Identities via Bell matrix and Fibonacci matrix

In this paper, we study the relations between the Bell matrix and the Fibonacci matrix, which provide a unified approach to some lower triangular matrices, such as the Stirling matrices of both kinds, the Lah matrix, and the generalized Pascal matrix. To make the results more general, the discussion is also extended to the generalized Fibonacci numbers and the corresponding matrix. Moreover, based on the matrix representations, various identities are derived.     

Schröder matrix as inverse of Delannoy matrix

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.     

Dispersion relations of elastic waves in three-dimensional cubical piezoelectric phononic crystal with initial stresses and mechanically and dielectrically imperfect interfaces

A three-dimensional cubical piezoelectric phononic crystal is theoretically studied in this paper, formed by pre-stressed piezoelectric rectangular blocks and imperfect interfaces. Firstly, transfer matrices of pre-stressed piezoelectric rectangular blocks and imperfect interfaces are obtained based on the structural characteristics. The Bloch waves consist of three groups of sub-coupled elastic waves corresponding with three orthogonal periodical directions in the three-dimensional periodical structure. Furthermore, based on the transfer matrices of typical single cell and the Bloch theorem, it is established that the theoretical model of above-mentioned three-dimensional cubical piezoelectric phononic crystal to obtain the dispersion relations of Bloch waves. Finally, the influences of non-dimensional geometrical parameters (structural and modal parameters) and physical parameters (initial stress and mechanically and dielectrically imperfect interface parameters) on the dispersion relations are discussed based on the graphically numerical results. Numerical calculation results show that the existences of initial stresses and mechanically and dielectrically imperfect interfaces equivalently lead to the alterations of structural flexibilities or rigidities. The theoretical model and numerical discussions will provide a direct guidance of multi-material additive manufacturing for pre-stressed and inhomogeneous periodic structures with partial and global dispersion properties.     

Polynomial Fibonacci–Hessenberg matrices

A Fibonacci–Hessenberg matrix with Fibonacci polynomial determinant is referred to as a polynomial Fibonacci–Hessenberg matrix. Several classes of polynomial Fibonacci–Hessenberg matrices are introduced. The notion of two-dimensional Fibonacci polynomial array is introduced and three classes of polynomial Fibonacci–Hessenberg matrices satisfying this property are given.     

Some combinatorial identities via Fibonacci numbers

The Pascal matrix and the Stirling matrices of the first kind and the second kind obtained from the Fibonacci matrix are studied, respectively. Also, we obtain combinatorial identities from the matrix representation of the Pascal matrix, the Stirling matrices of the first kind and the second kind and the Fibonacci matrix.     

The generalized relations among the code elements for Fibonacci coding theory

We have considered a class of square Fibonacci matrix of order (p + 1) whose elements are based on the Fibonacci p numbers with determinant equal to +1 or −1. There is a relation between Fibonacci numbers with initial terms which is known as cassini formula. Fibonacci series and the golden mean plays a very important role in the construction of a relatively new space–time theory, which is known as E-infinity theory. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. There already exists a relation between the code matrix elements for the case p = 1 [Stakhov AP. Fibonacci matrices, a generalization of the cassini formula and a new coding theory. Chaos, Solitons and Fractals 2006;30:56–66.]. In this paper, we have established generalized relations among the code matrix elements for all values of p. For p = 2, the correct ability of the method is 99.80%. In general, correct ability of the method increases as p increases.     

On the Transfer Matrix for the Two-Dimensional Ising Model

The eigenvalue problem for the transfer matrix of the two-dimensional Ising lattice, with free boundaries perpendicular to the transfer direction, is solved. The mathematical mechanism which produces the asymptotic degeneracy of the eigenvalue spectrum below the bulk transition temperature is different from the case with cyclic boundary conditions. The results are used to calculate the two-spin correlation function between two spins in the same edge of the lattice, and between two spins in opposite edges.     

State vector solutions for nonaxisymmetric problem of multilayered half space piezoelectric medium

A state space formulation is established for the nonaxisymmetric space problem of transversely isotropic piezoelectric media in a system of cylindrical coordinate by introducing the state vector. Using the Hankel transform and the Fourier series, the state vector equations are transformed into ordinary differential equations. By the use of the matrix methods, the analytical solutions of a single piezoelectric layer are presented in the form of the product of initial state variables and transfer matrix. The applications of state vector solutions are discussed. An analytical solution for a semiinfinite piezoelectric medium subjected to the vertical point forceP _z, horizontal point forceP _x along x-direction and point electric charge Q at the origin of the surface is presented. According to the continuity conditions at the interfaces, the general solution formulation forN-layered transversely isotropic piezoelectric media is given. Project supported by the National Natural Science Foundation of China (Grant No. 59648001).     

On Complex Fibonacci Quaternions

Horadam defined the Fibonacci quaternions and established a few relations for the Fibonacci quaternions. In this paper, we investigate the complex Fibonacci quaternions and give the generating function and Binet formula for these quaternions. Moreover, we also give the matrix representations of them.     

Dynamical bounds for Sturmian Schrödinger operators

The Fibonacci Hamiltonian, that is a Schrödinger operator associated to a Sturmian potential with respect to the golden number has been investigated intensively in recent years. Damanik and Tcheremchantsev developed a method and found a non-trivial dynamical upper bound for transport exponents for this model. This method can be generalized to obtain results for almost all irrational numbers. As a counter example, we exhibit a pathological irrational number with no possible better bound. Moreover, we establish a global lower bound for the lower box dimension of the spectrum that could be used to obtain a dynamical lower bound for irrational numbers with bounded density. To cite this article: L. Marin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Micro‐Macro Modelling of Piezoelectric Fiber Composites

The present work deals with the numerical modeling of 1‐3 periodic composites made of piezoceramic (PZT) fibers embedded in a soft non‐piezoelectric matrix. We especially focus on predicting the effective co‐efficients of the periodic transversely isotropic piezoelectric fiber composites using representative volume element method (unit cell method). The results which are obtained from the FEM technique are compared with analytical homogenization method for different volume fractions. The effective co‐efficients are obtained for rectangular and hexagonal arrangement of unidirectional piezoelectric fiber composites. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resistance distance in wheels and fans

The wheel graph is the join of a single vertex and a cycle, while the fan graph is the join of a single vertex and a path. The resistance distance between any two vertices of a wheel and a fan is obtained. The resistances are related to Fibonacci numbers and generalized Fibonacci numbers. The derivation is based on evaluating determinants of submatrices of the Laplacian matrix. A combinatorial argument is also illustrated. A connection with the problem of squaring a rectangle is described.     

Real matrix representations of complex split quaternions with applications

In this paper, we introduce 8×8 real matrix representations of complex split quaternions. Then, the relations between real matrix representations of split and complex split quaternions are stated. Moreover, we investigate some linear split and complex split quaternionic equations with split Fibonacci and complex split Fibonacci quaternion coefficients. Finally, we also give some numerical examples as applications of real matrix representation of complex split quaternions.     

Global Fibonacci nim

Fibonacci nim is a popular impartial combinatorial game, usually played with a single pile of stones: two players alternate in removing no more than twice the previous player’s removal. The game is appealing due to its surprising connections with the Fibonacci numbers and the Zeckendorf representation. In this article, we investigate some properties of a variant played with multiple piles of stones, and solve the 2-pile case. A player chooses one of the piles and plays as in Fibonacci nim, but here the move-size restriction is a global parameter, valid for any pile.     

Singularities at the tip of a crack on the interface of piezoelectric bodies

Singularities of elastic and electric fields are investigated at the tip of a crack on the interface of two anisotropic piezoelectric media under various boundary conditions on the crack surfaces. The singularity exponents form the spectrum of a certain polynomial pencil, and although explicit formulas are not available, this spectrum is described completely though. The mathematical results apply to problems in fracture mechanics. In this way the Griffith formulas are obtained for increments of energy functionals due to the growth of the crack, and the notion of energy release matrix is introduced. Normalization conditions for bases of singular solutions are proposed to adapt them to energy, stress, and deformation fracture criteria. Connections between these bases are determined, and additional properties of the deformation basis related to the notion of electric surface enthalpy are established. Bibliography: 44 titles. Dedicated to Vsevolod Alekseevich Solonnikov Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 241–271.     

Hexagonal piezoelectric half-space containing a crack

The elegant technique of piezoelectric dislocation layers combined with a method of images is used to consider an arbitrarily mechanically and electrically loaded Griffith-type antiplane shear crack situated parallel to the free-surface of a semi-infinite hexagonally symmetrical piezoelectric crystal. General closed-form representations are derived for the resulting stress and electric displacement fields. The global implications of these are considered. Some illustrative numerical results for particular loads are given and discussed.     

Fibonacci-type polynomial as a trajectory of a discrete dynamical system

Families of polynomials which obey the Fibonacci recursion relation can be generated by repeated iterations of a 2×2 matrix,Q ₂, acting on an initial value matrix,R ₂. One matrix fixes the recursion relation, while the other one distinguishes between the different polynomial families. Each family of polynomials can be considered as a single trajectory of a discrete dynamical system whose dynamics are determined byQ ₂. The starting point for each trajectory is fixed byR ₂(x). The forms of these matrices are studied, and some consequences for the properties of the corresponding polynomials are obtained. The main results generalize to the so-calledr-Bonacci polynomials.     

Real Matrix Representations for the Complex Quaternions

Starting from known results, due to Y. Tian in [5], referring to the real matrix representations of the real quaternions, in this paper we will investigate the left and right real matrix representations for the complex quaternions and we will give some examples in the special case of the complex Fibonacci quaternions.     

压电螺型位错与含非理想界面层圆形夹杂的干涉效应

研究了位于压电材料基体或夹杂中任意点的压电螺型位错与含非理想界面层圆形夹杂的电弹性干涉问题．运用复变函数方法,获得了复势函数的精确解．由广义Peach-Koehler公式,导出了作用在螺型位错上的像力的精确表达式．讨论了不同参数对压电螺型位错的运动和平衡位置的影响规律．研究结果表明,对某些材料组合,当界面层的内界面是非理想界面且界面的非理想度达到一定值时,在基体中靠近界面处会出现两个位错的平衡位置,此现象未在以往研究(不考虑非理想界面)中观察到．     

Dispersion relations of elastic waves in two-dimensional tessellated piezoelectric phononic crystals

A type of two-dimensional tessellated piezoelectric phononic crystal is theoretically studied in this paper, formed by homogeneous piezoelectric and inhomogeneous functionally graded rectangular columns. Firstly, the propagation properties of in-plane and anti-plane Bloch waves in each piezoelectric rectangular column are systematically investigated. Furthermore, the total transfer matrices of Bloch waves are obtained based on transfer matrices of homogeneous piezoelectric and inhomogeneous functionally graded rectangular columns. Finally, the Bloch theorem is used to obtain the dispersion relations of in-plane and anti-plane Bloch waves. The influences of non-dimensional geometrical parameters and gradient profile functions on the dispersion relations are discussed based on the graphically numerical results. Under a special value of modal parameter, the dispersion surfaces and curves of in-plane Bloch waves approximatively have plane and axis symmetries respectively, and an approximate total band-gap of anti-plane Bloch waves arise. The theoretical models and numerical discussions will provide a direct guidance of multi-material 3D printing for inhomogeneous periodic structures with dispersion and band-gaps properties.