In this article, we studied the Diophantine equation ax + 8y = z2, where a is a fixed positive integer with a ≡ 3 (mod 4) and x, y, z are non-negative integers. The results show all non-negative integer solutions of this Diophantine equation.

In this article partial modules over rings and tensor product of partial modules and its properties are studied. Left and right partial modules, partial bimodules and their homomorphisms are defined. Next, partial quotient modules are defined and the fundamental homomorphism theorem for partial modules is proven. Also, the tensor product of partial modules and the tensor product of homomorphisms of partial modules is defined. Some properties of the tensor product, the existence of hom-functors and tensor functors are proven. Finally it is shown that the hom-functor and the tensor functor are adjoint functors.

In this paper, we propose an efficient inertial iterative algorithm for solving a system of generalized quasi-variational inequalities (SGQVI) in Hilbert spaces. Using the projection operator technique, we establish an equivalence between SGQVI and fixed-point problems, thus developing a novel inertial method. The algorithm introduces an inertial term to accelerate convergence, and its performance is rigorously analyzed under some mild conditions, including relaxed co-coercivity and Lipschitz continuity of the involved mappings. Our framework unifies and extends several existing models, such as classical variational inequalities, quasi-variational inequalities, and related optimization problems. Some experiments demonstrate the effectiveness of the inertial method, which shows an improvement in convergence speed compared to noninertial methods. Our results generalize and enhance previous research results in the literature, making it more widely applicable in computational mathematics, engineering, and economics.

In connection to Brück conjecture we improve a uniqueness problem for entire functions that share a polynomial with linear differential polynomial.

This paper presents some nontrivial computational results on derived category and Fourier–Mukai technique in algebraic geometry. In particular, it aims at presenting calculations involving spherical twists as a certain class of Fourier–Mukai functors and its cohomological descent on the singular rational cohomology of smooth projective variety. The purpose of this investigation is to present a new perspective, based upon Fourier–Mukai technique, on solving classical problems involving characteristic classes: in particular, the Chern and the Euler characteristics.

We investigate hypersurfaces in the four-dimensional Thurston geometry Nil3 × R, by giving a complete classification of hypersurfaces whose second fundamental form is a Codazzi tensor, they are either parallel or totally geodesic. Furthermore, we prove that the totally umbilical hypersurfaces in Nil3 × R are totally geodesic.

In this paper, we have presented and explored the λ-ideal statistical convergence for sequences on fuzzy cone normed spaces. The related topological and geometrical properties are demonstrated with examples. Through analyzing the criteria based on λ-ideal statistical convergence on these spaces, we aim to establish a comprehensive set of equivalent conditions for sequences that exhibit λ-ideal statistical convergence.

In this paper, we design some generalized split problems which can be seen as an extended form of the split variational inequality problems. We present several iterative algorithms for solving generalized split problems and demonstrate the weak convergence results under some appropriate assumptions within the context of real Hilbert spaces. Finally, we support these results with the help of numerical examples in both the finite and infinite dimensional spaces. As a result of this work, a new direction will be opened in studying split problems.

In this study, we propose the idea of crossed homomorphisms between Lie–Yamaguti superalgebras and develop the Yamaguti cohomology theory of crossed homomorphisms. In light of this, we characterize linear deformations of crossing homomorphisms between Lie–Yamaguti superalgebras using this cohomology. We demonstrate that if two linear or formal deformations of a crossing homomorphism are similar, then their infinitesimals are in the same cohomology class in the first cohomology group. In addition, we show that an order n deformation of a crossing homomorphism can be extended to an order n+1 deformation if and only if the obstruction class in the second cohomology group is trivial.

The exploration of linear Euler harmonic sums had its beginnings with the works of Euler in the 18th century. In 1998 Flajolet and Salvy published a seminal paper in which they developed an approach to the evaluation of linear Euler harmonic sums and gave explicit formulae for several classes of Euler sums in terms of Riemann zeta and other special function values. In this paper, results given by Flajolet and Salvy are extended and generalized.