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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.

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.

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.

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.

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.