m-chain recurrent set; the m-chain recurrent set contain the non-wandering set, are proved. In some flows the non-wandering set is proper subset of the m-chain recurrent set; in the meantime the m-chain recurrent set is proper subset of the chain recurrent set. Moreover some criterions for the existence of trajectories joining singular points and a necessary and sufficient condition of the weakly gradient flow are also given here. At last the generalized Morse decomposition of the invariant set are discussed in the paper.

In this paper, we define the co-complex-type k -Fibonacci numbers and then give the relationships between the k -step Fibonacci numbers and the co-complex-type k -Fibonacci numbers. Also, we produce various properties of the co-complex-type k -Fibonacci numbers such as the generating matrices, the Binet formulas, the combinatorial, permanental and determinantal representations, and the finite sums by matrix methods. In addition, we study the co-complex-type k -Fibonacci sequence modulo m and then we give some results concerning the periods and the ranks of the co-complex-type k -Fibonacci sequences for any k and m . Furthermore, we extend the co-complex-type k -Fibonacci sequences to groups. Finally, we obtain the periods of the co-complex-type 2-Fibonacci sequences in the semidihedral group S D 2 m , ( m ≥ 4 ) with respect to the generating pair ( x , y ) .

Mutualistic networks are used to study the structure and processes inherent to mutualistic relationships. In this paper, we introduce a random matrix ensemble (RME) representing the adjacency matrices of mutualistic networks composed by two vertex sets of sizes n and m − n . Our RME depends on three parameters: the network size n , the size of the smaller set m , and the connectivity between the two sets α , where α is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the spectral, eigenvector and topological properties of the RME by computing, respectively, the ratio of consecutive eigenvalue spacings r , the Shannon entropy of the eigenvectors S , and the Randic index R . First, within a random matrix theory approach (i.e. a statistical approach), we identify a parameter ξ ≡ ξ ( n , m , α ) that scales the average normalized measures X ¯ > (with X representing r , S and R ). Specifically, we show that (i) ξ ∝ α n with a weak dependence on m , and (ii) for ξ 1 / 10 most vertices in the mutualistic network are isolated, while for ξ > 10 the network acquires the properties of a complete network, i.e., the transition from isolated vertices to a complete-like behavior occurs in the interval 1 / 10 ξ 10 . Then, we demonstrate that our statistical approach predicts reasonably well the properties of real-world mutualistic networks; that is, the universal curves X ¯ > vs. ξ show good correspondence with the properties of real-world networks.

The purpose of this paper is to introduce the concept of Cn WP-Bailey pairs. The Cn WP-Bailey transform is obtained by applying the Cn6ϕ5 summation formula. From this result, the Cn WP-Bailey lemma is deduced by making use of the Cn q-Dougall summation formula. Some applications are investigated. Finally, the case of elliptic Cn WP-Bailey pairs is discussed.

In this paper, dual-complex k-Fibonacci numbers are defined. Also, some algebraic properties of dual-complex k-Fibonacci numbers which are connected with dual-complex numbers and k-Fibonacci numbers are investigated. Furthermore, the Honsberger identity, the d’Ocagne’s identity, Binet’s formula, Cassini’s identity, Catalan’s identity for these numbers are given.

This paper presents detailed procedure for determining the formula for calculation Tribonacci sequence numbers with arbitrary initial numbers Ta,b,c,(n). Initial solution is based on the concept of damped oscillations of Lucas type series with initial numbers T3,1,3(n). Afterwards coefficient θ3 has been determined which reduces Lucas type Tribonacci series to Tribonacci sequence T0,0,1(n). Determined relation had to be corrected with a phase shift ω3. With known relations of unitary series T0,0,1(n) with remaining two equations of Tribonacci series sequence T1,0,0(n) and T0,1,0(n), Binet type equation of Tribonacci sequence that has initial numbers Ta,b,c(n) is obtained.

In this paper we apply the chaos game to n -sided regular polygons to generate fractals that are similar to the Sierpinski gasket. We show that for each n -gon, there is an exact ratio that will yield a perfect gasket. We then find a formula for this ratio that depends only on the angle π / n .

This paper deals with developing a new class of quaternions, octonions and sedenions called higher order Fibonacci 2 m -ions (or-higher order Fibonacci hyper complex numbers) whose components are higher order Fibonacci numbers. We give recurrence relation, Binet formula, generating function and exponential generating function of higher order Fibonacci 2 m -ions. We also derive some identities such as Vajda’s identity, Catalan’s identity, Cassini’s identity, and d’Ocagne’s identity with the aid of the Binet formula. Finally, we develop some matrix identities involving higher order Fibonacci 2 m -ions which allow us to obtain some properties of these higher order hyper complex numbers.

An Apollonian packing is one of the most beautiful circle packings based on an old theorem of Apollonius of Perga. Ford circles are important objects for studying the geometry of numbers and the hyperbolic geometry. In this paper we pursue a research on the Ford sphere packing, which is not only the three dimensional extension of Ford circle packing, but also a degenerated case of the Apollonian sphere packing. We focus on two interesting sequences in Ford sphere packings. One sequence converges slowly to an infinitesimal sphere touching the origin of the horizontal plane. The other sequence converges at fastest rate to an infinitesimal sphere in a particular position on the plane. All these sequences have their counterparts in Ford circle packings and keep similar features. For example, our finding shows that the x-coordinate of one Ford circle sequence converges to the golden ratio gracefully. We define a Ford sphere group to interpret the Ford sphere packing and its sequences finally.

The spatial distributions of cities fall into two groups: one is the simple distribution with characteristic scale (e.g. exponential distribution), and the other is the complex distribution without characteristic scale (e.g. power-law distribution). The latter belongs to scale-free distributions, which can be modeled with fractal geometry. However, fractal dimension is not suitable for the former distribution. In contrast, spatial entropy can be used to measure any types of urban distributions. This paper is devoted to generalizing multifractal parameters by means of dual relation between Euclidean and fractal geometries. The main method is mathematical derivation and empirical analysis, and the theoretical foundation is the discovery that the normalized fractal dimension is equal to the normalized entropy. Based on this finding, a set of useful spatial indexes termed dummy multifractal parameters are defined for geographical analysis. These indexes can be employed to describe both the simple distributions and complex distributions. The dummy multifractal indexes are applied to the population density distribution of Hangzhou city, China. The calculation results reveal the feature of spatio-temporal evolution of Hangzhou's urban morphology. This study indicates that fractal dimension and spatial entropy can be combined to produce a new methodology for spatial analysis of city development., Comment: 23 pages, 3 figures, 5 tables

This paper is concerned with the Diophantine properties of the sequence { ξ θ n } , where 1 ≤ ξ θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measures μ λ with λ = θ - 1 as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak, who consider the case of Bernoulli convolutions. As an application, we prove that μ λ almost every x is normal to any base b ≥ 2, which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.

In this paper, we study the following generalized quasilinear Schrodinger equations with critical or supercritical growths - div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = f ( x , u ) + λ | u | p - 2 u , x ∈ R N , where λ > 0 , N ≥ 3 , g : R → R + is a C1 even function, g ( 0 ) = 1 , g ′ ( s ) ≥ 0 for all s ≥ 0 , lim | s | → + ∞ g ( s ) | s | α - 1 : β > 0 for some α ≥ 1 and ( α - 1 ) g ( s ) > g ′ ( s ) s for all s > 0 and p ≥ α2*. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 using a change of variables and variational method.

In this paper, we investigate a horizontal Laplacian version of the clamped plate problem on Carnot groups and obtain some universal inequalities. Furthermore, for the lower order eigenvalues of this eigenvalue problem on carnot groups, we also give some universal inequalities.

In this paper, we consider a class of N -Laplacian equations involving critical growth { - Δ N u = λ | u | N - 2 u + f ( x , u ) , x ∈ Ω , u ∈ W 0 1 , N ( Ω ) , u ( x ) ≥ 0 , x ∈ Ω , where Ω is a bounded domain with smooth boundary in ℝ N ( N >2), f ( x, u ) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ > λ 1 , λ≠λ l (l = 2,3,ċċċ), and λ l is the eigenvalues of the operator ( - Δ N , W 0 1 , N ( Ω ) ) , which is defined by the ℤ 2 -cohomological index.

In this paper, we introduce h(x) – Fibonacci polynomials in an arbitrary finite-dimensional unitary algebra over a field K ( K = R , C ) . These polynomials generalize h(x) – Fibonacci quaternion polynomials andh(x) – Fibonacci octonion polynomials. For h(x) – Fibonacci polynomials in an arbitrary algebra, we provide generating function, Binet-style formula, Catalan-style identity, and d’Ocagne-type identity.

The main aim of this paper is to introduce new concept of bivariate Mersenne Lucas polynomials { m n ( x , y ) } n = 0 ∞ , we first give the recurrence relation of them. We then obtain Binet’s formula, generating function, Catalan’s identity and Cassini’s identity for this type of polynomials. After that, we give the symmetric function, explicit formula and d’Ocagne’s identity of bivariate Mersenne and bivariate Mersenne Lucas polynomials. By using the Binet’s formula we obtain some well-known identities of these bivariate polynomials. Also, some summation formulas of bivariate Mersenne and bivariate Mersenne Lucas polynomials are investigated.

In this paper, we introduced dual complex Fibonacci p -numbers and some properties of dual complex Fibonacci p -numbers which are related to complex Fibonacci numbers and complex Fibonacci p -numbers.

Let m ≥ 1 be an integer, 1 < β ≤ m + 1. A sequence ɛ1 ɛ2 ɛ3 with ɛi ∈ {0,1, … m} is called a β-expansion of a real number x if x = ∑ i ∈ i β i . It is known that when the base β is smaller than the generalized golden ration, any number has uncountably many expansions, while when β is larger, there are numbers which has unique expansion. In this paper, we consider the bases such that there is some number whose unique expansion is purely periodic with the given smallest period. We prove that such bases form an open interval, moreover, any two such open intervals have inclusion relationship according to the Sharkovskii ordering between the given minimal periods. We remark that our result answers an open question posed by Baker, and the proof for the case m = 1 is due to Allouche, Clarke and Sidorov.

Let m, r, k be three positive integers. Let G be a graph with vertex set V(G) and edge set E(G), and let f: V(G) → N be a function such that f ( x ) ≥ ( k + 2 ) r − 1 for any x ∈ V(G). Let H1, H2, … , Hk be k vertex disjoint mr-subgraphs of a graph G. In this paper, we prove that every ( 0 , m f − ( m − 1 ) r ) -graph admits a (0, f)-factorization randomly r-orthogonal to each Hi ( i = 1 , 2 , … , k ).

This paper establishes a high order condition on the restricted isometry property adapted to a frame D (D-RIP) for the signal recovery. It is shown that if the measurement matrix A satisfies the D-RIP condition δtk < t - 1 t for t > 1, then all signals f which are sparse in terms of a tight frame D can be recovered stably or exactly via the l1-analysis model based on y = Af + z in l2 and Dantzig selector bounded noise setting.

Let H 1 , H 2 , H 3 be real Hilbert spaces, let A : H 1 → H 3 , B : H 2 → H 3 be two bounded linear operators. The split equality common fixed point problem (SECFP) in the infinite-dimensional Hilbert spaces introduced by Moudafi (Alternating CQ-algorithm for convex feasibility and split fixed-point problems. Journal of Nonlinear and Convex Analysis) is (1) to find x ∈ F ( U ) , y ∈ F ( T ) such that A x = B y , where U : H 1 → H 1 and T : H 2 → H 2 are two nonlinear operators with nonempty fixed point sets F(U) = { x ∈ H 1 : Ux = x } and F ( T ) = { x ∈ H 2 : Tx = x }. Note that, by taking B = I and H 2 = H 3 in (1), we recover the split fixed point problem originally introduced in Censor and Segal. Recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms with weak convergence for the SECFP (1) of firmly quasi-nonexpansive operators. In this paper, we introduce two viscosity iterative algorithms for the SECFP (1) governed by the general class of quasi-nonexpansive operators. We prove the strong convergence of algorithms. Our results improve and extend previously discussed related problems and algorithms.

In this paper, we show that for log232log2≤β≤12, suppose S is an invariant subspace of the Hardy-Sobolev spaces Hβ2(Dn) for the n-tuple of multiplication operators (Mz1,⋯,Mzn). If (Mz1|S,⋯,Mzn|S) is doubly commuting, then for any non-empty sub-set α = {α1, …,αk} of {1, …, n}, wαS is a generating wandering subspace for Mα|S=(Mzα1|S,⋯,Mzαk|S), that is, [wαS]Mα|S=S, Where wαS=∩i=1k(SΘzαiS).

In the paper, we consider Moran-type sets Ea given by sequences { a k } k = 1 ∞ and { n k } k = 1 ∞ . we prove that Ea may be decompose into the disjoint union of level sets. Moreover, we define three type of equivalence between two dimension functions associated to two Moran-type sets, respectively, and we classify Moran-type sets by these equivalent relations.

In theoretical chemistry, the eccentric connectivity index ξ(G) of a molecular graph G was introduced as ξ ( G ) = ∑ v ∈ V ( G ) d ( v ) ɛ ( v ) where d(v) expresses the degree of vertex v and ɛ(v) is the largest distance between v and any other vertex of G. The corresponding eccentric connectivity polynomial is denoted by ξ ( G , x ) = ∑ v ∈ V ( G ) d ( v ) x ɛ ( v ) . In this paper, we present the exact expressions of eccentric connectivity polynomial for V-phenylenic nanotubes and Zig-Zag polyhex nanotubes.

A novel model is proposed to overcome the shortages of the classical hypothesis of the two-dimensional discrete hidden Markov model. In the proposed model, the state transition probability depends on not only immediate horizontal and vertical states but also on immediate diagonal state, and the observation symbol probability depends on not only current state but also on immediate horizontal, vertical and diagonal states. This paper defines the structure of the model, and studies the three basic problems of the model, including probability calculation, path backtracking and parameters estimation. By exploiting the idea that the sequences of states on rows or columns of the model can be seen as states of a one-dimensional discrete 1 × 2 order hidden Markov model, several algorithms solving the three questions are theoretically derived. Simulation results further demonstrate the performance of the algorithms. Compared with the two-dimensional discrete hidden Markov model, there are more statistical characteristics in the structure of the proposed model, therefore the proposed model theoretically can more accurately describe some practical problems.

The purpose of this paper is to study and analyze an iterative method for finding a common element of the solution set Ω of the split feasibility problem and the set F ( T ) of fixed points of a right Bregman strongly nonexpansive mapping T in the setting of p -uniformly convex Banach spaces which are also uniformly smooth. By combining Mann's iterative method and the Halpern's approximation method, we propose an iterative algorithm for finding an element of the set F( T )⋂ Ω; moreover, we derive the strong convergence of the proposed algorithm under appropriate conditions and give numerical results to verify the efficiency and implementation of our method. Our results extend and complement many known related results in the literature.

Let E , F ⊂ R be two given closed intervals, and let τ: E → F and θ: F → E be continuous maps. In this paper, we consider Koto’s chaos, sensitivity and accessibility of a given system Ψ ( u , v ) = ( θ ( v ) , τ ( u ) ) on a given product space E × F where u ∈ E and v ∈ F. In particular, it is proved that for any Cournot map Ψ ( u , v ) = ( θ ( v ) , τ ( u ) ) on the product space E × F, the following hold: (1) If Ψ satisfies Kato’s definition of chaos then at least one of Ψ 2 | Q 1 and Ψ 2 | Q 2 does, where Q 1 = { ( θ ( v ) , v ) : v ∈ F } and Q 2 = { ( u , τ ( u ) ) : u ∈ E } . (2) Suppose that Ψ 2 | Q 1 and Ψ 2 | Q 2 satisfy Kato’s definition of chaos, and that the maps θ and τ satisfy that for any e > 0, if ∣ ( τ ∘ θ ) n ( v 1 ) − ( τ ∘ θ ) n ( v 2 ) ∣ ɛ and ∣ ( θ ∘ τ ) m ( u 1 ) − ( θ ∘ τ ) m ( u 2 ) ∣ ɛ for some integers n, m > 0, then there is an integer l(n, m, e) > 0 with ∣ ( τ ∘ θ ) l ( n , m , ɛ ) ( v 1 ) − ( τ ∘ θ ) l ( n , m , ɛ ) ( v 2 ) ∣ ɛ and ∣ ( θ ∘ τ ) l ( n , m , ɛ ) ( u 1 ) − ( θ ∘ τ ) l ( n , m , ɛ ) ( u 2 ) ∣ ɛ . Then Ψ satisfies Kato’s definition of chaos.

Let (Gm)m ≥ 0 be an integer linear recurrence sequence (under some weak technical conditions) and let x ≥ 1 be an integer. In this paper, we are interested in the problem of finding combinations of the form x G n + G n − 1 which belongs to (Gm)m ≥ 0 for infinitely many positive integers n. In this case, we shall make explicit an upper bound for x which only depends on the roots of the characteristic polynomial of this recurrence. As application, we shall study the k-nacci case.

The Hofstadter Q-sequence is perhaps the most known example of meta-Fibonacci sequence. Many authors have been interested in meta-Fibonacci sequences related to this sequence. For example, recently, it was studied by A. Alkan, N. Fox, and O. Aybar the connection between the Q-sequence and the Hofstadter-Conway $ 10,000 sequence. Here, in the similar spirit as their paper, we study the interplay between the Hofstadter Q-sequence and one of the more important multiplicative arithmetic function, namely, the Mobius function.

In this paper, we introduce h ( x ) − Fibonacci quaternion polynomials that generalize the k − Fibonacci quaternion numbers, which in their turn are a generalization of the Fibonacci quaternion numbers. We also present a Binet-style formula, ordinary generating function and some basic identities for the h ( x ) − Fibonacci quaternion polynomial sequences.

Alternate Julia sets have been studied in Picard iterative procedures. The purpose of this paper is to study the quadratic and cubic maps using superior iterates to obtain Julia sets with different alternate structures. Analytically, graphically and computationally it has been shown that alternate superior Julia sets can be connected, disconnected and totally disconnected, and also fattier than the corresponding alternate Julia sets. A few examples have been studied by applying different type of alternate structures.

It is known that the whole space can be a Li–Yorke scrambled set in a compact dynamical system, but this does not hold for distributional chaos. In this paper we construct a noncompact weekly mixing dynamical system, and prove that the whole space is a transitive extremal distributionally scrambled set in this system.

Dehghan and Hajarian, [4], investigated the matrix equations A T X B + B T X T A = C and A T X B + B T X A = C providing inequalities for the determinant of the solutions of these equations. In the same paper, the authors presented a lower bound for the product of the eigenvalues of the solutions to these matrix equations. Inspired by their work, we give some generalizations of Dehghan and Hajarian results. Using the theory of the numerical ranges, we present an inequality involving the trace of C when A , B , X are normal matrices satisfying A T B = BA T .

Let ( T , d ) be a dendrite and f be a continuous map from T to T . Denote by ω ( x , f ) the ω -limit set of x under f . Write Ω ( x , f ) = { y | there exist a sequence of points x k ∈ T and a sequence of positive integers n 1 n 2 ⋯ such that lim k ⟶ ∞ x k = x and lim k ⟶ ∞ f n k ( x k ) = y } . In this paper, we show that if the cardinal number of the set of endpoints of T is less than the cardinal number c of the continuum, then f is equicontinuous if and only if Ω ( x , f n ) = ω ( x , f n ) for any x ∈ T and any positive integer n .

In this paper, we study the nonperiodic second order Hamiltonian systems u ¨ ( t ) - λ L ( t ) u ( t ) + ∇ W ( t , u ( t ) ) = 0 , ∀ t ∈ R , where λ ⩾ 1 is a parameter, the matrix L ( t ) is not necessarily positive definite for all t ∈ R nor coercive. Replacing the Ambrosetti–Rabinowitz condition by general superquadratic assumptions, we establish the existence and multiplicity results for the above system when λ > 1 large. We also consider the situation where W is a combination of subquadratic and superquadratic terms, and obtain infinitely many homoclinic solutions.

Let p j ∈ ℕ and p j ≥ 1 , j = 2 , … , k , k ≥ 2 , be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain Ω N = { z = ( z 1 , z ′ 2 , … , z ′ k ) ′ ∈ ℂ × ℂ n 2 × … × ℂ n k : | z 1 | 2 + | | z 2 | | 2 p 2 + … + | | z k | | k p k 1 } given by F ( z ) = ( f ( z 1 ) + f ′ ( z 1 ) ∑ j = 2 k P j ( z j ) , ( f ′ ( z 1 ) ) 1 p 2 z ′ 2 , … , ( f ′ ( z 1 ) 1 p k z ′ k ) ′ , , where f is a normalized biholomorphic function on the unit disc D , and for 2 ≤ j ≤ k , P j : ℂ n j → ℂ is a homogeneous polynomial of degree p j and z j = ( z j 1 , … , z j n j ) ′ ∈ ℂ n j , n j ≥ 1 , p j ≥ 1 , | | z j | | j = ( ∑ l = 1 n j | z j l | p j ) 1 p j . . In this paper, some conditions for p j are found under which the operator preserves the properties of almost starlikeness of order α, starlikeness of order α and strongly starlikeness of order α on Ω N , respectively.

Let B1 ⊂ ℝN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients: { u | ∂ B 1 = 0 , − d i v ( | ∇ u | p − 2 ∇ u ) = | x | s | u | p * ( s ) − 2 u + λ | x | t | u | p − 2 u , x ∈ B 1 , where t , s > − p , 2 ≤ p N , p * ( s ) = ( N + s ) p N − p and λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N > p ( p − 1 ) t + p ( p 2 − p + 1 ) and λ ∈ (0, λ1, t), where λ1, t is the first eigenvalue of -Δp with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ ( p s + p ) min { 1 , p + t p + s } + p 2 p − ( p − 1 ) min { 1 , p + t p + s } λ > 0 is small.

In this paper, we study the extension of isometries between the unit spheres of complex Banach spaces l p (Γ) and l p (Δ)( p > 1). We first derive the representation of isometries between the unit spheres of complex Banach spaces l p (Γ) and l p (Δ). Then we arrive at a conclusion that any surjective isometry between the unit spheres of complex Banach spaces l p (Γ)and l p (Δ) can be extended to be a linear isometry on the whole space.

The present paper formulates and solves a problem of dividing coins. The basic form of the problem seeks the set of the possible ways of dividing coins of face values 1,2,4,8,... between three people. We show that this set possesses a nested structure like the Sierpinski-gasket fractal. For a set of coins with face values power of r, the number of layer of the gasket becomes r. A higher-dimensional Sierpinski gasket is obtained if the number of people is more than three. In addition to Sierpinski-type fractals, the Cantor set is also obtained in dividing an incomplete coin set between two people., 12 pages, 7 figures

In [4], a new family W ( L p ( x ) , L m q ) of Wiener amalgam spaces was defined and investigated some properties of these spaces, where local component is a variable exponent Lebesgue space L p(x) (ℝ) and the global component is a weighted Lebesgue space L q m (ℝ). This present paper is a sequel to our work [4]. In Section 2, we discuss necessary and sufficient conditions for the equality W ( L p ( x ) , L m q ) = L q ( ℝ ) . Later we give some characterization of Wiener amalgam space W ( L p ( x ) , L m q ) . In Section 3 we define the Wiener amalgam space W ( F L p ( x ) , L m q ) and investigate some properties of this space, where F L p ( x ) is the image of L p(x) under the Fourier transform. In Section 4, we discuss boundedness of the Hardy-Littlewood maximal operator between some Wiener amalgam spaces.