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We develop a sparse hierarchical $hp$-finite element method ($hp$-FEM) for the Helmholtz equation with rotationally invariant variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and concentric annuli cells. The discretization preserves the Fourier mode decoupling of rotationally invariant operators, such as the Laplacian, which manifests as block diagonal mass and stiffness matrices. Moreover, the matrices have a sparsity pattern independent of the order of the discretization and admit an optimal complexity factorization. The sparse $hp$-FEM can handle radial discontinuities in the right-hand side and in rotationally invariant Helmholtz coefficients. We consider examples such as a high-frequency Helmholtz equation with radial discontinuities, the time-dependent Schr\"odinger equation, and an extension to a three-dimensional cylinder domain, with a quasi-optimal solve, via the Alternating Direction Implicit (ADI) algorithm.

We give an algorithm that given a graph $G$ with $n$ vertices and $m$ edges and an integer $k$, in time $O_k(n^{1+o(1)}) + O(m)$ either outputs a rank decomposition of $G$ of width at most $k$ or determines that the rankwidth of $G$ is larger than $k$; the $O_k(\cdot)$-notation hides factors depending on $k$. Our algorithm returns also a $(2^{k+1}-1)$-expression for cliquewidth, yielding a $(2^{k+1}-1)$-approximation algorithm for cliquewidth with the same running time. This improves upon the $O_k(n^2)$ time algorithm of Fomin and Korhonen [STOC 2022]. The main ingredient of our algorithm is a fully dynamic algorithm for maintaining rank decompositions of bounded width: We give a data structure that for a dynamic $n$-vertex graph $G$ that is updated by edge insertions and deletions maintains a rank decomposition of $G$ of width at most $4k$ under the promise that the rankwidth of $G$ never grows above $k$. The amortized running time of each update is $O_k(2^{\sqrt{\log n} \log \log n})$. The data structure furthermore can maintain whether $G$ satisfies some fixed ${\sf CMSO}_1$ property within the same running time. We also give a framework for performing ``dense'' edge updates inside a given set of vertices $X$, where the new edges inside $X$ are described by a given ${\sf CMSO}_1$ sentence and vertex labels, in amortized $O_k(|X| \cdot 2^{\sqrt{\log n} \log \log n})$ time. Our dynamic algorithm generalizes the dynamic treewidth algorithm of Korhonen, Majewski, Nadara, Pilipczuk, and Soko{\l}owski [FOCS 2023].

In this note, we give sufficient conditions for the almost sure and the convergence in $\mathbb{L}^p$ of a $U$-statistic of order $m$ built on a strictly stationary but not necessarily ergodic sequence.

Given a finite, simple, connected graph $G=(V,E)$ with $|V|=n$, we consider the associated graph Laplacian matrix $L = D - A$ with eigenvalues $0 = \lambda_1 < \lambda_2 \leq \dots \leq \lambda_n$. One can also consider the same graph equipped with positive edge weights $w:E \rightarrow \mathbb{R}_{> 0}$ normalized to $\sum_{e \in E} w_e = |E|$ and the associated weighted Laplacian matrix $L_w$. We say that $G$ is conformally rigid if constant edge-weights maximize the second eigenvalue $\lambda_2(w)$ of $L_w$ over all $w$, and minimize $\lambda_n(w')$ of $L_{w'}$ over all $w'$, i.e., for all $w,w'$, $$ \lambda_2(w) \leq \lambda_2(1) \leq \lambda_n(1) \leq \lambda_n(w').$$ Conformal rigidity requires an extraordinary amount of symmetry in $G$. Every edge-transitive graph is conformally rigid. We prove that every distance-regular graph, and hence every strongly-regular graph, is conformally rigid. Certain special graph embeddings can be used to characterize conformal rigidity. Cayley graphs can be conformally rigid but need not be, we prove a sufficient criterion. We also find a small set of conformally rigid graphs that do not belong into any of the above categories; these include the Hoffman graph, the crossing number graph 6B and others. Conformal rigidity can be certified via semidefinite programming, we provide explicit examples.

The Weisfeiler-Leman dimension of a graph $G$ is the least number $k$ such that the $k$-dimensional Weisfeiler-Leman algorithm distinguishes $G$ from every other non-isomorphic graph. The dimension is a standard measure of the descriptive complexity of a graph and recently finds various applications in particular in the context of machine learning. In this paper, we study the computational complexity of computing the Weisfeiler-Leman dimension. We observe that in general the problem of deciding whether the Weisfeiler-Leman dimension of $G$ is at most $k$ is NP-hard. This is also true for the more restricted problem with graphs of color multiplicity at most 4. Therefore, we study parameterized and approximate versions of the problem. We give, for each fixed $k\geq 2$, a polynomial-time algorithm that decides whether the Weisfeiler-Leman dimension of a given graph of color multiplicity at most $5$ is at most $k$. Moreover, we show that for these color multiplicities this is optimal in the sense that this problem is P-hard under logspace-uniform $\text{AC}_0$-reductions. Furthermore, for each larger bound $c$ on the color multiplicity and each fixed $k \geq 2$, we provide a polynomial-time approximation algorithm for the abelian case: given a relational structure with abelian color classes of size at most $c$, the algorithm outputs either that its Weisfeiler-Leman dimension is at most $(k+1)c$ or that it is larger than $k$.

For a graph $G$, $ mp(G) $ is the multipacking number, and $\gamma_b(G)$ is the broadcast domination number. It is known that $mp(G)\leq \gamma_b(G)$ and $\gamma_b(G)\leq 2mp(G)+3$ for any graph $G$, and it was shown that $\gamma_b(G)-mp(G)$ can be arbitrarily large for connected graphs. It is conjectured that $\gamma_b(G)\leq 2mp(G)$ for any general graph $G$. We show that, for any cactus graph $G$, $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$. We also show that $\gamma_b(G)-mp(G)$ can be arbitrarily large for cactus graphs and asteroidal triple-free graphs by constructing an infinite family of cactus graphs which are also asteroidal triple-free graphs such that the ratio $\gamma_b(G)/mp(G)=4/3$, with $mp(G)$ arbitrarily large. This result shows that, for cactus graphs, the bound $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$ cannot be improved to a bound in the form $\gamma_b(G)\leq c_1\cdot mp(G)+c_2$, for any constant $c_1<4/3$ and $c_2$. Moreover, we provide an $O(n)$-time algorithm to construct a multipacking of cactus graph $G$ of size at least $ \frac{2}{3}mp(G)-\frac{11}{3} $, where $n$ is the number of vertices of the graph $G$. The hyperbolicity of the cactus graph class is unbounded. For $0$-hyperbolic graphs, $mp(G)=\gamma_b(G)$. Moreover, $mp(G)=\gamma_b(G)$ holds for the strongly chordal graphs which is a subclass of $\frac{1}{2}$-hyperbolic graphs. Now it's a natural question: what is the minimum value of $\delta$, for which we can say that the difference $ \gamma_{b}(G) - mp(G) $ can be arbitrarily large for $\delta$-hyperbolic graphs? We show that the minimum value of $\delta$ is $\frac{1}{2}$ using a construction of an infinite family of cactus graphs with hyperbolicity $\frac{1}{2}$.

We present an efficient preconditioner for two-by-two block system of linear equations with the coefficient matrix $ \begin{pmatrix} F & -G^H G & F \end{pmatrix}$ where $F\in\mathbb{C}^{n\times n}$ is Hermitian positive definite and $G\in\mathbb{C}^{n\times n}$ is positive semidefinite. Spectral analysis of the preconditioned matrix is analyzed. In each iteration of a Krylov subspace method, like GMRES, for solving the preconditioned system in conjunction with proposed preconditioner two subsystems with Hermitian positive definite coefficient matrix should be solved which can be accomplished exactly using the Cholesky factorization or inexactly using the conjugate gradient method. Application of the proposed preconditioner to the systems arising from finite element discretization of PDE-constrained optimization problems is presented. Numerical results are given to demonstrate the efficiency of the preconditioner.

We propose a threshold-type algorithm to the $L^2$-gradient flow of the Canham-Helfrich functional generalized to $\mathbb{R}^N$. The algorithm to the Willmore flow is derived as a special case in $\mathbb{R}^2$ or $\mathbb{R}^3$. This algorithm is constructed based on an asymptotic expansion of the solution to the initial value problem for a fourth order linear parabolic partial differential equation whose initial data is the indicator function on the compact set $\Omega_0$. The crucial points are to prove that the boundary $\partial\Omega_1$ of the new set $\Omega_1$ generated by our algorithm is included in $O(t)$-neighborhood from $\partial\Omega_0$ for small time $t>0$ and to show that the derivative of the threshold function in the normal direction for $\partial\Omega_0$ is far from zero in the small time interval. Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our threshold-type algorithm.

In this paper, we examine the relationship between the stability of the dynamical system $x^{\prime}=f(x)$ and the computability of its basins of attraction. We present a computable $C^{\infty}$ system $x^{\prime}=f(x)$ that possesses a computable and stable equilibrium point, yet whose basin of attraction is robustly non-computable in a neighborhood of $f$ in the sense that both the equilibrium point and the non-computability of its associated basin of attraction persist when $f$ is slightly perturbed. This indicates that local stability near a stable equilibrium point alone is insufficient to guarantee the computability of its basin of attraction. However, we also demonstrate that the basins of attraction associated with a structurally stable - globally stable - planar system defined on a compact set are computable. Our findings suggest that the global stability of a system and the compactness of the domain play a pivotal role in determining the computability of its basins of attraction.

On a finite time interval $(0,T)$, we consider the multiresolution Galerkin discretization of a modified Hilbert transform $\mathcal H_T$ which arises in the space-time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in $(0,T)$ consisting of piecewise polynomials of degree $\geq 1$ with sufficiently many vanishing moments which constitute Riesz bases in the Sobolev spaces $ H^{s}_{0,}(0,T)$ and $ H^{s}_{,0}(0,T)$. These bases provide multilevel splittings of the temporal discretization spaces into "increment" or "detail" spaces of direct sum type. Via algebraic tensor-products of these temporal multilevel discretizations with standard, hierarchic finite element spaces in the spatial domain (with standard Lagrangian FE bases), sparse space-time tensor-product spaces are obtained, which afford a substantial reduction in the number of the degrees of freedom as compared to time-marching discretizations. In addition, temporal spline-wavelet bases allow to compress certain nonlocal integrodifferential operators which appear in stable space-time variational formulations of initial-boundary value problems, such as the heat equation and the acoustic wave equation. An efficient preconditioner is proposed that affords linear complexity solves of the linear system of equations which results from the sparse space-time Galerkin discretization.