Identification of a Novel Lipidomic Biomarker for Hepatocyte Carcinoma Diagnosis: Advanced Boosting Machine Learning Techniques Integrated with Explainable Artificial Intelligence. [PDF]
MetabolitesYagin FH +5 more
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Molecular Principles of Proton-Coupled Quinone Reduction in the Membrane-Bound Superoxide Oxidase. [PDF]
J Am Chem SocRiepl D +7 more
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Micro- and Nanoscale Flow Mechanisms in Shale Oil: A Fluid-Solid Coupling Model Integrating Adsorption, Slip, and Stress Sensitivity. [PDF]
Nanomaterials (Basel)Liu Z +5 more
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On an Elementary Proof of Some Asymptotic Formulas in the Theory of Partitions
The Annals of Mathematics, 1942Let \(p(n)\) be the number of partitions of the positive integer \(n\) and let \(p_k(n)\) be the number of partitions of \(n\) into exactly \(k\) summands. The author gives an elementary proof that \(\lim_{n \to \infty} n p(n) \exp\{-\pi(2n/3)^{1/2}\}\) exists and is positive, but does not determine its value (known to be \(48^{-1/2}\)).
openaire +2 more sources
Partitions of Positive Integers: An Elementary Topic of Number Theory
The Mathematics Teacher, 1974The author presents some interesting aspects of number theory and graphing that should be appealing to students.
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Elementary proofs of infinite families of congruences for Merca’s cubic partitions
Ramanujan Journal, 2022exaly
ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS
Bulletin of the Australian Mathematical Society, 2010Michael Hirschhorn
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Evaluating and Comparing Soft Partitions: An Approach Based on Dempster–Shafer Theory
IEEE Transactions on Fuzzy Systems, 2018Thierry Denoeux, Songsak Sriboonchitta
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