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Vol.46(6) (2022) page:
Author(s): Pan Shun Lau, K.P. Shum, Tin-Yau Tam and Ngai-Ching Wong  
Size:121KB   download
TitleSome Memories of Ky Fan in the Period of His Directorship at the Institute of Mathematics, Academia Sinica and His Later Visits
Vol.46(6) (2022) page:
Author(s): Jyh-Hao Lee  
Size:87KB   download
TitleA Short Biography of Professor Ky Fan, One of the Greatest Mathematicians of the 20th Century
Vol.46(6) (2022) page:
Author(s): Rugang Ye  
Size:144KB   download
TitleBest Approximation and Fixed Points
Vol.46(6) (2022) page: 691-704
Author(s): S. Chandok and T.D. Narang  

Picard sequences in the context of metric spaces are used to prove several results for the presence of a fixed point. We further extend Brosowski-Meinardus type results on invariant approximation in the case of normed linear spaces as an application of the findings obtained. Some examples are also given to show how the obtained results might be put to use. The findings presented improve on several previously published findings.

Keywords: Compact set; Convex set; Starshaped set; Best approximation.
Size:147KB   download
TitleGelfand Spaces of Uniform Algebras on $n$-torus
Vol.46(6) (2022) page: 705-712
Author(s): Y. Dong, L. Li and T. Liu  

In this paper, we give the classification of the Gelfand space of uniform algebra $\mathcal{A}_\alpha$ on $2$-torus. Moreover, we introduce the uniform algebra $\mathcal{A}_\mathbb{S}$ on $n$-torus $\mathbb{T}^n$. We give the Gelfand space of $\mathcal{A}_\mathbb{S}$.

Keywords: Uniform algebra; $n$-tours; Gelfand space.
Size:127KB   download
TitleInfimum of a Matrix Norm of $A$ Induced by an Absolute Vector Norm
Vol.46(6) (2022) page: 713-720
Author(s): S. Friedland  

We characterize the infimum of a matrix norm of a square matrix $A$ induced by an absolute norm, over the fields of real and complex numbers. Usually this infimum is greater than the spectral radius of $A$. If $A$ is sign equivalent to a nonnegative matrix $B$ then this infimum is the spectral radius of $B$.

Keywords: Absolute norm; Matrix norm; Nonnegative matrices; Spectral radius.
Size:102KB   download
TitleA Sharp Upper-Bound for the Norm of Positive Semi-Definite Block Matrices
Vol.46(6) (2022) page: 721-728
Author(s): M. Gumus and S. Raouafi  

This paper studies the positive semi-definite $2 \times 2$ block matrix $M=\left[\begin{array}{cc}
A & X \\
 X^* & B
\end{array}\right]$ $ \in \mathbb C^{2n \times 2n}$. A new sharp upper bound for $\norm{M}$ is provided under the condition of $X$ being normal, for any unitarily invariant norm $\norm{\cdot}$. A special pattern among the eigenvalues of $M$ when $A + B = kI$, for some $k>0$, is explored.

Keywords: Positive semi-definite matrix; Block matrix; Unitarily invariant norm.
Size:96KB   download
TitleExact Solution of the Bagley-Torvik Equation Using Laplace Transform Method
Vol.46(6) (2022) page: 729-736
Author(s): A.G. Kaplan and M.V. Ablay  

In this paper, the exact solution of the Bagley-Torvik equation which has an important role in fractional order differential equations has been investigated by Laplace transformation method. The Bagley-Torvik equation is transformed into an algebraic equation with Laplace transform. This algebraic equation is solved and the unknown function is found with inverse Laplace transformation. Caputo fractional derivative is considered throughout this work. The examples presented demonstrate the validity and applicability of the Laplace transformation method used to find the exact solution of the Bagley-Torvik equation.

Keywords: Laplace transform method; Fractional order differential equation; Bagley-Torvik equation.
Size:93KB   download
TitleOn the Largest Singular Value of a Matrix and Generalized Inverses
Vol.46(6) (2022) page: 737-748
Author(s): M. Niezgoda  

In this paper we study properties of the largest singular value $ s_1 (\cdot) $ of matrices viewed as a norm on the space of complex matrices. We give a refinement of the submultiplicativity inequality characterizing $ s_1 (\cdot) $. In our approach we use the equality case of the inequality. We introduce a corresponding preorder on the matrix space and show the monotonicity of a certain functional induced by $ s_1 (\cdot) $. We also provide some inequalities by using generalized inverses of matrices.

Keywords: Complex (real) matrix; Largest singular value; Monotone functional; Generalized inverse of matrix.
Size:116KB   download
TitleRelativistic Dissipatons in Integrable Nonlinear Majorana Type Spinor Model
Vol.46(6) (2022) page: 749-770
Author(s): O.K. Pashaev and J.-H. Lee  

By method of moving frame, the relativistic integrable nonlinear model for real, Majorana type spinor fields in 1+1 dimensions is introduced and gauge equivalence of this model with Papanicolau spin model on one sheet hyperboloid is established. In terms of the so called double numbers, the model is represented also as hyperbolic complex relativistic model, in the form similar to the massive Thirring model. By using Hirota's bilinear method, the one dissipaton solution of this model is constructed. We calculate first integrals of motion for this dissipaton and show that it represents a relativistic particle with highly nonlinear mass. Analyzing resonance conditions for scattering of two relativistic dissipatons, we find a solution describing resonant property of the dissipatons.

Keywords: Thirring model; JT gravity; Dissipative soliton; Hirota method; Relativistic particle; Double numbers.
Size:181KB   download
TitleOn Some $K$-g-Fusion Frames in Finite and Infinite Hilbert Spaces
Vol.46(6) (2022) page: 771-782
Author(s): V. Sadri  

In this note, we aim to present a method for reconstruction of generalized fusion frames for operators and error operator with its upper bound. Also, the approximation operator for these frames will be introduced and we study some results about them specially, a new identity about the norm of these frames is presented.

Keywords: Parseval frame; $K$-frame; $K$-g-fusion frame; Erasures.
Size:133KB   download
TitleThe Set of Automorphisms of Pell Forms and Pell Equations
Vol.46(6) (2022) page: 783-800
Author(s): A. Tekcan and G. Biberoglu  

Let $d=k^{2}+4$ for some integer $k\geq 2$. In this work, we first determined the set of automorphisms of the Pell form $F_{\Delta }(x,y)=x^{2}-dy^{2}$ of discriminant $\Delta =4d$. Later, we deduced the set of all integer solutions of the Pell equations $F_{\Delta }(x,y)=\pm 1$ and $ F_{\Delta }(x,y)=\pm k^{2}$.

Keywords: Quadratic form; Pell form; Automorphism; Pell equation.
Size:178KB   download
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