Research Article
Symmetry of Solutions of Integral Equation in the Heisenberg Group
Zhaobing Cui
,
Wei Shi*
Issue:
Volume 15, Issue 2, April 2026
Pages:
11-17
Received:
11 February 2026
Accepted:
25 February 2026
Published:
18 March 2026
Abstract: This paper investigates the existence and symmetry properties of solutions to a class of integral equations on the Heisenberg group. Building upon the moving plane method and Hardy-Littlewood-Sobolev type inequalities, we establish symmetry and monotonicity results for positive solutions of the integral equation. This paper extends classical Euclidean results to the Heisenberg group, highlighting profound interactions between geometry and analysis.
Abstract: This paper investigates the existence and symmetry properties of solutions to a class of integral equations on the Heisenberg group. Building upon the moving plane method and Hardy-Littlewood-Sobolev type inequalities, we establish symmetry and monotonicity results for positive solutions of the integral equation. This paper extends classical Euclid...
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Research Article
On the Representation of a Number as the Sum of Two Squares and a Prime
Shihui You*
Issue:
Volume 15, Issue 2, April 2026
Pages:
18-28
Received:
23 March 2026
Accepted:
1 April 2026
Published:
20 April 2026
DOI:
10.11648/j.pamj.20261502.12
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Abstract: This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for the linear Diophantine equations is comparatively easier. The methods involve the combinatorial approximation method to obtain the number of integer solutions of the Diophantine equation with the countable solution sets. First, the number of integer solutions to the linear Diophantine equations with integer sets or sequences as solution sets can be determined by combinatorial methods. Second, the approximate number of integer solutions of the linear Diophantine equations with integer sets or sequences as solution sets is obtained using the approximate method. Finally, for cases where the solution sets and the number of solution sets of the variables in the linear Diophantine equations are either the same or different, we propose the approximate combinatorial method to compute the approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets. The results are consistent with the existing conclusions. The purpose of this paper is to verify the adaptability and correctness of the approximate combinatorial methods for solving the number of integer solutions of the general Diophantine equations with countable solution sets.
Abstract: This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challe...
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Research Article
On (α, β)–Almost Similar Operators in Hilbert Spaces
Beatrice Obiero Adhiambo,
Victor Wanjala*
Issue:
Volume 15, Issue 2, April 2026
Pages:
29-34
Received:
16 March 2026
Accepted:
25 March 2026
Published:
24 April 2026
DOI:
10.11648/j.pamj.20261502.13
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Views:
Abstract: This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of bounded linear operators on a Hilbert space. Key results include the invariance of spectrum, point spectrum, and approximate point spectrum under this relation. The study also defines the class of (α,β)-𝔗 operators, an expansion of the classical 𝔗-operator concept, and explores its relationship with (α,β)--almost similarity. Furthermore, we analyze the connections between similarity, unitary equivalence, and (α,β)--almost similarity, providing conditions under which these relations coincide, particularly for self-adjoint and projection operators. The results contribute to the broader understanding of operator equivalence relations and their spectral implications.
Abstract: This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of boun...
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