Singular Boolean networks are introduced in this paper. Via semi-tensor product of matrices and the matrix expression of logical functions, two kinds of the condensed algebraic expressions of singular Boolean networks are obtained. The normalization problem of singular Boolean networks is addressed; that is, under what condition singular Boolean networks can be converted into normal Boolean networks with algebraic restrictions. Then one sufficient condition and one necessary and sufficient condition are derived for the normalization problem. Furthermore, the solvability of singular Boolean networks is discussed and the concept of admissible initial values of singular Boolean networks is presented. Finally, fixed points and cycles of singular Boolean networks are also investigated.

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We mainly aimed to preliminarily explore the prognostic values of nutrition-based prognostic scores in patients with advanced hilar cholangiocarcinoma (HCCA).We retrospectively analyzed 73 cases of HCCA, who underwent percutaneous transhepatic biliary stenting (PTBS) combined with I seed intracavitary irradiation from November 2012 to April 2017 in our department. The postoperative changes of total bilirubin (TBIL), direct bilirubin (DBIL), alanine aminotransferase (ALT), aspartate aminotransferase (AST), alkaline phosphatase (ALP), and albumin (ALB) were observed. The preoperative clinical data were collected to calculate the nutrition-based scores, including controlling nutritional status (CONUT), C-reactive protein/albumin ratio (CAR), and prognostic nutritional index (PNI). Kaplan-Meier curve and Cox regression model were used for overall survival (OS) analyses.The serum levels of TBIL, DBIL, ALT, AST, and ALP significantly reduced, and ALB significantly increased at 1 month and 3 months postoperatively. The median survival time of the cohort was 12 months and the 1-year survival rate was 53.1%. Univariate analysis revealed that the statistically significant factors related to OS were CA19-9, TBIL, ALB, CONUT, and PNI. Multivariate analysis further identified CA19-9, CONUT, and PNI as independent prognostic factors.Nutrition-based prognostic scores, CONUT and PNI in particular, can be used as predictors of survival in unresectable HCCA.