A mixed finite element method based on the H^1 -Galerkin approach is employed to address the Cahn-Hilliard (CH) equation, which is defined as a fourth-order, time-dependent, nonlinear partial differential equation. The implementation of a splitting technique is made possible through the introduction of an intermediate function, while cubic spline space is utilized as the test and trial space in the Galerkin framework. Error estimates of optimal order are established for both the semi-discrete and fully discrete schemes.
Let 𝕌𝕊ℂ^2n × 2n be the set of 2n × 2n unitary symplectic matrices, that is, 𝕌𝕊ℂ^2n × 2n={A ∈ℂ^2n × 2n | A^HA=I_2n, A^HJA=J . }, where J=[ [ 0 I_n; -I_n 0; ]] is a unit symplectic matrix. In this paper, we first consider an inverse problem (Problem I): Given X_2,Z_2∈ℂ^2n × k_2 and Y_2,M_2∈ℂ^2n × l_2, find a matrix A∈ S such that f(A)=‖ AX_2 -Z_2‖ ^2+‖ Y_2^H A-M_2^H‖ ^2 is minimized, where S={ A∈𝕌𝕊ℂ^2n × 2n| ‖ AX_1-Z_1‖ ^2+‖ Y_1^H A-M_1^H‖ ^2=min. } , and X_1,Z_1∈ℂ^2n × k_1, Y_1,M_1∈ℂ^2n × l_1. Further, we will consider a corresponding best approximate problem (Problem II): Given Ã∈ℂ^2n× 2n , find Â∈ S_E such that Â=min _A∈ S_E‖Ã- A‖ , where S_E is the solution set of Problem I. By applying the singular value decompositions of matrices, the solution set S_E is derived and the explicit formula of the solution to Problem II is provided. Finally, the correctness and effectiveness are verified by two numerical examples.
In this article, we study the family of elliptic curves given by E_p: y^2=x^3-5px for an odd prime p 5 and try to find the conditions under which E_p has rank one. Specifically, we have shown that if p ≡ 3, 27 40 such that 2(p+5) is a perfect square or if p ≡ 11, 19 40 such that (5+4p) is a perfect square, then the Mordell-Weil rank of E_p is exactly one. Finally, assuming the parity conjecture, we have proved that if p ≡ 3, 11, 19, 27 40 , then the rank of E_p is always one.
The aim of this paper is to establish the relation between the relative global dimension of an abelian category 𝒜 and that of the category Rep (Q,𝒜) of all representations of a quiver Q with values in 𝒜 . As an application, for a left rooted quiver Q, we obtain another version of relative global dimension of Rep (Q,𝒜) . In particular, taking 𝒜 to be the category of all left modules over a ring R, one can obtain the weak (resp., Gorenstein, Gorenstein weak, PGF) global dimension of Rep(Q, R). Moreover, we consider the case that Q is a quiver bound by some relations. A lower bound of the relative global dimension of Rep (Q,𝒜) is obtained in terms of that of 𝒜 .
A class of boundary blow-up elliptic problem with the weight function and the exponential term have been investigated. The existence, uniqueness and asymptotic estimates of the solution near the boundary have been obtained. Our proofs are mainly based on the method of upper and lower solutions and the comparison principle. And some known results have been generalized.
Expanders are highly connected sparse graphs which have a wide range of applications in computer science and mathematics. In this paper, we construct a family of expander graphs using the tensor product of cycle graph with complete graph both of which are non-expanders and prove that a particular case of this family of expanders is a bipartite expander. We also prove the sufficient condition for the tensor product of two regular graphs to be an edge-expander graph. Edge-expanders can be constructed using this result. Finally, we construct an unbalanced biregular bipartite expander using edge-vertex incidence graph. This newly constructed expander can be used as an error correcting code.
Recently, Mondal et al. introduced several novel topological indices based on neighborhood degree sum of vertices such as neighborhood version of the first, second and third Zagreb indices. The neighborhood version of the first, second and third Zagreb indices is defined as ∑ _x∈ V(G)S^2_G(x) , ∑ _xy∈ E(G)S_G(x) S_G(y) and ∑ _xy∈ E(G)[S_G(x)+S_G(y)] , respectively, where S_G(x) is the neighborhood degree sum of the vertex x which is the sum of degrees of all of its neighboring vertices. In this paper, the minimum values of neighborhood version of the first, second and third Zagreb indices are determined within the set of trees having given order and maximum vertex degree. We further establish the corresponding minimal trees.
Recently, a bipartite walk on a bipartite graph G was introduced, and the periodicity of its time evolution matrix was discussed. We present a formula for the time evolution matrix of the bipartite walk on a bipartite graph, and so give its spectra. As an application, we present a formula for the time evolution matrix of the bipartite walk on a semiregular bipartite graph, and so give its spectra. Furthermore, we study a relation between the Grover walk and the bipartite walk on a bipartite graph. Finally, we present a formula for the time evolution matrix of the bipartite walk on a regular covering of a bipartite graph.
In this article, we study the isoperimetric problem in two-dimensional spherically symmetric Randers space. We prove that circles centred at origin are local solutions of the isoperimetric problem with respect to Holmes–Thompson volume form.
In this article, we consider Kantorovich and Durrmeyer-type extensions of the discrete operators defined by İçöz et al. [Filomat 30 (2) (2016), 429–440], which are associated with Miller–Lee polynomials. The primary focus of our study is to obtain the complete asymptotic expansions for these operators. For this purpose, we derive explicit expressions for their moments and central moments in terms of Stirling numbers. Additionally, we examine their convergence behavior, derive quantitative Voronovskaya-type estimates, and establish weighted approximation results. The theoretical results are validated through graphical illustrations and numerical tables.
In this study, we investigate a two-point boundary value problem involving fractional derivatives (in Caputo sense) of different orders. To address this problem, we employ a finite difference scheme on a uniform mesh, where the fractional derivative of order α∈ (0,1) is approximated using the L1 scheme, while the derivative of order β∈ (1,2) is discretized using an explicit finite difference approximation. A rigorous analysis is carried out to establish the invertibility of the resulting discrete system and to derive theoretical error estimates for the proposed scheme. Numerical experiments are performed to assess its performance, indicating a linear rate of convergence consistent with the theoretical results. To further enhance accuracy, Richardson extrapolation is applied. In addition, the model is extended to incorporate nonlocal boundary conditions, and a corresponding numerical scheme is developed along with numerical experiment.
We study symmetric algebras of graded ideals of deviation three in a polynomial ring. For a class of such ideals, we construct explicit graded free resolutions of the symmetric algebra. Our results extend known explicit constructions, previously available for certain families of ideals of deviation at most two, to a new specific class of ideals of deviation three.
The growth of solutions of differential-difference equations is studied by using Nevanlinna theory. On the one hand, the growth of entire solutions of differential-difference equation f^n(z)f^(k)(z)+q(z)e^Q(z)f(z+c)=u(z)e^v(z) is obtained, where q, Q, u, v are polynomials such that Q(z) is not a constant and q(z)u(z)≢0 and c is a constant, n≥ 1 , k≥ 0 are integers, which improves the results of Chen et al. [Rocky Mountain J. Math. 52, 1251-1266 (2022)]. On the other hand, the growth and form of entire solutions of differential-difference equation f^n(z)+ω f^n-1(z)f^(k)(z)+q(z)e^Q(z)𝒟(z,f)=p_1(z)e^λ _1z+p_2(z)e^λ _2z are described, where 𝒟(z,f)=∑ _i=0^lb_if^(t_i)(z+c_i) , b_i , c_i∈ℂ , t_i(i=0,...,l) are non-negative integers, n, k(≥ 1) are integers, p_1, p_2 , λ _1, λ _2(λ _1λ _2) are non-zero constants, ω is a constant, and q(≢0) , Q(z) are polynomials such that Q(z) is non-constant, which extends the results of Erajikkappa et al. [Electron. J. Differ. Equations. 2025, 1-10 (2025)]. In addition, some examples are given to illustrate the accuracy of the results.
Let G be a finite, simple, undirected graph. The maximum degree matrix M(G) and the minimum degree matrix m(G) assign to each adjacent pair of vertices the maximum and minimum of their degrees, respectively, thereby encoding extremal degree interactions between vertices. In this paper, the spectral and energy properties of these degree-based matrices are examined in a unified framework. A complete spectral characterization of graph regularity via six equivalent conditions on M(G) and m(G) is established, together with results on irreducibility, inertia, spectral radius, diagonal dominance, and energy. Closed-form spectral and energy expressions are derived for complete graphs, wheel graphs, star graphs, paths, and trees. The central result proves that E_m(G) < E_M(G) for every connected non-regular graph, with equality if and only if G is regular; a quantitative lower bound on the energy gap in terms of edge-degree variance is also established. Sharp extremal bounds with unique equality cases are obtained for trees, and tight asymptotic formulas for all standard graph families are proved rigorously. A systematic four-way comparison of E_M(G) , E_m(G) , Laplacian energy LE(G), and eccentricity energy E_ε(G) demonstrates that these invariants capture genuinely complementary local and global structural information. These results contribute to the spectral study of degree-based extremal matrices and open several directions for further investigation.
Motivated by the results of [6], we introduce a new class of operators, called Q -symmetric operators, defined via the * -Duggal transform. We provide a characterization of this class and study its basic properties. We also show that it contains several well-known classes of operators, such as quasinormal operators, idempotents, partial isometries, contractions, cyclic subnormal operators, and generalized quasi-adjoint operators. In the case of hyponormal operators, the generalized quasi-adjoint property is equivalent to the corresponding property of their Duggal transform. Furthermore, we prove several results concerning the ultraweak closures of the ranges of some elementary operators.
