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In this paper, we consider how probability theory can be used to determine the survival strategy in two of the ``Squid Game" and ``Squid Game: The Challenge" challenges: the Hopscotch and the Warships. We show how Hopscotch can be easily tackled with the knowledge of the binomial distribution, taught in introductory statistics courses, while Warships is a much more complex problem, which can be tackled at different levels., Comment: 10 pages, 5 figures

Generative Artificial Intelligence image models have achieved outstanding performance in text-to-image generation and other tasks, such as inpainting that completes images with missing fragments. The performance of inpainting can be accurately measured by taking an image, removing some fragments, performing the inpainting to restore them, and comparing the results with the original image. Interestingly, inpainting can also be applied recursively, starting from an image, removing some parts, applying inpainting to reconstruct the image, and then starting the inpainting process again on the reconstructed image, and so forth. This process of recursively applying inpainting can lead to an image that is similar or completely different from the original one, depending on the fragments that are removed and the ability of the model to reconstruct them. Intuitively, stability, understood as the capability to recover an image that is similar to the original one even after many recursive inpainting operations, is a desirable feature and can be used as an additional performance metric for inpainting. The concept of stability is also being studied in the context of recursive training of generative AI models with their own data. Recursive inpainting is an inference-only recursive process whose understanding may complement ongoing efforts to study the behavior of generative AI models under training recursion. In this paper, the impact of recursive inpainting is studied for one of the most widely used image models: Stable Diffusion. The results show that recursive inpainting can lead to image collapse, so ending with a nonmeaningful image, and that the outcome depends on several factors such as the type of image, the size of the inpainting masks, and the number of iterations.

Open-source simulation tools play a crucial role for neuromorphic application engineers and hardware architects to investigate performance bottlenecks and explore design optimizations before committing to silicon. Reconfigurable Architecture for Neuromorphic Computing (RANC) is one such tool that offers ability to execute pre-trained Spiking Neural Network (SNN) models within a unified ecosystem through both software-based simulation and FPGA-based emulation. RANC has been utilized by the community with its flexible and highly parameterized design to study implementation bottlenecks, tune architectural parameters or modify neuron behavior based on application insights and study the trade space on hardware performance and network accuracy. In designing architectures for use in neuromorphic computing, there are an incredibly large number of configuration parameters such as number and precision of weights per neuron, neuron and axon counts per core, network topology, and neuron behavior. To accelerate such studies and provide users with a streamlined productive design space exploration, in this paper we introduce the GPU-based implementation of RANC. We summarize our parallelization approach and quantify the speedup gains achieved with GPU-based tick-accurate simulations across various use cases. We demonstrate up to 780 times speedup compared to serial version of the RANC simulator based on a 512 neuromorphic core MNIST inference application. We believe that the RANC ecosystem now provides a much more feasible avenue in the research of exploring different optimizations for accelerating SNNs and performing richer studies by enabling rapid convergence to optimized neuromorphic architectures., Comment: Accepted for publication in Neuro-Inspired Computational Elements (NICE) Workshop 2024

The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton-Watson process with migration introduced in [Yanev & Mitov (1980) C. R. Acad. Bulg. Sci. 33(4):473-475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation., Comment: 12 pages

The growing interest in Large Language Models (LLMs) and in particular in conversational models with which users can interact has led to the development of a large number of open-source chat LLMs. These models are evaluated on a wide range of benchmarks to assess their capabilities in answering questions or solving problems on almost any possible topic or to test their ability to reason or interpret texts. Instead, the evaluation of the knowledge that these models have of the languages has received much less attention. For example, the words that they can recognize and use in different languages. In this paper, we evaluate the knowledge that open-source chat LLMs have of Spanish words by testing a sample of words in a reference dictionary. The results show that open-source chat LLMs produce incorrect meanings for an important fraction of the words and are not able to use most of the words correctly to write sentences with context. These results show how Spanish is left behind in the open-source LLM race and highlight the need to push for linguistic fairness in conversational LLMs ensuring that they provide similar performance across languages., Comment: Procesamiento del Lenguaje Natural, 73, 95-108

Let $G$ be a complex semisimple Lie group and $\mathfrak g$ its Lie algebra. In this paper, we study a special class of cyclic Higgs bundles constructed from a $\mathbb Z$-grading $\mathfrak g = \bigoplus_{j=1-m}^{m-1}\mathfrak g_j$ by using the natural representation $G_0 \to GL(\mathfrak g_1 \oplus \mathfrak g_{1-m})$, where $G_0 \le G$ is the connected subgroup corresponding to $\mathfrak g_0$. The resulting Higgs pairs include $G^{\mathbb R}$-Higgs bundles for $G^{\mathbb R} \le G$ a real form of Hermitian type (in the case $m=2$) and fixed points of the ${\mathbb C}^*$-action on $G$-Higgs bundles (in the case where the Higgs field vanishes along $\mathfrak g_{1-m}$). In both of these situations a topological invariant with interesting properties, known as the Toledo invariant, has been defined and studied in the literature. This paper generalises its definition and properties to the case of arbitrary $(G_0,\mathfrak g_1 \oplus \mathfrak g_{1-m})$-Higgs pairs, which give rise to families of cyclic Higgs bundles. The results are applied to the example with $m=3$ that arises from the theory of quaternion-K\"ahler symmetric spaces., Comment: 31 pages

A discrete time branching process where the offspring distribution is generation-dependent, and the number of reproductive individuals is controlled by a random mechanism is considered. This model is a Markov chain but, in general, the transition probabilities are non-stationary. Under not too restrictive hypotheses, this model presents the classical duality of branching processes: either becomes extinct almost surely or grows to infinity. Sufficient conditions for the almost sure extinction and for a positive probability of indefinite growth are provided. Finally rates of growth of the process provided the non-extinction are studied., Comment: 15 pages. This is a plain preprint version of the article that was published, after peer review, and is subject to Springer Nature's terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s10986-015-9265-0

A branching process in varying environment with generation-dependent immigration is a modification of the standard branching process in which immigration is allowed and the reproduction and immigration laws may vary over the generations. This flexibility makes the process more appropriate to model real populations due to the fact that the stability in the reproductive capacity and in the immigration laws are not usually fulfilled. In this setting, we study the extinction problem, providing a necessary and sufficient condition for the certain extinction of the population. The asymptotic behaviour of the model is analysed for those processes with critical offspring distributions, according with the classification established in Kersting (2017), and when the immigration means stabilize to a positive value. More specifically, we establish that the asymptotic distribution of the process -- under a suitable normalization -- belongs to the gamma distribution family., Comment: 18 pages, 2 figures. This is the preprint version of the paper published in the journal Stochastic Models (see the official journal website at https://doi.org/10.1080/15326349.2019.1575754)

Models of invasive species spread often assume that landscapes are spatially homogeneous; thus simplifying analysis but potentially reducing accuracy. We extend a recently developed partial differential equation model for invasive conifer spread to account for spatial heterogeneity in parameter values and introduce a method to obtain key outputs (e.g. spread rates) from computational simulations. Simulations produce patterns of spatial spread remarkably similar to observed patterns in grassland ecosystems invaded by exotic conifers, validating our spatially explicit strategy. We find that incorporating spatial variation in different parameters does not significantly affect the evolution of invasions (which are characterised by a long quiescent period followed by rapid evolution towards to a constant rate of invasion) but that distributional assumptions can have a significant impact on the spread rate of invasions. Our work demonstrates that spatial variation in site-suitability or other parameters can have a significant impact on invasions, Comment: 13 pages, 18 figures

Invasive pine trees pose a threat to biodiversity in a variety of Southern Hemisphere countries, but understanding of the dynamics of invasions and the factors that retard or accelerate spread is limited. Here, we consider the past models of wilding pine spread and develop a new model of pine invasion. We show that many prior models feature parameter estimates which are not biologically supported and rely on a conjecture to obtain an asymptotic spread speed of invasive pine populations, the main output of these models. In contrast to prior approaches, we use partial differential equations to model an invasion. We show that invasions are almost static for a significant period of time before rapidly accelerating to spread at a constant rate, matching observed behaviour in at least some field sites. Our work suggests that prior methods for estimating invasion speeds may not accurately predict spread and are sensitive to assumptions about the distribution of parameters. However, we present alternative estimation methods and suggest directions for further research., Comment: 36 pages, 9 figures

(Photo)electrocatalysts capture sunlight and use it to drive chemical reactions such as water splitting to produce H2. A major factor limiting photocatalyst development is their large heterogeneity which spatially modulates reactivity and precludes establishing robust structure-function relationships. To make such links requires simultaneously probing of the electrochemical environment at microscopic length scales (nm to um) and broad timescales (ns to s). Here, we address this challenge by developing and applying in-situ steady-state and transient optical microscopies to directly map and correlate local electrochemical activity with hole lifetimes, oxygen vacancy concentration and the photoelectrodes crystal structure. Using this combined approach alongside spatially resolved X-Ray absorption measurements, we study microstructural and point defects in prototypical hematite (Fe2O3) photoanodes. We demonstrate that regions of Fe2O3, adjacent to microstructural cracks have a better photoelectrochemical response and reduced back electron recombination due to an optimal oxide vacancy concentration, with the film thickness and carbon impurities also dramatically influencing activity in a complex manner. Our work highlights the importance of microscopic mapping to understand activity and the impact of defects in even, seemingly, homogeneous solid-state metal oxide photoelectrodes.

Branching processes form an important family of stochastic processes that have been successfully applied in many fields. In this paper, we focus our attention on controlled multi-type branching processes (CMBPs). A Feller-type diffusion approximation is derived for some critical CMBPs. Namely, we consider a sequence of appropriately scaled random step functions formed from a critical CMBP with control distributions having expectations that satisfy a kind of linearity assumption. It is proved that such a sequence converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of a matrix related to the offspring mean matrix and the control distributions of the branching process in question. As applications, among others, we derive Feller-type diffusion approximations of critical, primitive multi-type branching processes with immigration and some two-sex branching processes. We also describe the asymptotic behaviour of the relative frequencies of distinct types of individuals for critical CMBPs., Comment: 41 pages