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Objective: Sciatic scoliosis can be seen in patients with lumbar disc herniation. Percutaneous endoscopic lumbar discectomy (PELD) is a common surgical method for the treatment of lumbar disc herniation. The difference between single-segment lumbar disc herniation and double-segment lumbar disc herniation with Sciatic Scoliosis in adults after PELD needs further study. The aim of this study was to compare the imaging features of single-segment and double-segment lumbar disc herniation with Sciatic Scoliosis in adults and to further explore the clinical outcomes of functional improvement and scoliosis imaging parameters of the two groups after PELD. Methods: Adult patients with lumbar disc herniation with sciatic scoliosis who received PELD from January 2019 to June 2022 were analyzed retrospectively. According to the number of operative segments, the patients were divided into a single-segment group and a double-segment group. Perioperative parameters were observed and compared between the two groups. The Visual Analogue Scale (VAS) score, Oswestry dysfunction index (ODI), Japanese Orthopaedic Association scores (JOA) and imaging parameters of the two groups were recorded and compared before the operation and during the follow-up. Results: A total of 53 patients with single segments and 21 patients with double segments were included in this study. During the follow-up, the VAS score, ODI index and JOA score of the two groups were significantly improved as compared with those before the operation(P < 0. 05). Ninety-two point five percent of single-segment patients and 90.5% of double segment patients returned to normal scoliosis within 12 months after the operation. The operation time, number of intraoperative fluoroscopy times and the amount of intraoperative blood loss in single-segment patients were better than those in double-segment group(P < 0. 05). At the last follow-up, the AVT, CBD and SVA in the double-segment group were 5.2 ± 2.3, 5.1 ± 1.0 and 12.2 ± 3.0 mm, respectively, which were higher than those in the single-segment group (1.9 ± 0.4, 1.1 ± 1.6 and 3.9 ± 2.1 mm) (P < 0. 05). Conclusion: PELD is an effective treatment for single-segment and double-segment lumbar disc herniation with Sciatic scoliosis. Double-segment patients can enjoy similar clinical efficacy to single-segment patients, avoiding complications caused by decompression, fusion, and internal fixation. Scoliosis was corrected spontaneously within 12 months after operation, and the sagittal curve was significantly improved in both groups. The improvement of coronal and sagittal balance in double -segment patients may take longer. [ABSTRACT FROM AUTHOR]

This document discusses the importance of optimization in various fields such as statistics, biology, finance, economics, and control. It highlights the advancements made in optimization theory and methods, including first-order methods and augmented Lagrangian methods. The document also introduces a special issue on "Symmetry in Optimization Theory, Algorithms and Applications," which features five papers on topics such as ensemble learning, ant colony system algorithms, subspace algorithms, consensus problems in multi-agent systems, and dynamic conditional correlation models. The authors express their gratitude to the contributors and reviewers of the special issue. [Extracted from the article]

The covariance matrix estimation plays an important role in portfolio optimization and risk management. It is well-known that portfolio is essentially a convex quadratic programming problem, which is also a special case of symmetric cone optimization. Accurate covariance matrix estimation will lead to more reasonable asset weight allocation. However, some existing methods do not consider the influence of time-varying factor on the covariance matrix estimations. To remedy this, in this article, we propose an improved dynamic conditional correlation model (DCC) by using nonconvex optimization model under smoothly clipped absolute deviation and hard-threshold penalty functions. We first construct a nonconvex optimization model to obtain the optimal covariance matrix estimation, and then we use this covariance matrix estimation to replace the unconditional covariance matrix in the DCC model. The result shows that the loss of the proposed estimator is smaller than other variants of the DCC model in numerical experiments. Finally, we apply our proposed model to the classic Markowitz portfolio. The results show that the improved dynamic conditional correlation model performs better than the current DCC models. [ABSTRACT FROM AUTHOR]

In this paper,we state and prove an analog of Lie product formula in the setting of Euclidean Jordan algebras.

The estimation of the covariance matrix is an important topic in the field of multivariate statistical analysis. In this paper, we propose a new estimator, which is a convex combination of the linear shrinkage estimation and the rotation-invariant estimator under the Frobenius norm. We first obtain the optimal parameters by using grid search and cross-validation, and then, we use these optimal parameters to demonstrate the effectiveness and robustness of the proposed estimation in the numerical simulations. Finally, in empirical research, we apply the covariance matrix estimation to the portfolio optimization. Compared to the existing estimators, we show that the proposed estimator has better performance and lower out-of-sample risk in portfolio optimization. [ABSTRACT FROM AUTHOR]