scholarly journals Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 339 ◽  
Author(s):  
Constantin Buşe ◽  
Donal O’Regan ◽  
Olivia Saierli
Keyword(s):  
Positive Integer ◽  
Unit Circle ◽  
Nonnegative Integer ◽  
Monodromy Matrix ◽  
Time Dependent ◽  
Linear Recurrence ◽  
Ulam Stability ◽  
Periodic Coefficients ◽  
Periodic Sequences

Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) and ( c j ) (with j nonnegative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is defined below. Assume that the eigenvalues x , y , z of the “monodromy matrix” A ( q ) verify the condition ( x − y ) ( y − z ) ( z − x ) ≠ 0 . We prove that the linear recurrence in C x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers–Ulam stable if and only if ( | x | − 1 ) ( | y | − 1 ) ( | z | − 1 ) ≠ 0 , i.e., the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } .

scholarly journals Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 512 ◽  
Author(s):  
Constantin Buşe ◽  
Donal O’Regan ◽  
Olivia Saierli
Keyword(s):  
Positive Integer ◽  
Unit Circle ◽  
Monodromy Matrix ◽  
Time Dependent ◽  
Multiple Eigenvalue ◽  
Linear Recurrence ◽  
Ulam Stability ◽  
Periodic Coefficients ◽  
Periodic Sequences ◽  
Linear Scalar

Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) , and ( c j ) (with j a non-negative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is as is given below. Assuming that the “monodromy matrix” A ( q ) has at least one multiple eigenvalue, we prove that the linear scalar recurrence x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } . Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle.

Hyers–Ulam stability for equations with differences and differential equations with time-dependent and periodic coefficients

2019 ◽  
Vol 150 (5) ◽  
pp. 2175-2188 ◽  
Author(s):  
Constantin Buşe ◽  
Vasile Lupulescu ◽  
Donal O'Regan
Keyword(s):  
Monodromy Matrix ◽  
Time Dependent ◽  
Linear Recurrence ◽  
Ulam Stability ◽  
Identity Matrix ◽  
Periodic Functions ◽  
Periodic Sequences ◽  
First Order ◽  
Linear Differential ◽  
Image Position

AbstractLetqbe a positive integer and let (an) and (bn) be two given ℂ-valued andq-periodic sequences. First we prove that the linear recurrence in ℂ0.1$$x_{n + 2} = a_nx_{n + 1} + b_nx_n,\quad n\in {\open Z}_+ $$is Hyers–Ulam stable if and only if the spectrum of the monodromy matrixTq: =Aq−1· · ·A0(i.e. the set of all its eigenvalues) does not intersect the unit circle Γ = {z∈ ℂ: |z| = 1}, i.e.Tqis hyperbolic. Here (and in as follows) we let0.2$$A_n = \left( {\matrix{ 0 & 1 \cr {b_n} & {a_n} \cr } } \right)\quad n\in {\open Z}_+ .$$Secondly we prove that the linear differential equation0.3$${x}^{\prime \prime}(t) = a(t){x}^{\prime}(t) + b(t)x(t),\quad t\in {\open R},$$(wherea(t) andb(t) are ℂ-valued continuous and 1-periodic functions defined on ℝ) is Hyers–Ulam stable if and only ifP(1) is hyperbolic; hereP(t) denotes the solution of the first-order matrix 2-dimensional differential system0.4$${X}^{\prime}(t) = A(t)X(t),\quad t\in {\open R},\quad X(0) = I_2,$$whereI2is the identity matrix of order 2 and0.5$$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$

scholarly journals Extension of Strongly Regular Graphs

10.37236/878 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ralucca Gera ◽  
Jian Shen
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Regularity Property ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Vertex Degrees

The Friendship Theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let $\lambda$ be any nonnegative integer and $\mu$ be any positive integer. Suppose each pair of friends have exactly $\lambda$ common friends and each pair of strangers have exactly $\mu$ common friends in a party. The corresponding graph is a generalization of strongly regular graphs obtained by relaxing the regularity property on vertex degrees. We prove that either everyone has exactly the same number of friends or there exists a politician who is a friend of everybody. As an immediate consequence, this implies a recent conjecture by Limaye et. al.

scholarly journals Stability of a Functional Equation Deriving from Quadratic and Additive Functions in Non-Archimedean Normed Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Abasalt Bodaghi ◽  
Sang Og Kim
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
General Solution ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Additive Functions ◽  
Normed Spaces

We obtain the general solution of the generalized mixed additive and quadratic functional equationfx+my+fx−my=2fx−2m2fy+m2f2y,mis even;fx+y+fx−y−2m2−1fy+m2−1f2y,mis odd, for a positive integerm. We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces whenmis an even positive integer orm=3.

A NOTE ON ERDŐS–STRAUS AND ERDŐS–GRAHAM DIVISIBILITY PROBLEMS (WITH AN APPENDIX BY ANDRZEJ SCHINZEL)

2013 ◽  
Vol 09 (03) ◽  
pp. 583-599 ◽  
Author(s):  
MACIEJ ULAS ◽  
ANDRZEJ SCHINZEL
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Positive Answer ◽  
Computational Results

In this paper we are interested in two problems stated in the book of Erdős and Graham. The first problem was stated by Erdős and Straus in the following way: Let n ∈ ℕ+ be fixed. Does there exist a positive integer k such that [Formula: see text] The second problem is similar and was formulated by Erdős and Graham. It can be stated as follows: Can one show that for every nonnegative integer n there is an integer k such that [Formula: see text] The aim of this paper is to give some computational results related to these problems. In particular we show that the first problem has positive answer for each n ≤ 20. Similarly, we show the existence of desired n in the second problem for all n ≤ 9. We also note some interesting connections between these two problems.

On the Fourier series of a finitely described convex curve and a conjecture of H. S. Shapiro

1985 ◽  
Vol 98 (3) ◽  
pp. 513-527 ◽  
Author(s):  
T. Sheil-Small
Keyword(s):  
Fourier Series ◽  
Analytic Function ◽  
Positive Integer ◽  
Unit Circle ◽  
Fourier Coefficients ◽  
Convex Curve

AbstractLet F(eis) denote a homeomorphism of the positively oriented unit circle onto a convex curve Γ and let f (eit) = F(eiΦ(t)), where Φ(t) is a non-decreasing function such that Φ(2π) – Φ(0) ≤ 2πN (N a positive integer). If f (eit) has Fourier coefficients cn, we show that is either constant or an N -valent analytic function in {|z| < 1}. We prove that where d is the distance from 0 to Γ and δ(N) > 0 depends only on N. This settles affirmatively a conjecture of H. S. Shapiro.

scholarly journals Hyers–Ulam Stability of Additive Functional Equation Using Direct and Fixed-Point Methods

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
K. Tamilvanan ◽  
G. Balasubramanian ◽  
Nazek Alessa ◽  
K. Loganathan
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Banach Spaces ◽  
Nonnegative Integer ◽  
Additive Functional ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
Fixed Point Methods ◽  
Stability Results

In this present work, we obtain the solution of the generalized additive functional equation and also establish Hyers–Ulam stability results by using alternative fixed point for a generalized additive functional equation χ ∑ g = 1 l v g = ∑ 1 ≤ g < h < i ≤ l χ v g + v h + v i − ∑ 1 ≤ g < h ≤ l χ v g + v h − l 2 − 5 l + 2 / 2 ∑ g = 1 l χ v g − χ − v g / 2 . where l is a nonnegative integer with ℕ − 0,1,2,3,4 in Banach spaces.

scholarly journals Mathematical Modeling of a Class of Symmetrical Islamic Design

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 517
Author(s):  
Mostafa ZAHRI
Keyword(s):  
Mathematical Modeling ◽  
Unit Circle ◽  
Convergence Properties ◽  
Periodic Sequences ◽  
New Model ◽  
One Dimensional ◽  
Geometric Patterns ◽  
Interesting Class ◽  
The One

In this paper, we present a new model for simulating an interesting class of Islamic design. Based on periodic sequences on the one-dimensional manifolds, and from emerging numbers, we construct closed graphs with edges on the unit circle. These graphs build very nice shapes and lead to a symmetrical class of geometric patterns of so-called Islamic design. Moreover, we mathematically characterize and analyze some convergence properties of the used up-down sequences. Finally, four planar type of patterns are simulated.

Small Sets of k-th Powers

1994 ◽  
Vol 37 (2) ◽  
pp. 168-173
Author(s):  
Ping Ding ◽  
A. R. Freedman
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Nonnegative Integer ◽  
Large Integer

AbstractLet k ≥ 2 and q = g(k) — G(k), where g(k) is the smallest possible value of r such that every natural number is the sum of at most r k-th powers and G(k) is the minimal value of r such that every sufficiently large integer is the sum of r k-th powers. For each positive integer r ≥ q, let Then for every ε > 0 and N ≥ N(r, ε), we construct a set A of k-th powers such that |A| ≤ (r(2 + ε)r + l)N1/(k+r) and every nonnegative integer n ≤ N is the sum of k-th powers in A. Some related results are also obtained.

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