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We develop a new, powerful method for counting elements in a {\em multiset.} As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and the general behaviour of these is reasonably well-known. We slightly extend some of the known results in that case, and exhaustively study the case of patterns of length 4, about which there is little previous knowledge. For such patterns, there are seven Wilf classes, and based on extensive enumerations and careful series analysis, we have conjectured the asymptotic behaviour for all classes. Finally, we investigate a proposal of Blitvi\'c and Steingr\'imsson as to the range of a parameter for which a particular generating function formed from the occurrence sequences is itself a Stieltjes moment sequence.

We examine the effect of noise on societies of agents using an agent-based model of evolutionary norm emergence. Generally, we see that noisy societies are more selfish, smaller and discontent, and are caught in rounds of perpetual punishment preventing them from flourishing. Surprisingly, despite the effect of noise on the population, it does not seem to evolve away. In fact, in some cases it seems that the level of noise increases. We carry out further analysis and provide reasons for why this may be the case. Furthermore, we claim that our framework that evolves the noise/ambiguity of norms may be a new way to model the tight/loose framework of norms, suggesting that despite ambiguous norms detrimental effect on society, evolution does not favour clarity.

Stick-breaking (SB) processes are often adopted in Bayesian mixture models for generating mixing weights. When covariates influence the sizes of clusters, SB mixtures are particularly convenient as they can leverage their connection to binary regression to ease both the specification of covariate effects and posterior computation. Existing SB models are typically constructed based on continually breaking a single remaining piece of the unit stick. We view this from a dyadic tree perspective in terms of a lopsided bifurcating tree that extends only in one side. We show that several unsavory characteristics of SB models are in fact largely due to this lopsided tree structure. We consider a generalized class of SB models with alternative bifurcating tree structures and examine the influence of the underlying tree topology on the resulting Bayesian analysis in terms of prior assumptions, posterior uncertainty, and computational effectiveness. In particular, we provide evidence that a balanced tree topology, which corresponds to continually breaking all remaining pieces of the unit stick, can resolve or mitigate several undesirable properties of SB models that rely on a lopsided tree.

The principle of maximum entropy, as introduced by Jaynes in information theory, has contributed to advancements in various domains such as Statistical Mechanics, Machine Learning, and Ecology. Its resultant solutions have served as a catalyst, facilitating researchers in mapping their empirical observations to the acquisition of unbiased models, whilst deepening the understanding of complex systems and phenomena. However, when we consider situations in which the model elements are not directly observable, such as when noise or ocular occlusion is present, possibilities arise for which standard maximum entropy approaches may fail, as they are unable to match feature constraints. Here we show the Principle of Uncertain Maximum Entropy as a method that both encodes all available information in spite of arbitrarily noisy observations while surpassing the accuracy of some ad-hoc methods. Additionally, we utilize the output of a black-box machine learning model as input into an uncertain maximum entropy model, resulting in a novel approach for scenarios where the observation function is unavailable. Previous remedies either relaxed feature constraints when accounting for observation error, given well-characterized errors such as zero-mean Gaussian, or chose to simply select the most likely model element given an observation. We anticipate our principle finding broad applications in diverse fields due to generalizing the traditional maximum entropy method with the ability to utilize uncertain observations.

In extreme value theory and other related risk analysis fields, probability weighted moments (PWM) have been frequently used to estimate the parameters of classical extreme value distributions. This method-of-moment technique can be applied when second moments are finite, a reasonable assumption in many environmental domains like climatological and hydrological studies. Three advantages of PWM estimators can be put forward: their simple interpretations, their rapid numerical implementation and their close connection to the well-studied class of U-statistics. Concerning the later, this connection leads to precise asymptotic properties, but non asymptotic bounds have been lacking when off-the-shelf techniques (Chernoff method) cannot be applied, as exponential moment assumptions become unrealistic in many extreme value settings. In addition, large values analysis is not immune to the undesirable effect of outliers, for example, defective readings in satellite measurements or possible anomalies in climate model runs. Recently, the treatment of outliers has sparked some interest in extreme value theory, but results about finite sample bounds in a robust extreme value theory context are yet to be found, in particular for PWMs or tail index estimators. In this work, we propose a new class of robust PWM estimators, inspired by the median-of-means framework of Devroye et al. (2016). This class of robust estimators is shown to satisfy a sub-Gaussian inequality when the assumption of finite second moments holds. Such non asymptotic bounds are also derived under the general contamination model. Our main proposition confirms theoretically a trade-off between efficiency and robustness. Our simulation study indicates that, while classical estimators of PWMs can be highly sensitive to outliers.

A general class of the almost instantaneous fixed-to-variable-length (AIFV) codes is proposed, which contains every possible binary code we can make when allowing finite bits of decoding delay. The contribution of the paper lies in the following. (i) Introducing $N$-bit-delay AIFV codes, constructed by multiple code trees with higher flexibility than the conventional AIFV codes. (ii) Proving that the proposed codes can represent any uniquely-encodable and uniquely-decodable variable-to-variable length codes. (iii) Showing how to express codes as multiple code trees with minimum decoding delay. (iv) Formulating the constraints of decodability as the comparison of intervals in the real number line. The theoretical results in this paper are expected to be useful for further study on AIFV codes.

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

Closed combustion devices like gas turbines and rockets are prone to thermoacoustic instabilities. Design engineers in the industry need tools to accurately identify and remove instabilities early in the design cycle. Many different approaches have been developed by the researchers over the years. In this work we focus on the Helmholtz wave equation based solver which is found to be relatively fast and accurate for most applications. This solver has been a subject of study in many previous works. The Helmholtz wave equation in frequency space reduces to a nonlinear eigenvalue problem which needs to be solved to compute the acoustic modes. Most previous implementations of this solver have relied on linearized solvers and iterative methods which as shown in this work are not very efficient and sometimes inaccurate. In this work we make use of specialized algorithms implemented in SLEPc that are accurate and efficient for computing eigenvalues of nonlinear eigenvalue problems. We make use of the n-tau model to compute the reacting source terms in the Helmholtz equation and describe the steps involved in deriving the Helmholtz eigenvalue equation and obtaining its solution using the SLEPc library.

When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant.

Artificial Intelligence (AI) is rapidly becoming integrated into military Command and Control (C2) systems as a strategic priority for many defence forces. The successful implementation of AI is promising to herald a significant leap in C2 agility through automation. However, realistic expectations need to be set on what AI can achieve in the foreseeable future. This paper will argue that AI could lead to a fragility trap, whereby the delegation of C2 functions to an AI could increase the fragility of C2, resulting in catastrophic strategic failures. This calls for a new framework for AI in C2 to avoid this trap. We will argue that antifragility along with agility should form the core design principles for AI-enabled C2 systems. This duality is termed Agile, Antifragile, AI-Enabled Command and Control (A3IC2). An A3IC2 system continuously improves its capacity to perform in the face of shocks and surprises through overcompensation from feedback during the C2 decision-making cycle. An A3IC2 system will not only be able to survive within a complex operational environment, it will also thrive, benefiting from the inevitable shocks and volatility of war.