Current Issue : January-March Volume : 2026 Issue Number : 1 Articles : 5 Articles
This paper introduces a specific matrix spectral problem involving four potentials and derives an associated soliton hierarchy using the zero-curvature formulation. The bi- Hamiltonian formulation is derived via the trace identity, thereby establishing the hierarchy’s Liouville integrability. This is exemplified through two systems: generalized combined NLS-type equations and modified KdV-type equations. Owing to Liouville integrability, each member of the hierarchy admits a bi-Hamiltonian structure and, consequently, possesses infinitely many symmetries and conservation laws....
The quadratic regularization technique is widely used in the literature for constructing efficient algorithms, particularly for solving nonsmooth optimization problems. We propose an inexact nonsmooth quadratic regularization algorithm for solving large-scale optimization, which involves a large-scale smooth separable item and a nonsmooth one. The main difference between our algorithm and the (exact) quadratic regularization algorithm is that it employs inexact gradients instead of the full gradients of the smooth item. Also, a slightly different update rule for the regularization parameters is adopted for easier implementation. Under certain assumptions, it is proved that the algorithm achieves a first-order approximate critical point of the problem, and the iteration complexity of the algorithm is O(ε −2). In the end, we apply the algorithm to solve LASSO problems. The numerical results show that the inexact algorithm is more efficient than the corresponding exact one in large-scale cases....
Let Q be the finite classical polar space of rank ν ≥ 1 over Fq, and Qm be the set of all m-dimensional subspaces of Q. In this paper, we introduce the (m, 2)-graph with Qm as its vertex set, and two vertices P,Q are adjacent if and only if P + Q is an (m +2)-dimensional subspace of Q. The full automorphism group of (m, 2)-graph is determined....
The oscillatory behavior of a class of third-order hybrid-type delay differential equations— used to model various real-world phenomena in fluid dynamics, control systems, biology, and beam deflection—is investigated in this study. A novel method is proposed, whereby these complex trinomial equations are reduced to a simpler binomial form by employing solutions of the corresponding linear differential equations. Through the use of comparison techniques and integral averaging methods, new oscillation criteria are derived to ensure that all solutions exhibit oscillatory behavior. These results are shown to extend and enhance existing theories in the oscillation analysis of functional differential equations. The effectiveness and originality of the proposed approach are illustrated by means of two representative examples....
Hyers and Ulam considered the problem of whether there is a true isometry that approximates the ε-isometry defined on a Hilbert space with a stability constant 10ε. Subsequently, Fickett considered the same question on a bounded subset of the n-dimensional Euclidean space Rn with a stability constant of 27ε1/2n . And Vestfrid gave a stability constant of 27nε as the answer for bounded subsets. In this paper, by applying singular value decomposition, we improve the previous stability constants by C √ nε for bounded subsets, where the constant C depends on the approximate linearity parameter K, which is defined later....
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