Tamkang Journal of Mathematics

Inverse problem for Dirac operators with a small delay

2024-09-18T04:55:00+00:00

This paper addresses inverse spectral problems associated with Dirac-type operators with a constant delay, specifically when this delay is less than one-third of the interval length. Our research focuses on eigenvalue behavior and operator recovery from spectra. We find that two spectra alone are insufficient to fully recover the potentials. Additionally, we consider the Ambarzumian-type inverse problem for Dirac-type operators with a delay. Our results have significant implications for the study of inverse problems related to the differential operators with a constant delay and may inform future research directions in this field.

Antimagic graph constructions with triangle and three-path unions

2024-09-16T05:23:53+00:00

Let $G=(V,E)$ be a graph with $p$ edges and let $f$ be a bijective function from $E(G)$ to $\{1,2,\dots ,p\}$.
For any vertex $v$, let $\phi_f(v)$ denote the sum of $f(e)$ over all edges $e$ incident to $v$.
If $\phi_f(v)\not=\phi_f(u)$ holds for any two distinct vertices $u$ and $v$, then $f$ is called an antimagic labelling of $G$.
A graph $G$ is deemed antimagic if it admits such a labelling.
In this study, we investigate the antimagic properties of graph unions, particularly focusing on structures composed of multiple triangles and 3-paths. We employ Skolem sequences and extended Skolem sequences to construct antimagic labelling for these graph unions. Specifically, we demonstrate that for any integer $n\geq 9$, the graph formed by the disjoint union of $m$ copies of the triangle $C_3$ and $n$ copies of the path $P_3$ is antimagic for $m\geq \lceil\frac{n}{3}\rceil$.

Application of Lipschitz viscosity solutions for higher-order partial differential equations containing the special Lagrangian operator

2025-01-05T09:13:52+00:00

Using the Lipschitz continuity of a class of viscosity solutions‎, ‎we find a kind of viscosity solution for some higher-order partial differential equations containing the special Lagrangian operator‎. ‎Additionally‎, ‎we extend this analysis to equations that simultaneously contain the special Lagrangian and some other operators including ‎Laplacian.

New studies on a family of $q$-weighted Bergman spaces on the unit disk and applications

2025-05-07T05:34:34+00:00

In this paper, we give a family of $q$-weighted
Bergman spaces
$\left\{\mathcal{A}_{{\alpha,n,q}}\right\}_{n\in\N}$ which
satisfies the continuous inclusion
$\mathcal{A}_{{\alpha,n,q}}\subset...\subset\mathcal{A}_{{\alpha,1,q}}\subset
\mathcal{A}_{{\alpha,0,q}}=\mathcal{A}_{{\alpha,q}}$, where
$\mathcal{A}_{{\alpha,q}}$ the $q$-weighted Bergman space.
Moreover, a more general uncertainty inequality of the
Heisenberg-type for the space $\mathcal{A}_{{\alpha,n,q}}$ is
given by considering the operators
$\nabla_{\alpha,n,q}:=\nabla^n_{\alpha,q}$ and
$L_{\alpha,n,q}:=L^n_{\alpha,q}$. Also, we study on
$\mathcal{A}_{{\alpha,q}}$ the $q$-Toeplitz operators, the
$q$-Hankel operators and the $q$-Berezin operators. Finally, an
application of the theory of extremal function and reproducing
kernel of Hilbert space is given and we use it to establish the
extremal function associated to an bounded linear operator
$T:\mathcal{A}_{{\alpha,q}}\rightarrow H$, for any Hilbert space
$H$. As application, we come up with some results regarding the
extremal functions associated to the difference operator
$Tf(z):=\frac{1}{z}(f(z)-f(0))$ and
$Tf(z):=\frac{1}{1+q}(f(z)-f(-z)).$

Inequalities for generalized normalized δ-Casorati curvatures of Quasi Bi-Slant Submanifolds of Generalized Complex Space Forms

2024-10-15T09:37:09+00:00

In this article, we establish sharp inequalities involving generalized normalized δ-Casorati curvatures for quasi bi-slant submanifolds
in generalized complex space forms and also characterize the submanifolds for which the equality holds. In addition, we’ve extended the sam inequalities to other types of submanifolds within the same geometric space. These include slant, invariant, anti-invariant, semi-slant, hemislant and bi-slant submanifolds.

A study on reaction-diffusion singular perturbation problems with non-classical conditions using collocation method

2025-04-11T03:00:18+00:00

This article discuss about a numerical study to find the solution of second order reaction diffusion singular perturbation problem with non-local boundary conditions using cubic B-spline functions and collocation technique. Shishkin mesh is used to construct layer adapted meshes. The non-local boundary conditions are discretized using Trapezoidal rule. The study establishes that the discussed scheme's result is uniformly convergent up to second order in the supremum norm. To establish the efficiency of the discussed method, two numerical examples are presented along with their results in the form of tables and figures.