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In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with tempered fractional derivative of order $\alpha$. Although some of its variants are considered in many recent numerical analysis papers, there are still some significant differences. Here we first provide the regularity estimates of the solution. And then a modified $L$1 scheme inspired by the middle rectangle quadrature formula on graded meshes is employed to compensate for the singularity of the solution at $t\rightarrow 0^{+}$, while the five-point difference scheme is used in space. Stability and convergence are proved in the sence of $L^{\infty}$ norm, then a sharp error estimate $\mathscr{O}(\tau^{\min\{2-\alpha, r\alpha\}})$ is derived on graded meshes. Furthermore, unlike the bounds proved in the previous works, the constant multipliers in our analysis do not blow up as the Caputo fractional derivative $\alpha$ approaches the classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and convergence order of the presented algorithms.

We propose a space-time scheme that combines an unfitted finite element method in space with a discontinuous Galerkin time discretisation for the accurate numerical approximation of parabolic problems with moving domains or interfaces. We make use of an aggregated finite element space to attain robustness with respect to the cut locations. The aggregation is performed slab-wise to have a tensor product structure of the space-time discrete space, which is required in the numerical analysis. We analyse the proposed algorithm, providing stability, condition number bounds and anisotropic \emph{a priori} error estimates. A set of numerical experiments confirm the theoretical results for a parabolic problem on a moving domain. The method is applied for a mass transfer problem with changing topology.

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189-202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613-624]. We exploit the full symmetry of the problem by developing a block-diagonalization of the underlying matrix algebra and use it to improve bounds on several concrete instances. Our results imply that $\text{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\text{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $ \text{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\text{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all $n$. The latter three bounds are computed using a new relaxation of the original semidefinite programming bound, by only requiring one small matrix block to be positive semidefinite. Our lower bound on $K_{13,n}$ implies that for each fixed $m \geq 13$, $\lim_{n \to \infty} \text{cr}(K_{m,n})/Z(m,n) \geq 0.8878 m/(m-1)$. Here $Z(m,n)$ is the Zarankiewicz number: the conjectured crossing number of $K_{m,n}$.

We introduce a new type of Krasnoselskii's result. Using a simple differentiability condition, we relax the nonexpansive condition in Krasnoselskii's theorem. More clearly, we analyze the convergence of the sequence $x_{n+1}=\frac{x_n+g(x_n)}{2}$ based on some differentiability condition of $g$ and present some fixed point results. We introduce some iterative sequences that for any real differentiable function $g$ and any starting point $x_0\in \mathbb [a,b]$ converge monotonically to the nearest root of $g$ in $[a,b]$ that lay to the right or left side of $x_0$. Based on this approach, we present an efficient and novel method for finding the real roots of real functions. We prove that no root will be missed in our method. It is worth mentioning that our iterative method is free from the derivative evaluation which can be regarded as an advantage of this method in comparison with many other methods. Finally, we illustrate our results with some numerical examples.

Two-dimensional finite element complexes with various smoothness, including the de Rham complex, the curldiv complex, the elasticity complex, and the divdiv complex, are systematically constructed in this work. First smooth scalar finite elements in two dimensions are developed based on a non-overlapping decomposition of the simplicial lattice and the Bernstein basis of the polynomial space. Smoothness at vertices is more than doubled than that at edges. Then the finite element de Rham complexes with various smoothness are devised using smooth finite elements with smoothness parameters satisfying certain relations. Finally, finite element elasticity complexes and finite element divdiv complexes are derived from finite element de Rham complexes by using the Bernstein-Gelfand-Gelfand (BGG) framework. Additionally, some finite element divdiv complexes are constructed without BGG framework. Dimension count plays an important role for verifying the exactness of two-dimensional finite element complexes.

We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in $\mathbb{R}^d$ that is covered by constant-sized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., $2^{-{\operatorname{polylog}(d)}}$) fraction of the incidences. Alternatively, our conjecture may be interpreted linear-algebraically as follows: Any rank-$d$ matrix containing at most $O(1)$ distinct entries in each column contains a submatrix of fractional size $2^{-{\operatorname{polylog}(d)}}$, in which each column contains one distinct entry. We prove that our conjecture is equivalent to the log-rank conjecture. Motivated by the connections above, we revisit well-studied questions in point-hyperplane incidence geometry without structural assumptions (i.e., the existence of partitions). We give an elementary argument for the existence of complete bipartite subgraphs of density $\Omega(\epsilon^{2d}/d)$ in any $d$-dimensional configuration with incidence density $\epsilon$. We also improve an upper-bound construction of Apfelbaum and Sharir (SIAM J. Discrete Math. '07), yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is $\Omega(1/\sqrt d)$. Finally, we discuss various constructions (due to others) which yield configurations with incidence density $\Omega(1)$ and bipartite subgraph density $2^{-\Omega(\sqrt d)}$. Our framework and results may help shed light on the difficulty of improving Lovett's $\tilde{O}(\sqrt{\operatorname{rank}(f)})$ bound (J. ACM '16) for the log-rank conjecture; in particular, any improvement on this bound would imply the first bipartite subgraph size bounds for parallel $3$-partitioned configurations which beat our generic bounds for unstructured configurations.

Linear error-correcting codes can be used for constructing secret sharing schemes; however finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult. Here we investigate the properties of certain algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane only takes a few values; these varieties give rise to $q$-divisible linear codes with at most $5$ weights. Furthermore, for $q$ odd these codes turn out to be minimal and we characterize the access structures of the secret sharing schemes based on their dual codes. Indeed, the secret sharing schemes thus obtained are democratic, that is each participant belongs to the same number of minimal access sets and can easily be described.

For the evolution of a closed surface under anisotropic surface diffusion with a general anisotropic surface energy $\gamma(\boldsymbol{n})$ in three dimensions (3D), where $\boldsymbol{n}$ is the unit outward normal vector, by introducing a novel symmetric positive definite surface energy matrix $\boldsymbol{Z}_k(\boldsymbol{n})$ depending on a stabilizing function $k(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbol{\xi}$-vector, we present a new symmetrized variational formulation for anisotropic surface diffusion with weakly or strongly anisotropic surface energy, which preserves two important structures including volume conservation and energy dissipation. Then we propose a structural-preserving parametric finite element method (SP-PFEM) to discretize the symmetrized variational problem, which preserves the volume in the discretized level. Under a relatively mild and simple condition on $\gamma(\boldsymbol{n})$, we show that SP-PFEM is unconditionally energy-stable for almost all anisotropic surface energies $\gamma(\boldsymbol{n})$ arising in practical applications. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed SP-PFEM for solving anisotropic surface diffusion in 3D.

We study the dynamics of (synchronous) one-dimensional cellular automata with cyclical boundary conditions that evolve according to the majority rule with radius $ r $. We introduce a notion that we term cell stability with which we express the structure of the possible configurations that could emerge in this setting. Our main finding is that apart from the configurations of the form $ (0^{r+1}0^* + 1^{r+1}1^*)^* $, which are always fixed-points, the other configurations that the automata could possibly converge to, which are known to be either fixed-points or 2-cycles, have a particular spatially periodic structure. Namely, each of these configurations is of the form $ s^* $ where $ s $ consists of $ O(r^2) $ consecutive sequences of cells with the same state, each such sequence is of length at most $ r $, and the total length of $ s $ is $ O(r^2) $ as well. We show that an analogous result also holds for the minority rule.

In this paper we study colorings (or tilings) of the two-dimensional grid $\mathbb{Z}^2$. A coloring is said to be valid with respect to a set $P$ of $n\times m$ rectangular patterns if all $n\times m$ sub-patterns of the coloring are in $P$. A coloring $c$ is said to be of low complexity with respect to a rectangle if there exist $m,n\in\mathbb{N}$ and a set $P$ of $n\times m$ rectangular patterns such that $c$ is valid with respect to $P$ and $|P|\leq nm$. Open since it was stated in 1997, Nivat's conjecture states that such a coloring is necessarily periodic. If Nivat's conjecture is true, all valid colorings with respect to $P$ such that $|P|\leq mn$ must be periodic. We prove that there exists at least one periodic coloring among the valid ones. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Then, we use our result to show that Nivat's conjecture holds for uniformly recurrent configurations. These results also extend to other convex shapes in place of the rectangle.\\ After that, we prove that the $nm$ bound is multiplicatively optimal for the decidability of the domino problem, as for all $\varepsilon>0$ it is undecidable to determine if there exists a valid coloring for a given $m,n\in \mathbb{N}$ and set of rectangular patterns $P$ of size $n\times m$ such that $|P|\leq (1+\varepsilon)nm$. We prove a slightly better bound in the case where $m=n$, as well as constructing aperiodic SFTs of pretty low complexity.\\ This paper is an extended version of a paper published in STACS 2020.