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In 1972 Mykkeltveit proved that the maximum number of vertex-disjoint cycles in the de Bruijn graphs of order $n$ is attained by the pure cycling register rule, as conjectured by Golomb. We generalize this result to the tensor product of the de Bruijn graph of order $n$ and a simple cycle of size $k$, when $n$ divides $k$ or vice versa. We also develop counting formulae for a large family of cycling register rules, including the linear register rules proposed by Golomb.

In this research, new concepts of existential granules that determine themselves are invented, and are characterized from algebraic, topological, and mereological perspectives. Existential granules are those that determine themselves initially, and interact with their environment subsequently. Examples of the concept, such as those of granular balls, though inadequately defined, algorithmically established, and insufficiently theorized in earlier works by others, are already used in applications of rough sets and soft computing. It is shown that they fit into multiple theoretical frameworks (axiomatic, adaptive, and others) of granular computing. The characterization is intended for algorithm development, application to classification problems and possible mathematical foundations of generalizations of the approach. Additionally, many open problems are posed and directions provided.

The laborious and costly nature of affect annotation is a key detrimental factor for obtaining large scale corpora with valid and reliable affect labels. Motivated by the lack of tools that can effectively determine an annotator's reliability, this paper proposes general quality assurance (QA) tests for real-time continuous annotation tasks. Assuming that the annotation tasks rely on stimuli with audiovisual components, such as videos, we propose and evaluate two QA tests: a visual and an auditory QA test. We validate the QA tool across 20 annotators that are asked to go through the test followed by a lengthy task of annotating the engagement of gameplay videos. Our findings suggest that the proposed QA tool reveals, unsurprisingly, that trained annotators are more reliable than the best of untrained crowdworkers we could employ. Importantly, the QA tool introduced can predict effectively the reliability of an affect annotator with 80% accuracy, thereby, saving on resources, effort and cost, and maximizing the reliability of labels solicited in affective corpora. The introduced QA tool is available and accessible through the PAGAN annotation platform.

Pearl and Verma developed d-separation as a widely used graphical criterion to reason about the conditional independencies that are implied by the causal structure of a Bayesian network. As acyclic ground probabilistic logic programs correspond to Bayesian networks on their dependency graph, we can compute conditional independencies from d-separation in the latter. In the present paper, we generalize the reasoning above to the non-ground case. First, we abstract the notion of a probabilistic logic program away from external databases and probabilities to obtain so-called program structures. We then present a correct meta-interpreter that decides whether a certain conditional independence statement is implied by a program structure on a given external database. Finally, we give a fragment of program structures for which we obtain a completeness statement of our conditional independence oracle. We close with an experimental evaluation of our approach revealing that our meta-interpreter performs significantly faster than checking the definition of independence using exact inference in ProbLog 2.

Kirigami are part of the larger class of mechanical metamaterials, which exhibit exotic properties. This article focuses on rhombi-slits, which is a specific type of kirigami. A nonlinear kinematics model was previously proposed as a second order divergence form PDE with a possibly degenerate, and sign-changing coefficient matrix. We first propose to study the existence solutions of this equation by using the limiting absorption principle. Then, we propose a numerical method based on adding a complex dissipation to approximate the solutions. Finally, comparisons of simulations with experiments are performed.

For an arbitrary finite family of graphs, the distance labeling problem asks to assign labels to all nodes of every graph in the family in a way that allows one to recover the distance between any two nodes of any graph from their labels. The main goal is to minimize the number of unique labels used. We study this problem for the families $\mathcal{C}_n$ consisting of cycles of all lengths between 3 and $n$. We observe that the exact solution for directed cycles is straightforward and focus on the undirected case. We design a labeling scheme requiring $\frac{n\sqrt{n}}{\sqrt{6}}+O(n)$ labels, which is almost twice less than is required by the earlier known scheme. Using the computer search, we find an optimal labeling for each $n\le 17$, showing that our scheme gives the results that are very close to the optimum.

In this paper the authors propose a polynomial algorithm which allows the computation of the farthest in an intersection of balls to a given point under three additional hypothesis: the farthest is unique, the distance to it is known and its magnitude is known. As a use case the authors analyze the subset sum problem SSP(S,T) for a given $S\in \mathbb{R}^n$ and $T \in \mathbb{R}$. The proposed approach is to write the SSP as a distance maximization over an intersection of balls. It was shown that the SSP has a solution if and only if the maximum value of the distance has a predefined value. This together with the fact that a solution is a corner of the unit hypercube, allows the authors to apply the proposed geometry results to find a solution to the SSP under the hypothesis that is unique.

Conjugate gradient is an efficient algorithm for solving large sparse linear systems. It has been utilized to accelerate the computation in Bayesian analysis for many large-scale problems. This article discusses the applications of conjugate gradient in Bayesian computation, with a focus on sparse regression and spatial analysis. A self-contained introduction of conjugate gradient is provided to facilitate potential applications in a broader range of problems.

Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.

Although measuring held-out accuracy has been the primary approach to evaluate generalization, it often overestimates the performance of NLP models, while alternative approaches for evaluating models either focus on individual tasks or on specific behaviors. Inspired by principles of behavioral testing in software engineering, we introduce CheckList, a task-agnostic methodology for testing NLP models. CheckList includes a matrix of general linguistic capabilities and test types that facilitate comprehensive test ideation, as well as a software tool to generate a large and diverse number of test cases quickly. We illustrate the utility of CheckList with tests for three tasks, identifying critical failures in both commercial and state-of-art models. In a user study, a team responsible for a commercial sentiment analysis model found new and actionable bugs in an extensively tested model. In another user study, NLP practitioners with CheckList created twice as many tests, and found almost three times as many bugs as users without it.