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Mock‐Lie algebras, a unique class of commutative algebras that satisfy the Jacobi identity, are gaining attention for their potential applications in various mathematical contexts. This study presents a matrix representation of the centroid for Mock‐Lie algebras in dimensions up to four, offering insights into their structure and interactions. By applying the centroid concept and utilizing matrix operations, we explore the relationships between matrix elements, leading to a better understanding of these algebras. Our findings aim to contribute to the growing interest in Mock‐Lie algebras and their role within the broader algebraic landscape. [ABSTRACT FROM AUTHOR]

Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

This paper is aimed at determining the derivation superalgebra of modular Lie superalgebra K ¯ (n , m). To that end, we first describe the Z -homogeneous derivations of K ¯ (n , m). Then we obtain the derivation superalgebra D e r (K ¯). Finally, we partly determine the derivation superalgebra D e r (K) by virtue of the invariance of K (n , m) under D e r (K ¯). [ABSTRACT FROM AUTHOR]

In order to improve the mechanical strength of bituminous briquette, the modified humic acid was adopted as binder. The influence of heating time and temperature, mass fraction of NaOH, humic acid addition on compressive strength and dropping strength of bri-quette was investigated through the single element experiment and orthogonal experiment. The results showed that the modified briquette had better mechanical strength, the briquetting ratio of fine coal was high. The main influencing factor was humic acid addition, followed by mass fraction of NaOH, heating time and temperature. The optimum technological condition was that the heating time was 130 minutes, the mass fraction of NaOH was 6%, the heating temperature was 90 ²C, the addition of modified humic acid was 15%. Under the optimum condition, the compressive strength and dropping strength of briquette was 1620.2 N, 99.0%. The bituminous briquette bonded by modified humic acid was suitable for industrial production. [ABSTRACT FROM AUTHOR]

Objective: To evaluate the clinical value of the four regions 12-core systemic prostate biopsy., Methods: Ninety-one patients suspected of prostate cancer received four regions 12-core systemic prostate biopsy, and the detection rate and complications were compared with those of the sextant method., Results: More positive results were obtained in detecting prostate cancer with the four regions 12-core method, and the complications were similar to those of the sextant biopsy., Conclusion: The four regions 12-core biopsy was suggested for patients suspected of prostate cancer with enlarged prostate gland above 40 ml.