Inspired by the $r$-refinement method in isogeometric analysis, in this paper, the authors propose a curvature-based $r$-adaptive isogeometric method for planar multi-sided computational domains parameterized by toric surface patches. The authors construct three absolute curvature metrics of isogeometric solution surface to characterize its gradient information, which is more straightforward and effective. The proposed method takes the internal weights as optimization variables and the resulting parameterization is analysis-suitable and injectivity-preserving with a theoretical guarantee. Several PDEs are solved over multi-sided computational domains parameterized by toric surface patches to demonstrate the effectiveness and efficiency of the proposed method

In this paper, the $L_{2,\infty}$ normalization of the weight matrices is used to enhance the robustness and accuracy of the deep neural network (DNN) with Relu as activation functions. It is shown that the $L_{2,\infty}$ normalization leads to large dihedral angles between two adjacent faces of the DNN function graph and hence smoother DNN functions, which reduces over-fitting of the DNN. A global measure is proposed for the robustness of a classification DNN, which is the average radius of the maximal robust spheres with the training samples as centers. A lower bound for the robustness measure in terms of the $L_{2,\infty}$ norm is given. Finally, an upper bound for the Rademacher complexity of DNNs with $L_{2,\infty}$ normalization is given. An algorithm is given to train DNNs with the $L_{2,\infty}$ normalization and numerical experimental results are used to show that the $L_{2,\infty}$ normalization is effective in terms of improving the robustness and accuracy.

This paper investigates the equivalence problem of bivariate polynomial matrices. A necessary and sufficient condition for the equivalence of a square matrix with the determinant being some power of a univariate irreducible polynomial and its Smith form is proposed. Meanwhile, the authors present an algorithm that reduces this class of bivariate polynomial matrices to their Smith forms, and an example is given to illustrate the effectiveness of the algorithm. In addition, the authors generalize the main result to the non-square case.

Elias, et al. (2016) conjectured that the Kazhdan-Lusztig polynomial of any matroid is logconcave. Inspired by a computer proof of Moll’s log-concavity conjecture given by Kauers and Paule, the authors use a computer algebra system to prove the conjecture for arbitrary uniform matroids.

The Smith form of a matrix plays an important role in the equivalence of matrix. It is known that some multivariate polynomial matrices are not equivalent to their Smith forms. In this paper, the authors investigate mainly the Smith forms of multivariate polynomial triangular matrices and testify two upper multivariate polynomial triangular matrices are equivalent to their Smith forms respectively.

Porous structures widely exist in nature and artifacts, which can be exploited to reduce structural weight and material usage or improve damage tolerance and energy absorption. In this study, the authors develop an approach to design optimized porous structures with Triply Periodic Minimal Surfaces (TPMSs) in the framework of isogeometric analysis (IGA)-based topological optimization. In the developed method, by controlling the density distribution, the designed porous structures can achieve the optimal mechanical performance without increasing the usage of materials. First, the implicit functions of the TPMSs are adopted to design several types of porous elements parametrically. Second, to reduce the cost of computation, the authors propose an equivalent method to forecast the elastic modulus of these porous elements with different densities. Subsequently, the relationships of different porous elements between the elastic modulus and the relative density are constructed. Third, the IGA-based porous topological optimization is developed to obtain an optimal density distribution, which solves a volume constrained compliance minimization problem based on IGA. Finally, an optimum heterogeneous porous structure is generated based on the optimized density distribution. Experimental results demonstrate the effectiveness and efficiency of the proposed method.

Chinese Reminder Theorem (CRT) for integers has been widely used to construct secret sharing schemes for different scenarios, but these schemes have lower information rates than that of Lagrange interpolation-based schemes. In ASIACRYPT 2018, Ning, et al. constructed a perfect $(r,n)$-threshold scheme based on CRT for polynomial ring over finite field, and the corresponding information rate is one which is the greatest case for a $(r,n)$-threshold scheme. However, for many practical purposes, the information rate of Ning, et al. scheme is low and perfect security is too much security. In this work, the authors generalize the Ning, et al. $(r,n)$-threshold scheme to a $(t,r,n)$-ramp scheme based on CRT for polynomial ring over finite field, which attains the greatest information rate $(r-t)$ for a $(t,r,n)$-ramp scheme. Moreover, for any given $2\leq r_1 < r_2\leq n$, the ramp scheme can be used to construct a $(r_1,n)$-threshold scheme that is threshold changeable to $(r',n)$-threshold scheme for all $r'\in \{r_1+1,r_1+2,\cdots,r_2\}$. The threshold changeable secret sharing (TCSS) scheme has a greater information rate than other existing TCSS schemes of this type.

In this paper, by means of the classical Lagrange inversion formula, the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper [J. Wang, Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula, J. Integer Seq., Vol. 22 (2019), Article 19.3.8]. As applications of this inverse relation, the authors not only find a short proof of another nonlinear inverse relation due to Birmajer, et al. (2012), but also set up a few convolution identities concerning the Mina polynomials.