Carlitz and Riordan introduced the q-analogue f_q\left(n,k\right) of ballot numbers. In this paper, using the combinatorial interpretation of f_q\left(n,k\right) and constructing injections, we prove that f_q\left(n,k\right) is q-log-concave with respect to n and k, i.e., all coefficients of the polynomials f_q{\left(n,k\right)}^2-f_q\left(n+1,k\right)f_q\left(n-1,k\right) and f_q{\left(n,k\right)}^2-f_q\left(n,k+1\right)f_q\left(n,k-1\right) are non-negative for .

Linear model is a kind of important model in statistics. The estimation and prediction of unknown parameters in the model is a basic problem in statistical inference. According to different evaluation criteria, different forms of predictors are obtained. The simple projection prediction (SPP) boasts simple form while the best linear unbiased prediction (BLUP) enjoys very excellent properties. Naturally, the relationship between the two predictors are considered. We give the necessary and sufficient conditions for the equality of SPP and BLUP. Finally, we study the robustness of SPP and BLUP with respect to covariance matrix and illustrate the application of equivalence conditions by use of error uniform correlation models.

This paper studies a class of Q-type spaces Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)related to logarithmic functions. We first investigate some basic properties of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right). Further, by the aid of Poisson integral and harmonic function spaces H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right), the harmonic extension of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)and the boundary value problem of H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right)are obtained.

[European J. Combin., 2020, 86: Paper No. 103084, 20 pp.] introduced the concept of partial-dual Euler-genus polynomial in the ribbon graphs and gave the interpolation conjecture. That is, the partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating. In fact, [European J. Combin., 2022, 102: Paper No. 103493, 7 pp.] gave two classes of counterexamples to deny the conjecture, and only one or two of the side loops contained in the two classes of bouquets were non-orientable. On the basis of [European J. Combin., 2022, 102: Paper No. 103493, 7 pp.], we further calculate the partial-dual Euler-genus polynomials of two other classes of bouquets. One is non-interpolating, whose side loop has an arbitrary number of non-orientable loops. The other is interpolating, whose side loop has an arbitrary number of both non-orientable loops and orientable loops.