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In the analysis of the $h$-version of the finite-element method (FEM), with fixed polynomial degree $p$, applied to the Helmholtz equation with wavenumber $k\gg 1$, the $\textit{asymptotic regime}$ is when $(hk)^p C_{\rm sol}$ is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here $C_{\rm sol}$ is the norm of the Helmholtz solution operator, normalised so that $C_{\rm sol} \sim k$ for nontrapping problems. In the $\textit{preasymptotic regime}$, one expects that if $(hk)^{2p}C_{\rm sol}$ is sufficiently small, then (for physical data) the relative error of the Galerkin solution is controllably small. In this paper, we prove the natural error bounds in the preasymptotic regime for the variable-coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition $\textit{either}$ realised exactly using the Dirichlet-to-Neumann map on the boundary of a ball $\textit{or}$ approximated either by a radial perfectly-matched layer (PML) or an impedance boundary condition. Previously, such bounds for $p>1$ were only available for Dirichlet obstacles with the radiation condition approximated by an impedance boundary condition. Our result is obtained via a novel generalisation of the "elliptic-projection" argument (the argument used to obtain the result for $p=1$) which can be applied to a wide variety of abstract Helmholtz-type problems.

The paper studies nonstationary high-dimensional vector autoregressions of order $k$, VAR($k$). Additional deterministic terms such as trend or seasonality are allowed. The number of time periods, $T$, and the number of coordinates, $N$, are assumed to be large and of the same order. Under this regime the first-order asymptotics of the Johansen likelihood ratio (LR), Pillai-Bartlett, and Hotelling-Lawley tests for cointegration are derived: the test statistics converge to nonrandom integrals. For more refined analysis, the paper proposes and analyzes a modification of the Johansen test. The new test for the absence of cointegration converges to the partial sum of the Airy$_1$ point process. Supporting Monte Carlo simulations indicate that the same behavior persists universally in many situations beyond those considered in our theorems. The paper presents empirical implementations of the approach for the analysis of S$\&$P$100$ stocks and of cryptocurrencies. The latter example has a strong presence of multiple cointegrating relationships, while the results for the former are consistent with the null of no cointegration.

In this paper, the stability of $\theta$-methods for delay differential equations is studied based on the test equation $y'(t)=-A y(t) + B y(t-\tau)$, where $\tau$ is a constant delay and $A$ is a positive definite matrix. It is mainly considered the case where the matrices $A$ and $B$ are not simultaneosly diagonalizable and the concept of field of values is used to prove a sufficient condition for unconditional stability of these methods and another condition which also guarantees their stability, but according to the step size. The results obtained are also simplified for the case where the matrices $A$ and $B$ are simultaneously diagonalizable and compared with other similar works for the general case. Several numerical examples in which the theory discussed here is applied to parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented, too.

We consider the gradient descent flow widely used for the minimization of the $\mathcal{L}^2$ cost function in Deep Learning networks, and introduce two modified versions; one adapted for the overparametrized setting, and the other for the underparametrized setting. Both have a clear and natural invariant geometric meaning, taking into account the pullback vector bundle structure in the overparametrized, and the pushforward vector bundle structure in the underparametrized setting. In the overparametrized case, we prove that, provided that a rank condition holds, all orbits of the modified gradient descent drive the $\mathcal{L}^2$ cost to its global minimum at a uniform exponential convergence rate. We point out relations of the latter to sub-Riemannian geometry.

A one-dimensional sequence $u_0, u_1, u_2, \ldots \in [0, 1)$ is said to be completely uniformly distributed (CUD) if overlapping $s$-blocks $(u_i, u_{i+1}, \ldots , u_{i+s-1})$, $i = 0, 1, 2, \ldots$, are uniformly distributed for every dimension $s \geq 1$. This concept naturally arises in Markov chain quasi-Monte Carlo (QMC). However, the definition of CUD sequences is not constructive, and thus there remains the problem of how to implement the Markov chain QMC algorithm in practice. Harase (2021) focused on the $t$-value, which is a measure of uniformity widely used in the study of QMC, and implemented short-period Tausworthe generators (i.e., linear feedback shift register generators) over the two-element field $\mathbb{F}_2$ that approximate CUD sequences by running for the entire period. In this paper, we generalize a search algorithm over $\mathbb{F}_2$ to that over arbitrary finite fields $\mathbb{F}_b$ with $b$ elements and conduct a search for Tausworthe generators over $\mathbb{F}_b$ with $t$-values zero (i.e., optimal) for dimension $s = 3$ and small for $s \geq 4$, especially in the case where $b = 3, 4$, and $5$. We provide a parameter table of Tausworthe generators over $\mathbb{F}_4$, and report a comparison between our new generators over $\mathbb{F}_4$ and existing generators over $\mathbb{F}_2$ in numerical examples using Markov chain QMC.

We provide a simple $(1-O(\frac{1}{\sqrt{k}}))$-selectable Online Contention Resolution Scheme for $k$-uniform matroids against a fixed-order adversary. If $A_i$ and $G_i$ denote the set of selected elements and the set of realized active elements among the first $i$ (respectively), our algorithm selects with probability $1-\frac{1}{\sqrt{k}}$ any active element $i$ such that $|A_{i-1}| + 1 \leq (1-\frac{1}{\sqrt{k}})\cdot \mathbb{E}[|G_i|]+\sqrt{k}$. This implies a $(1-O(\frac{1}{\sqrt{k}}))$ prophet inequality against fixed-order adversaries for $k$-uniform matroids that is considerably simpler than previous algorithms [Ala14, AKW14, JMZ22]. We also prove that no OCRS can be $(1-\Omega(\sqrt{\frac{\log k}{k}}))$-selectable for $k$-uniform matroids against an almighty adversary. This guarantee is matched by the (known) simple greedy algorithm that accepts every active element with probability $1-\Theta(\sqrt{\frac{\log k}{k}})$ [HKS07].

Based on the theory of homogeneous spaces we derive \textit{geometrically optimal edge attributes} to be used within the flexible message passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions $\mathbb{R}^3$, position and orientations $\mathbb{R}^3 {\times} S^2$, and the group SE$(3)$ itself. Among these, $\mathbb{R}^3 {\times} S^2$ is an optimal choice due to the ability to represent directional information, which $\mathbb{R}^3$ methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE$(3)$ group. We empirically support this claim by reaching state-of-the-art results -- in accuracy and speed -- on three different benchmarks: interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

We present an algorithm that finds a maximum cardinality $f$-matching of a simple graph in time $O(n^{2/3} m)$. Here $f:V\to \mathbb{N}$ is a given function, and an $f$-matching is a subgraph wherein each vertex $v\in V$ has degree $\le f(v)$. This result generalizes a string of algorithms, concentrating on simple bipartite graphs. The bipartite case is based on the notion of level graph, introduced by Dinic for network flow. For general graphs the ``level'' of a vertex is unclear: A given vertex can occur on many different levels in augmenting trails. In fact there does not seem to be a unique level graph, our notion of level graph depends on the trails being analyzed. Our analysis presents new properties of blossoms of shortest augmenting trails. Our algorithm, unmodified, is also efficient on multigraphs, achieving time $O(\min \{\sqrt {f(V)}, n\}\,m)$, for $f(V)=\sum_vf(v)$.

$k$-clique listing is a vital graph mining operator with diverse applications in various networks. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial $k$-clique with a vertex. These algorithms have the time complexity of $O(k m (\delta/2)^{k-2})$, where $m$ is the number of edges in the graph and $\delta$ is the degeneracy of the graph. In this paper, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial $k$-clique with two vertices that connect each other (which correspond to an edge). We explore various edge orderings for EBBkC such that it achieves a time complexity of $O(\delta m + k m (\tau/2)^{k-2})$, where $\tau$ is an integer related to the maximum truss number of the graph and we have $\tau < \delta$. The time complexity of EBBkC is better than that of VBBkC algorithms for $k>3$ since both $O(\delta m)$ and $O(k m (\tau/2)^{k-2})$ are bounded by $O(k m (\delta/2)^{k-2})$. Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC-based algorithms with the early termination technique consistently and largely outperform the state-of-the-art (VBBkC-based) algorithms.

A recent line of work has shown the unconditional advantage of constant-depth quantum computation, or $\mathsf{QNC^0}$, over $\mathsf{NC^0}$, $\mathsf{AC^0}$, and related models of classical computation. Problems exhibiting this advantage include search and sampling tasks related to the parity function, and it is natural to ask whether $\mathsf{QNC^0}$ can be used to help compute parity itself. We study $\mathsf{AC^0\circ QNC^0}$ -- a hybrid circuit model where $\mathsf{AC^0}$ operates on measurement outcomes of a $\mathsf{QNC^0}$ circuit, and conjecture $\mathsf{AC^0\circ QNC^0}$ cannot even achieve $\Omega(1)$ correlation with parity. As evidence for this conjecture, we prove: $\bullet$ When the $\mathsf{QNC^0}$ circuit is ancilla-free, this model achieves only negligible correlation with parity. $\bullet$ For the general (non-ancilla-free) case, we show via a connection to nonlocal games that the conjecture holds for any class of postprocessing functions that has approximate degree $o(n)$ and is closed under restrictions, even when the $\mathsf{QNC^0}$ circuit is given arbitrary quantum advice. By known results this confirms the conjecture for linear-size $\mathsf{AC^0}$ circuits. $\bullet$ Towards the a switching lemma for $\mathsf{AC^0\circ QNC^0}$, we study the effect of quantum preprocessing on the decision tree complexity of Boolean functions. We find that from this perspective, nonlocal channels are no better than randomness: a Boolean function $f$ precomposed with an $n$-party nonlocal channel is together equal to a randomized decision tree with worst-case depth at most $\mathrm{DT}_\mathrm{depth}[f]$. Our results suggest that while $\mathsf{QNC^0}$ is surprisingly powerful for search and sampling, that power is "locked away" in the global correlations of its output, inaccessible to simple classical computation for solving decision problems.