Open Access

Articles

Article ID: 4409

On numerical radius inequalities via McCarthy inequality

by Muhammad Fazeel Anwar, Saira Iqbal, Muhammad Saeed Akram

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This paper aims to establish new inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space. By employing a generalized form of the McCarthy inequality, we derive several upper bounds for the numerical radius of a single operator as well as for expressions involving sums and products of operators. The obtained results extend and refine a number of existing inequalities in the literature. In particular, many known numerical radius inequalities are recovered as special cases of our results, thereby providing a unified framework for their analysis. The refinement is based on the Akkouchi-Ighachane version of the Hölder-McCarthy inequality, which allows the usual McCarthy term to be replaced by a smaller parameter-dependent expression before taking the supremum. This gives a common refinement mechanism for estimates involving a single operator, Cartesian decompositions, finite sums, block matrices and the Euclidean operator radius. We also state explicitly the parameter choices which recover the earlier inequalities and include simple finite-dimensional comparisons showing that, for suitable non-normal matrices, the refined bounds may be strictly sharper than the corresponding classical estimates. These comparisons confirm that the refinement provides a usable quantitative improvement, rather than only a formal parameter extension of existing bounds in concrete cases.

Open Access

Article

Article ID: 4390

Not loss of stability but steady-state shift: A systems biology explanation for elevated fasting blood glucose in type 2 diabetes

by Guanyu Wang

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Elevated fasting blood glucose is a hallmark clinical feature of prediabetes and type 2 diabetes, reflecting underlying pathological alterations in the body’s glucose-insulin regulatory system. This study employs a validated mathematical model of the glucose-insulin negative feedback loop to investigate the fundamental mechanisms of this elevation from a systems biology perspective. We specifically analyzed whether hyperglycemia arises from decreased system stability or an upward shift in the steady-state set-point. Our findings demonstrate that the intrinsic stability of the glucose-insulin regulatory system remains largely unchanged from healthy states through early-stage diabetes. Contrary to the hypothesis of stability decay, the essence of elevated fasting glucose is an upward shift of the steady-state level driven primarily by hepatic insulin resistance, which increases basal hepatic glucose output. While system stability is preserved during early progression, the dynamic coupling between glucose and insulin weakens; low-frequency oscillations inherent to the healthy system gradually diminish and eventually disappear as insulin resistance intensifies, signaling a decoupling of dynamics before stability loss. Significant weakening of system stability, characterized by a markedly reduced convergence rate following perturbation, occurs only in late-stage diabetes accompanied by severe pancreatic damage and compromised insulin secretory capacity. These results redefine the pathophysiological understanding of fasting hyperglycemia, suggesting that early therapeutic strategies should target recalibrating the glucose set point and hepatic sensitivity rather than bolstering system stability. This research provides valuable theoretical insights into the pathogenesis of diabetes and highlights potential dynamic biomarkers for monitoring disease progression.

Open Access

Article

Article ID: 4327

Channeling paths identification and mitigation by using integrated profile control and flooding based on meshless connection element method

by Wei Yong, Zhijie Wei, Jian Zhang, Wensheng Zhou, Yuyang Liu, Wentao Zhan, Wei Liu, Yixin Zhang, Jiaxu Mei

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Addressing the challenge of quantitatively identifying deep thief zones in mature oilfields during the high water-cut stage, this study proposes a robust quantitative characterization model for thief channels based on the non-Euclidean, meshless Connection Element Method (CEM) to directly guide in-depth fluid diversion and integrated profile control and flooding treatments through automated flow path tracking rooted in a directed-graph depth-first search algorithm. To systematically capture the complex subsurface topological network, a comprehensive multi-parameter connectivity parameter system was constructed by integrating key dynamic indicators, including connection conductivity, connection volume, and path splitting coefficients. By dynamically coupling these parameters with the field-wide Lorentz coefficient, a dimensionless channeling factor was defined to establish a rigorous four-level quantitative standard—ranging from extreme channeling to matrix seepage—thereby successfully advancing preferential pathway evaluation from qualitative inference to spatial grading and precise localization. Quantitative validation against conventional commercial grid-based compositional simulators demonstrates the superior fidelity and performance forecasting efficiency of the proposed method: the CEM framework achieves an exceptionally accurate water-cut prediction Root Mean Square Error (RMSE) of approximately 3.8%. Crucially, by abstracting continuous domains into streamlined networks, it drastically compresses structural degrees of freedom, successfully accelerating the operational execution runtime from 7.3 s to a mere 1.6 s. Ultimately, this work provides a computationally highly efficient, physically sound novel approach for the reliable mapping and graded quantification of deep dominant channeling pathways in mature heterogeneous oilfields.