Cloud computing has become prevalent in the sharing of outsourced data due to its strong computing power and storage capacity. Ensuring data security is vitally important when sharing data in the cloud. Recently, numerous broadcast proxy re-encryption (BPRE) schemes have been designed to address the data security issues of such applications. However, there are no any BPRE schemes that have been designed to address the issue of updating the re-encryption key in a dynamic cloud environment. Therefore, the authors propose a lightweight dynamic broadcast proxy re-encryption scheme (LD-BPRE) to address this issue in dynamic settings where the data owner can dynamically change the set of data users and does not need to update the re-encryption key for the new set of data users. In other words, the proxy can reset a re-encryption ciphertext for the new set of data users using the original re-encryption key. This is significant in a dynamic cloud setting and provides convenience for cloud users. The proposed LD-BPRE is lightweight for users with low-power devices as most of the computing overhead is offloaded to the cloud. The authors formally define the LD-BPRE scheme and prove its security under a decision $n$-BDHE assumption in the standard model. Finally, extensive comparisons and experiments indicate that LD-BPRE is efficient and practical.

Characteristic pairs consist of lexicographical Gröbner bases and the minimal triangular sets, called W-characteristic sets, contained in them, and they are good representations of multivariate polynomial ideals in terms of Gröbner bases and triangular sets simultaneously. In this paper, it is studied how to decompose a polynomial set of arbitrary dimensions into characteristic pairs with simple W-characteristic sets, and two algorithms are proposed over fields of characteristic zero and over finite fields respectively. Both of the algorithms rely on the concept of strong regular characteristic divisors, and the one for fields of characteristic zero also uses Lazard lemma to test whether an ideal is radical. Experimental results are presented to illustrate the effectiveness of the proposed algorithms

Recently, linear complementary dual (LCD) codes have garnered substantial interest within coding theory research due to their diverse applications and favorable attributes. This paper directs its attention to the construction of binary and ternary LCD codes leveraging curiosity-driven reinforcement learning (RL). By establishing reward and devising well-reasoned mappings from actions to states, it aims to facilitate the successful synthesis of binary or ternary LCD codes. Experimental results indicate that LCD codes constructed using RL exhibit slightly superior error-correction performance compared to those conventionally constructed LCD codes and those developed via standard RL methodologies. The paper introduces novel binary and ternary LCD codes with enhanced minimum distance bounds. Finally, it showcases how random network distillation aids agents in exploring beyond local optima, enhancing the overall performance of the models without compromising convergence.

Let $\beta$ be an integer and satisfy $0\leq\beta\leq5$. In this paper, the authors prove that the partition polynomial $$\prod_{k=1}^{n}[1+(2+\beta)q^k+q^{2k}]$$ is symmetric and unimodal for $n\geq1$.

The problem of nonconvex and nonsmooth optimization (NNO) has been extensively studied in the machine learning community, leading to the development of numerous fast and convergent numerical algorithms. Existing algorithms typically employ unified iteration schemes and require explicit solutions to subproblems for ensuring convergence. However, these inflexible iteration schemes overlook task-specific details and may encounter difficulties in providing explicit solutions to subproblems. In contrast, there is evidence suggesting that practical applications can benefit from approximately solving subproblems; however, many existing works fail to establish the theoretical validity of such approximations. In this paper, the authors propose a hybrid inexact proximal alternating method (hiPAM), which addresses a general NNO problem with coupled terms while overcoming all aforementioned challenges. The proposed hiPAM algorithm offers a flexible yet highly efficient approach by seamlessly integrating any efficient methods for approximate subproblem solving that cater to specificities. Additionally, the authors have devised a simple yet implementable stopping criterion that generates a Cauchy sequence and ultimately converges to a critical point of the original NNO problem. The proposed numerical experiments using both simulated and real data have demonstrated that hiPAM represents an exceedingly efficient and robust approach to NNO problems.

Direct modeling in CAD offers a promising avenue for model modification through direct interaction. However, a critical impediment to the advancement of direct modeling technology is the inconsistency between modified geometry and unaltered constraints. While several methods have been proposed to address this challenge, they often fail to provide effective solutions, particularly in contexts demanding high model precision. In this paper, the authors introduce a novel approach by integrating tolerance analysis into the constraint update system, followed by the proposal of a constraint update scheme utilizing the extreme value tolerance model and the probability tolerance model. This innovative tolerance-based scheme adeptly resolves the inconsistency problem prevalent in direct modeling while satisfying the requisites of high-precision modeling. A comparative analysis against established methodologies is conducted to demonstrate the advantages of the proposed approach.

Isogeometric collocation method (IGC) shows high computational efficiency compared with isogeometric Galerkin method (IGG) when solving partial differential equations (PDEs). However, few studies about IGC have focused on multi-sided physical domains. In this paper, the authors propose a new IGC method based on toric parameterization (IGCT) for the multi-sided planar physical domains. Due to the high order continuity of toric basis functions, the IGCT method shows more accurate numerical approximation. Moreover, the authors generalize the adaptive $w$-refinement method into IGCT (IGCT-$w$), in which the weights of basis functions in physical domains are optimized independently for geometry representation. The numerical accuracy of IGCT-$w$ is significantly improved by an order of magnitude in comparison with IGCT method. To save the computational cost of IGCT-$w$, the authors devise a selection of weights scheme according to relative residuals. Finally, several numerical examples demonstrate the effectiveness and robustness of the proposed method.

The authors extend the shell and kernel reductions for hyperexponential functions over the field of rational functions to a monomial extension. Both of the reductions are incorporated into one algorithm. As an application, the authors present an additive decomposition in rationally hyperexponential towers. The decomposition yields an alternative algorithm for computing elementary integrals over such towers. The alternative can find some elementary integrals that are unevaluated by the integrators in the latest versions of MAPLE and MATHEMATICA.

Riemman metric tensor (Rmt) plays a significant role in deducing basic formulas and equations arising in differential geometry and (pseudo-) Riemannian manifolds. It is a fundamental and challenging problem to determine the equivalence of indexed differential Rmt polynomials. This paper solves the problem by extending Gröbner basis theory and the previous work on the computational theory for indexed differentials. $L$-expansion of an indexed differential Rmt polynomial is defined. Then a decomposed form of the Gröbner basis of defining syzygies of the polynomial ring is presented, based on a partition of elementary indexed monomials. Meanwhile, the upper bound of the dummy index numbers of sim-monomials of the elements in each disjoint elementary indexed monomial subset is found. Finally, a DST-fundamental restricted ring is constructed, and the canonical form of a polynomial is confirmed to be the normal form with respect to the Gröbner basis in the DST-fundamental restricted ring.

The minimal basis of a univariate polynomial matrix $M(s)\in K[s]^{m\times n}$ is a basis of the syzygies of the polynomial matrix $M(s)$ with lowest possible degree, where $K[s]$ is the univariate polynomial ring over the field of $K$. It provides an efficient tool to compute the moving planes and moving quadratics of a rational parametric surface, which are employed to implicitize the parametric surface as a powerful implicitization method. In this paper, the authors develop two improved algorithms for computing the minimal bases of polynomial matrices. The algorithms are based on efficient methods to reduce the degrees of a set of univariate polynomial vectors. It is shown that the computational complexities of the two algorithms are $\mathcal{O}\big(m^{2}n^{3}d^2+d^2n^5-\big(2mn^4d^2-\frac{1}{6}m^3nd\big)\big)$, and $\mathcal{O}\big(m^2nd^2+(n-m)n^3d^2+\frac{m^2n^2d^2}{n-m}\big)$ respectively, where $m,n$ are the sizes of the polynomial matrix $M(s)$ and $d$ is the degree of each entry of the matrix. The new algorithms are faster than the state-of-the-art methods by experimental examples. Some properties about the degree of the minimal basis are also provided.

In this paper, the authors aim to study Kronecker canonical form theory for T-type digraphs, which can be used to construct trees by tensor product with some directed paths. Firstly, the authors show that some bicyclic digraphs and multicyclic digraphs are T-type digraphs. Secondly, the authors provide a characterization for T-type digraphs by their Kronecker canonical form. Moreover, the authors present an algorithm for computing the Kronecker canonical form, which can be used to determine whether or not a digraph is a T-type digraph. Lastly, for a class of T-type digraphs, the authors show that their incidence matrix pair can be transformed into Kronecker canonical form using unimodular matrices. The authors also present an algorithm related to finding such unimodular matrices.