Under mild conditions on n, p, we give a lower bound on the number of n-variable balanced symmetric polynomials over finite fields GF (p), where p is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we conjecture that X(2 t , 2 t+1 l − 1) are the only nonlinear balanced elementary symmetric polynomials over GF (2), whereand we prove various results in support of this conjecture.

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ObjectiveThe definite prognostic role of p-STAT3 has not been well defined. We performed a meta-analysis evaluating the prognostic role of p-STAT3 expression in patients with digestive system cancers.MethodsWe searched the available articles reporting the prognostic value of p-STAT3 in patients with cancers of the digestive system, mainly including colorectal cancer, gastric cancer, hepatocellular carcinoma, esophagus cancer and pancreatic cancer. The pooled hazard ratios (HRs) with 95 % confidence intervals (95 % CIs) of overall survival (OS) and disease-free survival (DFS) were used to assess the prognostic role of p-STAT3 expression level in cancer tissues. And the association between p-STAT3 expression and clinicopathological characteristics was evaluated.ResultsA total of 22 studies with 3585 patients were finally enrolled in the meta-analysis. The results showed that elevated p-STAT3 expression level predicted inferior OS (HR=1.809, 95% CI: 1.442-2.270, P<0.001) and DFS (HR=1.481, 95% CI: 1.028-2.133, P= 0.035) in patients with malignant cancers of the digestive system. Increased expression of p-STAT3 is significantly related with tumor cell differentiation (Odds ratio (OR) =1.895, 95% CI: 1.364-2.632, P<0.001) and lymph node metastases (OR=2.108, 95% CI: 1.104-4.024, P=0.024). Sensitivity analysis suggested that the pooled HR was stable and omitting a single study did not change the significance of the pooled HR. Funnel plots and Egger’s tests revealed there was no significant publication bias in the meta-analysis.ConclusionPhospho-STAT3 might be a prognostic factor of patients with digestive system cancers. More well designed studies with adequate follow-up are needed to gain a thorough understanding of the prognostic role of p-STAT3.

Recently, Tu and Deng proposed a combinatorial conjecture about binary strings, and, on the assumption that the conjecture is correct, they obtained two classes of Boolean functions which are both algebraic immunity optimal, the first of which are also bent functions. The second class gives balanced functions, which have optimal algebraic degree and the best nonlinearity known up to now. In this paper, by using three different approaches, we prove the conjecture for many cases. We also propose some problems about the weight equations which are related to this conjecture. Because of the scattered distribution, we predict that an exact count is difficult to obtain, in general.