Quantifying Post-peak Behavior of Rocks with Type-I, Type-II, and Mixed Fractures by Developing a Quasi-State-Based Peridynamics

Tian, Feng; Liu, Zaobao; Zhou, Jinxin; Chen, Lin; Feng, Xia-ting

doi:10.1007/s00603-024-03788-8

Cited by 6 publications

(1 citation statement)

References 62 publications

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“…Convex functions have been a central topic in mathematics for a significant period. Their growing popularity is likely due to their essential role in optimization, machine learning, and various applications across scientific and engineering disciplines [7][8][9][10][11]. There is a profound and intricate relationship between the theory of inequalities and convex functions.…”

Section: Introductionmentioning

confidence: 99%

Some New Estimations of Left and Right Interval Fractional Pachpatte’s Type Integral Inequalities via Rectangle Plane

Alshehry,

Ciurdariu,

Saber

et al. 2024

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Inequalities involving fractional operators have been an active area of research, which is crucial in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, we initially presented a new class that is known as coordinated left and right ℏ-pre-invex interval-valued mappings (C·L·R-ℏ-pre-invex Ι·V-M), as well classical convex and nonconvex are also obtained. This newly defined class enabled us to derive novel inequalities, such as Hermite–Hadamard and Pachpatte’s type inequalities. Furthermore, the obtained results allowed us to recapture several special cases of known results for different parameter choices, which can be applications of the main results. Finally, we discussed the validity of the main outcomes.

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