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Directional interpolation is a fast and efficient compression technique for high-frequency Helmholtz boundary integral equations, but it requires a very large amount of storage in its original form. Algebraic recompression can significantly reduce the storage requirements and speed up the solution process accordingly. During the recompression process, weight matrices are required to correctly measure the influence of different basis vectors on the final result, and for highly accurate approximations, these weight matrices require more storage than the final compressed matrix. We present a compression method for the weight matrices and demonstrate that it introduces only a controllable error to the overall approximation. Numerical experiments show that the new method leads to a significant reduction in storage requirements.

This tutorial gives an advanced introduction to string diagrams and graph languages for higher-order computation. The subject matter develops in a principled way, starting from the two dimensional syntax of key categorical concepts such as functors, adjunctions, and strictification, and leading up to Cartesian Closed Categories, the core mathematical model of the lambda calculus and of functional programming languages. This methodology inverts the usual approach of proceeding from syntax to a categorical interpretation, by rationally reconstructing a syntax from the categorical model. The result is a graph syntax -- more precisely, a hierarchical hypergraph syntax -- which in many ways is shown to be an improvement over the conventional linear term syntax. The rest of the tutorial focuses on applications of interest to programming languages: operational semantics, general frameworks for type inference, and complex whole-program transformations such as closure conversion and automatic differentiation.

Higher-order regularization problem formulations are popular frameworks used in machine learning, inverse problems and image/signal processing. In this paper, we consider the computational problem of finding the minimizer of the Sobolev $\mathrm{W}^{1,p}$ semi-norm with a data-fidelity term. We propose a discretization procedure and prove convergence rates between our numerical solution and the target function. Our approach consists of discretizing an appropriate gradient flow problem in space and time. The space discretization is a nonlocal approximation of the p-Laplacian operator and our rates directly depend on the localization parameter $\epsilon_n$ and the time mesh-size $\tau_n$. We precisely characterize the asymptotic behaviour of $\epsilon_n$ and $\tau_n$ in order to ensure convergence to the considered minimizer. Finally, we apply our results to the setting of random graph models.

Motivated by demand-responsive parking pricing systems, we consider posted-price algorithms for the online metric matching problem. We give an $O(\log n)$-competitive posted-price randomized algorithm in the case that the metric space is a line. In particular, in this setting we show how to implement the ubiquitous guess-and-double technique using prices.

We study the question of which visibly pushdown languages (VPLs) are in the complexity class $\mathsf{AC}^0$ and how to effectively decide this question. Our contribution is to introduce a particular subclass of one-turn VPLs, called intermediate VPLs, for which the raised question is entirely unclear: to the best of our knowledge our research community is unaware of containment or non-containment in $\mathsf{AC}^0$ for any intermediate VPL. Our main result states that there is an algorithm that, given a visibly pushdown automaton, correctly outputs either that its language is in $\mathsf{AC}^0$, outputs some $m\geq 2$ such that $\mathsf{MOD}_m$ is constant-depth reducible to $L$ (implying that $L$ is not in $\mathsf{AC}^0$), or outputs a finite disjoint union of intermediate VPLs that $L$ is constant-depth equivalent to. In the latter case one can moreover effectively compute $k,l\in\mathbb{N}_{>0}$ with $k\not=l$ such that the concrete intermediate VPL $L(S\rightarrow\varepsilon\mid a c^{k-1} S b_1\mid ac^{l-1}Sb_2)$ is constant-depth reducible to the language $L$. Due to their particular nature we conjecture that either all intermediate VPLs are in $\mathsf{AC}^0$ or all are not. As a corollary of our main result we obtain that in case the input language is a visibly counter language our algorithm can effectively determine if it is in $\mathsf{AC}^0$ -- hence our main result generalizes a result by Krebs et al. stating that it is decidable if a given visibly counter language is in $\mathsf{AC}^0$ (when restricted to well-matched words). For our proofs we revisit so-called Ext-algebras (introduced by Czarnetzki et al.), which are closely related to forest algebras (introduced by Boja\'nczyk and Walukiewicz), and use Green's relations.

We propose CAPGrasp, an $\mathbb{R}^3\times \text{SO(2)-equivariant}$ 6-DoF continuous approach-constrained generative grasp sampler. It includes a novel learning strategy for training CAPGrasp that eliminates the need to curate massive conditionally labeled datasets and a constrained grasp refinement technique that improves grasp poses while respecting the grasp approach directional constraints. The experimental results demonstrate that CAPGrasp is more than three times as sample efficient as unconstrained grasp samplers while achieving up to 38% grasp success rate improvement. CAPGrasp also achieves 4-10% higher grasp success rates than constrained but noncontinuous grasp samplers. Overall, CAPGrasp is a sample-efficient solution when grasps must originate from specific directions, such as grasping in confined spaces.

Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on $\mathcal{L}_2$-optimal reduced-order modeling of stationary parametric problems, in this paper we develop and investigate optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of structured LTI systems, in particular, for second-order, port-Hamiltonian, and time-delay systems. We show that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.

The development of secure cryptographic protocols and the subsequent attack mechanisms have been placed in the literature with the utmost curiosity. While sophisticated quantum attacks bring a concern to the classical cryptographic protocols present in the applications used in everyday life, the necessity of developing post-quantum protocols is felt primarily. In post-quantum cryptography, elliptic curve-base protocols are exciting to the researchers. While the comprehensive study of elliptic curves over finite fields is well known, the extended study over finite rings is still missing. In this work, we generalize the study of Weierstrass elliptic curves over finite ring $\mathbb{Z}_n$ through classification. Several expressions to compute critical factors in studying elliptic curves are conferred. An all-around computational classification on the Weierstrass elliptic curves over $\mathbb{Z}_n$ for rigorous understanding is also attached to this work.

This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously $O(1)$-approximate for all $\ell_p$-norms of the disagreement vector. This proves that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. Our algorithm is the first combinatorial approximation algorithm for the $\ell_2$-norm objective, and more generally the first combinatorial algorithm for the $\ell_p$-norm objective when $2 \leq p < \infty$. It is also faster than all previous algorithms that minimize the $\ell_p$-norm of the disagreement vector, with run-time $O(n^\omega)$, where $O(n^\omega)$ is the time for matrix multiplication on $n \times n$ matrices. When the maximum positive degree in the graph is at most $\Delta$, this can be improved to a run-time of $O(n\Delta^2 \log n)$.