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In this short note, we show that the higher order derivatives of the adjugate matrix $\mbox{Adj}(z-A)$, are related to the nilpotent matrices and projections in the Jordan decomposition of the matrix $A$. These relations appear as a factorization of the derivative of the adjugate matrix as a product of factors related to the eigenvalues, nilpotent matrices and projectors. The novel relations are obtained using the Riesz projector and functional calculus. The results presented here can be considered a generalization of the Thompson and McEnteggert theorem that relates the adjugate matrix with the orthogonal projection on the eigenspace of simple eigenvalues for symmetric matrices. They can also be viewed as a complement to some previous results by B. Parisse, M. Vaughan that related derivatives of the adjugate matrix with the invariant subspaces associated with an eigenvalue. Our results can also be interpreted as a general eigenvector-eigenvalue identity. Many previous works have dealt with relations between the projectors on the eigenspaces and derivatives of the adjugate matrix with the characteristic spaces but it seems there is no explicit mention in the literature of the factorization of the higher-order derivatives of the adjugate matrix as a product involving nilpotent and projector matrices that appears in the Jordan decomposition theorem.

New emerging technologies powered by Artificial Intelligence (AI) have the potential to disruptively transform our societies for the better. In particular, data-driven learning approaches (i.e., Machine Learning (ML)) have been a true revolution in the advancement of multiple technologies in various application domains. But at the same time there is growing concern about certain intrinsic characteristics of these methodologies that carry potential risks to both safety and fundamental rights. Although there are mechanisms in the adoption process to minimize these risks (e.g., safety regulations), these do not exclude the possibility of harm occurring, and if this happens, victims should be able to seek compensation. Liability regimes will therefore play a key role in ensuring basic protection for victims using or interacting with these systems. However, the same characteristics that make AI systems inherently risky, such as lack of causality, opacity, unpredictability or their self and continuous learning capabilities, may lead to considerable difficulties when it comes to proving causation. This paper presents three case studies, as well as the methodology to reach them, that illustrate these difficulties. Specifically, we address the cases of cleaning robots, delivery drones and robots in education. The outcome of the proposed analysis suggests the need to revise liability regimes to alleviate the burden of proof on victims in cases involving AI technologies.

Understanding player shooting profiles is an essential part of basketball analysis: knowing where certain opposing players like to shoot from can help coaches neutralize offensive gameplans from their opponents; understanding where their players are most comfortable can lead them to developing more effective offensive strategies. An automatic tool that can provide these performance profiles in a timely manner can become invaluable for coaches to maximize both the effectiveness of their game plan as well as the time dedicated to practice and other related activities. Additionally, basketball is dictated by many variables, such as playstyle and game dynamics, that can change the flow of the game and, by extension, player performance profiles. It is crucial that the performance profiles can reflect the diverse playstyles, as well as the fast-changing dynamics of the game. We present a tool that can visualize player performance profiles in a timely manner while taking into account factors such as play-style and game dynamics. Our approach generates interpretable heatmaps that allow us to identify and analyze how non-spatial factors, such as game dynamics or playstyle, affect player performance profiles.

While the majority of autonomous driving research has concentrated on everyday driving scenarios, further safety and performance improvements of autonomous vehicles require a focus on extreme driving conditions. In this context, autonomous racing is a new area of research that has been attracting considerable interest recently. Due to the fact that a vehicle is driven by its perception, planning, and control limits during racing, numerous research and development issues arise. This paper provides a comprehensive overview of the autonomous racing system built by team KAIST for the Indy Autonomous Challenge (IAC). Our autonomy stack consists primarily of a multi-modal perception module, a high-speed overtaking planner, a resilient control stack, and a system status manager. We present the details of all components of our autonomy solution, including algorithms, implementation, and unit test results. In addition, this paper outlines the design principles and the results of a systematical analysis. Even though our design principles are derived from the unique application domain of autonomous racing, they can also be applied to a variety of safety-critical, high-cost-of-failure robotics applications. The proposed system was integrated into a full-scale autonomous race car (Dallara AV-21) and field-tested extensively. As a result, team KAIST was one of three teams who qualified and participated in the official IAC race events without any accidents. Our proposed autonomous system successfully completed all missions, including overtaking at speeds of around $220 km/h$ in the IAC@CES2022, the world's first autonomous 1:1 head-to-head race.

We evaluated the capability of generative pre-trained transformers (GPT), to pass assessments in introductory and intermediate Python programming courses at the postsecondary level. Discussions of potential uses (e.g., exercise generation, code explanation) and misuses (e.g., cheating) of this emerging technology in programming education have intensified, but to date there has not been a rigorous analysis of the models' capabilities in the realistic context of a full-fledged programming course with diverse set of assessment instruments. We evaluated GPT on three Python courses that employ assessments ranging from simple multiple-choice questions (no code involved) to complex programming projects with code bases distributed into multiple files (599 exercises overall). Further, we studied if and how successfully GPT models leverage feedback provided by an auto-grader. We found that the current models are not capable of passing the full spectrum of assessments typically involved in a Python programming course (<70% on even entry-level modules). Yet, it is clear that a straightforward application of these easily accessible models could enable a learner to obtain a non-trivial portion of the overall available score (>55%) in introductory and intermediate courses alike. While the models exhibit remarkable capabilities, including correcting solutions based on auto-grader's feedback, some limitations exist (e.g., poor handling of exercises requiring complex chains of reasoning steps). These findings can be leveraged by instructors wishing to adapt their assessments so that GPT becomes a valuable assistant for a learner as opposed to an end-to-end solution.

The organiser of the UEFA Champions League, one of the most prestigious football tournaments in the world, faces a non-trivial mechanism design problem each autumn: how to choose a perfect matching in a balanced bipartite graph randomly. For the sake of credibility and transparency, the Round of 16 draw consists of some discrete uniform choices from two urns whose compositions are dynamically updated with computer assistance. Even though the adopted mechanism is unevenly distributed over all valid assignments, it resembles the fairest possible lottery according to a recent result. We challenge this finding by analysing the effect of reversing the order of the urns. The optimal draw procedure is shown to be primarily dependent on the lexicographic order of degree sequences for the two sets of teams. An example is provided where exchanging the urns can reduce unfairness by one-third on average and almost halve the worst bias for all pairs of teams. Nonetheless, the current policy of starting the draw with the runners-up remains the best option if the draw order should be determined before the national associations of the clubs are known.

Visualization researchers and visualization professionals seek appropriate abstractions of visualization requirements that permit considering visualization solutions independently from specific problems. Abstractions can help us design, analyze, organize, and evaluate the things we create. The literature has many task structures (taxonomies, typologies, etc.), design spaces, and related ``frameworks'' that provide abstractions of the problems a visualization is meant to address. In this viewpoint, we introduce a different one, a problem space that complements existing frameworks by focusing on the needs that a visualization is meant to solve. We believe it provides a valuable conceptual tool for designing and discussing visualizations.

This paper presents a succinct review of attempts in the literature to use game theory to model decision making scenarios relevant to defence applications. Game theory has been proven as a very effective tool in modelling decision making processes of intelligent agents, entities, and players. It has been used to model scenarios from diverse fields such as economics, evolutionary biology, and computer science. In defence applications, there is often a need to model and predict actions of hostile actors, and players who try to evade or out-smart each other. Modelling how the actions of competitive players shape the decision making of each other is the forte of game theory. In past decades, there have been several studies which applied different branches of game theory to model a range of defence-related scenarios. This paper provides a structured review of such attempts, and classifies existing literature in terms of the kind of warfare modelled, the types of game used, and the players involved. The presented analysis provides a concise summary about the state-of-the-art with regards to the use of game theory in defence applications, and highlights the benefits and limitations of game theory in the considered scenarios.

Over the past few years, we have seen fundamental breakthroughs in core problems in machine learning, largely driven by advances in deep neural networks. At the same time, the amount of data collected in a wide array of scientific domains is dramatically increasing in both size and complexity. Taken together, this suggests many exciting opportunities for deep learning applications in scientific settings. But a significant challenge to this is simply knowing where to start. The sheer breadth and diversity of different deep learning techniques makes it difficult to determine what scientific problems might be most amenable to these methods, or which specific combination of methods might offer the most promising first approach. In this survey, we focus on addressing this central issue, providing an overview of many widely used deep learning models, spanning visual, sequential and graph structured data, associated tasks and different training methods, along with techniques to use deep learning with less data and better interpret these complex models --- two central considerations for many scientific use cases. We also include overviews of the full design process, implementation tips, and links to a plethora of tutorials, research summaries and open-sourced deep learning pipelines and pretrained models, developed by the community. We hope that this survey will help accelerate the use of deep learning across different scientific domains.

In this monograph, I introduce the basic concepts of Online Learning through a modern view of Online Convex Optimization. Here, online learning refers to the framework of regret minimization under worst-case assumptions. I present first-order and second-order algorithms for online learning with convex losses, in Euclidean and non-Euclidean settings. All the algorithms are clearly presented as instantiation of Online Mirror Descent or Follow-The-Regularized-Leader and their variants. Particular attention is given to the issue of tuning the parameters of the algorithms and learning in unbounded domains, through adaptive and parameter-free online learning algorithms. Non-convex losses are dealt through convex surrogate losses and through randomization. The bandit setting is also briefly discussed, touching on the problem of adversarial and stochastic multi-armed bandits. These notes do not require prior knowledge of convex analysis and all the required mathematical tools are rigorously explained. Moreover, all the proofs have been carefully chosen to be as simple and as short as possible.