scholarly journals k-Step Sum and m-Step Gap Fibonacci Sequence

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Maria Adam ◽  
Nicholas Assimakis
Keyword(s):  
Spectral Radius ◽  
Recurrence Formula ◽  
Fibonacci Sequence

For two given integers k, m, we introduce the k-step sumand m-step gap Fibonacci sequence by presenting a recurrence formula that generates the nth term as the sum of k successive previous terms starting the sum at the mth previous term. Known sequences, like Fibonacci, tribonacci, tetranacci, and Padovan sequences, are derived for specific values of k, m. Two limiting properties concerning the terms of the sequence are presented. The limits are related to the spectral radius of the associated {0,1}-matrix.

A MINIMAL LATTICE MODEL RELATED TO PHYLLOTAXY

1996 ◽  
Vol 04 (02) ◽  
pp. 199-218
Author(s):  
X.D. FAN
Keyword(s):  
Growth Rate ◽  
Analytical Solution ◽  
Lattice Model ◽  
Lattice Theory ◽  
Initial Conditions ◽  
Recurrence Formula ◽  
Fibonacci Sequence ◽  
Divergence Angle ◽  
Asymptotic Values ◽  
Minimal Lattice

Structures in phyllotaxy are generalized using some concepts from lattice theory. A minimal lattice model is introduced to simulate the dynamics of the morphology of phyllotaxis. Numerical results indicate that the lattice divergences, which are equivalent to the divergence angle as used in phyllotaxy, converge to constant numbers for most of the growth rate and initial conditions. These numbers are associated with the noble numbers which are related to the general Fibonacci sequence. A method is developed to derive the lattice divergence as a sequence expressed in a recurrence formula. An analytical solution of the latter is obtained which reveals the convergence and asymptotic values of the sequence.

SPECTRAL RADIUS OF NONSINGULAR TRANSFORMATIONS

1989 ◽  
Vol 15 (1) ◽  
pp. 275
Author(s):  
NADKARNI ◽  
ROBERTSON
Keyword(s):  
Spectral Radius

scholarly journals On Gibbs Measures and Spectra of Ruelle Transfer Operators

2017 ◽  
Vol 60 (2) ◽  
pp. 411-421
Author(s):  
Luchezar Stoyanov
Keyword(s):  
Spectral Radius ◽  
Spectral Properties ◽  
Gibbs Measures ◽  
Transfer Operator ◽  
Frobenius Theorem ◽  
Transfer Operators

AbstractWe prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Convergence analysis of gradient-based iterative algorithms for a class of rectangular Sylvester matrix equations based on Banach contraction principle

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adisorn Kittisopaporn ◽  
Pattrawut Chansangiam ◽  
Wicharn Lewkeeratiyutkul
Keyword(s):  
Spectral Radius ◽  
Matrix Equation ◽  
Iterative Algorithms ◽  
Banach Contraction Principle ◽  
Contraction Principle ◽  
Convergence Factor ◽  
Sylvester Matrix ◽  
Iteration Matrix ◽  
Banach Contraction ◽  
The Matrix

AbstractWe derive an iterative procedure for solving a generalized Sylvester matrix equation $AXB+CXD = E$ A X B + C X D = E , where $A,B,C,D,E$ A , B , C , D , E are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation $AXB=C$ A X B = C , the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

Sign in / Sign up

Export Citation Format

Share Document