Abstract. A set of lines in R n is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in R n , using semidefinite programming to improve the upper bounds on this quantity. Improvements are obtained in dimensions 24 ≤ n ≤ 136. In particular, we show that the maximum number of equiangular lines in R n is 276 for all 24 ≤ n ≤ 41 and is 344 for n = 43. This provides a partial resolution of the conjecture set forth by Lemmens and Seidel (1973).

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Maximum size of equiangular lines in R 19 has been known in the range between 72 to 76 since 1973. Acoording to the nonexistence of strongly regular graph (75, 32,10,16) [1], Larmen-Rogers-Seidel Theorem [20] and Lemmen-Seidel bounds on equiangular lines with common angle 1 3 [22], we can prove that there are no 76 equiangular lines in R 19 . As a corollary, there is no strongly regular graph (76, 35,18,14). Similar discussion can prove that there are no 96 equiangular lines in R 20 .

Tongue diagnosis, a noninvasive examination, is an essential step for syndrome differentiation and treatment in traditional Chinese medicine (TCM). Sublingual vein (SV) is examined to determine the presence of blood stasis and blood stasis syndrome. Many studies have shown that the degree of SV stasis positively correlates with disease severity. However, the diagnoses of SV examination are often subjective because they are influenced by factors such as physicians’ experience and color perception, resulting in different interpretations. Therefore, objective and scientific diagnostic approaches are required to determine the severity of sublingual varices. This study aims at developing a computer-assisted system based on machine learning (ML) techniques for diagnosing the severity of sublingual varicose veins. We conducted a comparative study of the performance of several supervised ML models, including the support vendor machine, K-neighbor, decision tree, linear regression, and Ridge classifier and their variants. The main task was to differentiate sublingual varices into mild and severe by using images of patients’ SVs. To improve diagnostic accuracy and to accelerate the training process, we proposed using two model reduction techniques, namely, the principal component analysis in conjunction with the slice inverse regression and the convolution neural network (CNN), to extract valuable features during the preprocessing of data. Our results showed that these two extraction methods can reduce the training time for the ML methods, and the Ridge-CNN method can achieve an accuracy rate as high as 87.5%, which is similar to that of experienced TCM physicians. This computer-aided tool can be used for reference clinical diagnosis. Furthermore, it can be employed by junior physicians to learn and to use in clinical settings.

Obesity is a prevalent metabolic disease that increases the risk of other diseases, such as hypertension, diabetes, hyperlipidemia, cardiovascular disease, and certain cancers. A meta-analysis of 11 randomized sham-controlled trials indicates that acupuncture had adjuvant benefits in improving simple obesity, and previous studies have reported that acupoint combinations were more useful than single-acupoint therapy. The Apriori algorithm, a data mining-based analysis that finds potential correlations in datasets, is broadly applied in medicine and business. This study, based on the Apriori algorithm-based association rule analysis, found the association rules of acupoints among 11 randomized controlled trials (RCTs). There were 23 acupoints extracted from 11 RCTs. We used Python to calculate the association between acupoints and disease. We found the top 10 frequency acupoints were Extra12, TF4, LI4, LI11, ST25, ST36, ST44, CO4, CO18, and CO1. We investigated the 1118 association rule and found that {LI4, ST36} ≥ {ST44}, {LI4, ST44} ≥ {ST36}, and {ST36, ST44} ≥ {LI4} were the most associated rules in the data. Acupoints, including LI4, ST36, and ST44, are the core acupoint combinations in the treatment of simple obesity.