Generalizations of Choi's Orthogonal Latin Squares and Their Magic Squares

Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on two $3 \times 3$ orthogonal Latin squares. In this paper, we give a new generalization of Choi's orthogonal Latin squares of order 9 to orthogonal Latin squares of size $n^2$ using the Kronecker product including Lih's construction. We find a geometric description of Chois' orthogonal Latin squares of order 9 using the dihedral group $D_8$. We also give a new way to construct magic squares from two orthogonal non-diagonal Latin square, which explains why Choi's Latin squares produce a magic square of order 9.Comment: 18 pages revised slightly from Dec. 5, 2018 versio

Recent results on Choi's orthogonal Latin squares

Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on the two $3 \times 3$ orthogonal Latin squares. In this paper, we give a new generalization of Choi's orthogonal Latin squares of order 9 to orthogonal Latin squares of size $n^2$ using the Kronecker product including Lih's construction. We find a geometric description of Choi's orthogonal Latin squares of order 9 using the dihedral group $D_8$. We also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi's Latin squares produce a magic square of order 9

Relationships between cognitive function and body composition among community-dwelling older adults: a cross-sectional study

Abstract Background Previous studies reported mixed results regarding the association between cognition and body weight in late life. We evaluated the relationships between cognitive function and body composition among community-dwelling older adults. Methods Three hundred twenty subjects (≥65 years, women 53%) with available data of cognitive function and body composition from 2010 Hallym Aging Study. Cognitive function was assessed using Korean Mini-Mental State Examination (K-MMSE). Dual-energy X-ray absorptiometry (DEXA) was used for measuring body composition including body fat and lean body mass. Anthropometric measurements and laboratory data were collected in clinical examination. Body composition variables were divided into sex-specific tertiles, and examined by multivariable logistic regression. Results Among female, the highest tertile group of fat mass and second tertile group of total lean body mass were associated with lower risk for cognitive impairment compared to the respective first tertile groups (odds ratios, 0.23 and 0.09, respectively; 95% confidence intervals, 0.04–0.88 and 0.01–0.44, respectively) after adjusting for confounding factors. In male, higher arm bone mineral content was associated with lower risk for cognitive impairment, but significance was lost after adjusting for adiponectin, age, and education. Conclusions Higher fat mass and lean body mass were associated with lower risk of cognitive impairment in older women. These observations suggest that body fat and lean mass later in life might be beneficial for cognition

Additional file 1: Figure S1. of Relationships between cognitive function and body composition among community-dwelling older adults: a cross-sectional study

Flowchart of the selection of the study population. (DOCX 23 kb

High prevalence of TP53 mutations is associated with poor survival and an EMT signature in gliosarcoma patients.

Gliosarcoma (GS) is a rare variant (2%) of glioblastoma (GBM) that poses clinical genomic challenges because of its poor prognosis and limited genomic information. To gain a comprehensive view of the genomic alterations in GS and to understand the molecular etiology of GS, we applied whole-exome sequencing analyses for 28 GS cases (6 blood-matched fresh-frozen tissues for the discovery set, 22 formalin-fixed paraffin-embedded tissues for the validation set) and copy-number variation microarrays for 5 blood-matched fresh-frozen tissues. TP53 mutations were more prevalent in the GS cases (20/28, 70%) compared to the GBM cases (29/90, 32%), and the GS patients with TP53 mutations showed a significantly shorter survival (multivariate Cox analysis, hazard ratio=23.9, 95% confidence interval, 2.87-199.63, P=0.003). A pathway analysis showed recurrent alterations in MAPK signaling (EGFR, RASGRF2 and TP53), phosphatidylinositol/calcium signaling (CACNA1s, PLCs and ITPRs) and focal adhesion/tight junction (PTEN and PAK3) pathways. Genomic profiling of the matched recurrent GS cases detected the occurrence of TP53 mutations in two recurrent GS cases, which suggests that TP53 mutations play a role in treatment resistance. Functionally, we found that TP53 mutations are associated with the epithelial-mesenchymal transition (EMT) process of sarcomatous components of GS. We provide the first comprehensive genome-wide genetic alternation profiling of GS, which suggests novel prognostic subgroups in GS patients based on their TP53 mutation status and provides new insight in the pathogenesis and targeted treatment of GS. Exp Mol Med 2017 Apr 14; 49(4):e317