In this paper, we derive the exact formula for the probability that three randomly and uniformly selected points from the interior of the unit cube form vertices of an obtuse triangle.

This article critically examines how Web3 decentralization policy trends impact global digital governance, questioning whether they genuinely distribute power or merely shift influence to a new, tech-savvy elite. Based on fieldwork in Silicon Valley since August 2022 and engagement with scholars and practitioners up to December 2025, the article provides a conceptual analysis with emerging empirical insights around the nascent global Web3 movement. While Web3 advocates challenge centralized data monopolies and traditional state structures, this analysis critiques the assumption that Web3 democratizes power, highlighting both its potential for inclusion and risks of exclusion, insofar as it may reinforce hierarchies rooted in technical expertise and digital access. While acknowledging the broader landscape of Web3 governance (including hybrid and federated models) and scoping the Global North and Global South contexts considering global adoption cases, the article particularly focuses on three post-Westphalian paradigms: (i) Network States, (ii) Network Sovereignties, and (iii) Algorithmic Nations. While Network States advocate for crypto-libertarian governance, Network Sovereignties and Algorithmic Nations emphasize cooperative governance aimed at empowering minority communities, such as indigenous groups, stateless nations, and e-diasporas, through decentralized, data-driven systems. By engaging with both the limitations and some promises, prospects, and pitfalls of Web3, this article questions whether Web3 can create a more inclusive global order or if influence is increasingly concentrated among a new elite. This article contributes to debates on sovereignty, governance, and citizenship by advocating hybrid policy frameworks that balance global and local dynamics, emphasizing solidarity, digital justice, and international cooperation for equitable Web3 governance.

We consider general discrete-time multitype branching processes on a countable set X. According to these processes, a particle of type $x\in X$ generates a random number of children and chooses their type in X, not necessarily independently nor with the same law for different parent types. We introduce a new type of stochastic ordering of multitype branching processes, generalising the germ order introduced by Hutchcroft, which relies on the generating function of the process. We prove that given two multitype branching processes with laws ${\boldsymbol{\mu}}$ and ${\boldsymbol{\nu}}$ respectively, with ${\boldsymbol{\mu}}\ge{\boldsymbol{\nu}}$, then in every set where there is survival according to ${\boldsymbol{\nu}}$, there is also survival according to ${\boldsymbol{\mu}}$. Moreover, in every set where there is strong survival according to ${\boldsymbol{\nu}}$, there is also strong survival according to ${\boldsymbol{\mu}}$, provided that the supremum of the global extinction probabilities for the ${\boldsymbol{\nu}}$ process, taken over all starting points x, is strictly smaller than 1. New conditions for survival and strong survival for inhomogeneous multitype branching processes are provided. We also extend a result of Moyal which claims that, under some conditions, the global extinction probability for a multitype branching process is the only fixed point of its generating function, whose supremum over all starting coordinates may be smaller than 1.

The purpose of this paper is to analyze the degree index and the clustering index in dense random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the corresponding theoretical analysis in a random graph setup turns out to be tractable. On the other hand, the clustering index, based on a similar reasoning, is first introduced in this paper. Computing exact expressions for the expected clustering index turns out to be more challenging even in the case of Erdős–Rényi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. In addition to the Erdős–Rényi case, we also present a simulation-based analysis for random regular graphs, the Barabási–Albert model, and the Watts–Strogatz model.

Given a sequence of graphs $G_n$ and a fixed graph H, denote by $T(H, G_n)$ the number of monochromatic copies of the graph H in a uniformly random c-coloring of the vertices of $G_n$. In this paper we study the joint distribution of a finite collection of monochromatic graph counts in networks with multiple layers (multiplex networks). Specifically, given a finite collection of graphs $H_1, H_2, \ldots, H_d$ we derive the joint distribution of $(T(H_1, G_n^{(1)}), T(H_2, G_n^{(2)}), \ldots, T(H_d, G_n^{(d)}))$, where $\mathbf{G}_n = (G_n^{(1)}, G_n^{(2)}, \ldots, G_n^{(d)})$ is a collection of dense graphs on the same vertex set converging in the multiplex cut-metric. The limiting distribution is the sum of two independent components: a multivariate Gaussian and a sum of independent bivariate stochastic integrals. This extends previous results on the marginal convergence of monochromatic subgraphs in a sequence of graphs to the joint convergence of a finite collection of monochromatic subgraphs in a sequence of multiplex networks. Several applications and examples are discussed.

We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.

The goal of the Paris Agreement is to prevent global temperatures from rising by more than 2°C above pre-industrial levels and pursue efforts to limit them to 1.5°C above pre-industrial levels. This requires a significant reduction in global greenhouse gas emissions and achieving net zero emissions by 2050. Portfolio alignment metrics are forward-looking metrics intended to help investors understand whether their investment portfolios are on track to meet the Paris Agreement goals. They also aim to encourage capital flows towards activities needed for a net zero transition. Since 2020, several metrics have been put forward by industry groups and explored in technical papers. Companies and actuaries have been exploring the practicalities of these metrics and starting to incorporate them into investment reporting and design. But this has not been without key challenges. The Net Zero and Implications for Investment Portfolios working party aims to help actuaries improve their understanding of what net zero means for an investment portfolio and what the key mechanisms are to achieve this, as well as key challenges to date and the outlook for development.

In this paper we propose a refracted skew Brownian motion as a risk model with endogenous regime switching, which generalizes the refracted diffusion risk process introduced by Gerber and Shiu. We consider an optimal dividend problem for the refracted skew Brownian risk model and identify sufficient conditions, respectively, for barrier strategy, band strategy, and their variants to be optimal.

Extropy-based divergence measures offer distinct advantages over entropy-based counterparts, owing to their mathematical simplicity and enhanced interpretability. Relative extropy by Lad et al. [5] is a symmetric divergence measure between two probability distributions, and Mohammadi et al. [8] introduced the asymmetric divergence between two distributions based on extropy. We further study these measures, their properties, and interrelationships in this article. To address the divergence between truncated lifetime distributions, we define dynamic relative extropy for residual and past lifetime scenarios. Exploring the interrelationships of dynamic cases of relative extropy, extropy divergence, and extropy inaccuracy, we derive some unique properties and characterizations for the exponential distribution. A nonparametric estimator for relative extropy is developed, and its performance is assessed through numerical simulation studies. The practical applicability of relative extropy is used to analyze the divergence in lifetime patterns of mice under a lifetime feeding experiment and the shopping patterns of customers based on age and income groups. Further, the application of relative extropy is also applied to find the dissimilarity between two images.

Past research has indicated that the covariance of the stochastic gradient descent (SGD) error done via minibatching plays a critical role in determining its regularization and escape from low potential points. Motivated by some new research in this area, we prove universality results by showing that noise classes that have the same mean and covariance structure of SGD via minibatching have similar properties. We mainly consider the SGD algorithm, with multiplicative noise, introduced in previous work (Wu et al (2016) Int. Conf. on Machine Learning, PMLR, pp. 10367–10376), which has a much more general noise class than the SGD algorithm done via minibatching. We establish non-asymptotic bounds for the multiplicative SGD algorithm in the Wasserstein distance. We also show that the error term for the algorithm is approximately a scaled Gaussian distribution with mean 0 at any fixed point.

We consider the asymptotic behaviour of the expectation of the perimeter deviation of a uniform random spherical disc–polygon in a spherical spindle convex disc with smooth boundary. We also introduce the notion of duality on the sphere, define a model of random circumscribed disc–polygons, and determine some asymptotic results about them.

Consecutive $k$-type systems have become important in both reliability theory and applications; in spite of a large literature existing on them, three-dimensional consecutive $k$-type systems have not yet been studied for multi-state case. In this paper, we introduce several different types of multi-state linear three-dimensional consecutive $k$-type systems for the first time, with due consideration to possible overlapping of failure blocks. The finite Markov chain imbedding approach is then used for the derivation of their reliability functions with state spaces and transition matrices provided in a novel way, and the involved computational process is illustrated through several numerical examples. Finally, some possible applications of the work and potential extensions are pointed out.

We consider a critical bisexual branching process in a random environment generated by independent and identically distributed random variables. Assuming that the process starts with a large number of pairs N, we prove that its extinction time is of order $\ln^2 N$. Interestingly, this result is valid for a general class of mating functions. Among these are the functions describing the monogamous and polygamous behavior of couples, as well as the function reducing the bisexual branching process to the simple one.

This article studies uniform inference on a function $g(\cdot )$ and its functionals in a nonparametric panel data model with fixed effects. The nonparametric panel model relaxes restrictions on time-series behavior by allowing for arbitrary types of stationary or nonstationary dependence (e.g., stationary mixingale, mildly stationary, or local-to-unity process). After removing the fixed effects via transformations, a sieve estimator is proposed, accompanied by Yurinskii’s coupling principle of Gaussian processes and uniform confidence bands (UCBs) that rely on the sieve score bootstrap method to test for linear functionals of $g(\cdot )$. Under the asymptotic framework of an increasing cross-sectional dimension and either a fixed or diverging time dimension, we prove that the bootstrapping Kolmogorov–Smirnov (sup-type) test has asymptotic uniform size controls. This article shows that our uniform inference procedure can be extended to the two-way fixed-effects nonparametric panel model with stationary mixingale regressors. Extensive simulations confirm that our sieve estimators and their UCBs work well in finite samples. The present article further applies the above methods to empirical settings and finds some interesting results in nonlinear patterns of consumption concerning income shocks and asset holdings.

In this paper, we study the self-normalized Cramér-type moderate deviation of the empirical measure of the stochastic gradient Langevin dynamics (SGLD). Consequently, we also derive the Berry–Esseen bound for the SGLD. Our approach is by constructing a stochastic differential equation to approximate the SGLD and then applying Stein’s method to decompose the empirical measure into a martingale difference series sum and a negligible remainder term.

The COVID-19 pandemic and associated non-pharmaceutical interventions (NPIs) reduced transmission of other infections. We quantified changes in hospital admission rates for respiratory and gastrointestinal infections among young children in England during and after implementation of NPIs, compared to pre-pandemic, and variations by sociodemographic and clinical characteristics. Children aged <5 years at any time between 1 January 2017 and 31 January 2022 were followed from birth or 1 January 2017, until their 5th birthday, death, or 31 January 2022, within a birth cohort based on Hospital Episode Statistics data. Quarterly emergency admission rates for respiratory and gastrointestinal infections from April-June 2020 onwards were compared to corresponding quarters in 2017–2019 using Poisson regression, with and without interaction terms for time period and sociodemographic/clinical characteristics. Admission rates for respiratory and gastrointestinal infections were lower in April–June 2020 compared to this quarter pre-pandemic (incidence rate ratio (99% CI) 0.17 (0.17–0.18) for respiratory; 0.29 (0.28–0.31) for gastrointestinal). Rates remained below pre-pandemic levels until April–June 2021 (respiratory infections) and July–September 2021 (gastrointestinal infections), subsequently increasing above the corresponding pre-pandemic quarters. Changes in rates did not differ by sociodemographic/clinical characteristics. These results can inform planning for future pandemics and their aftermath.

Antimicrobial resistance (AMR) is a pressing global health challenge, with sub-Saharan Africa experiencing the highest burden of AMR-related deaths. Inappropriate prescribing and rising antibiotic consumption drive AMR, while limited local data hampers antimicrobial stewardship efforts. This study analysed Global Point Prevalence Survey of Antimicrobial Consumption and Resistance (Global-PPS) data from Tygerberg Hospital to identify antimicrobial use trends and inform stewardship priorities. Standard Global-PPS methodology was employed at three distinct time points. All inpatients prescribed at least one antimicrobial on the day of each survey were included in the analysis. Among 3,524 hospitalized patients, 25.9% (911/3,524) received antimicrobial therapy. Overall antimicrobial use decreased significantly (p < 0.05), with the largest reduction among paediatric patients (p < 0.01). Community-acquired infections accounted for the majority of prescriptions (50.7%; 483/952) and empirical antibiotic use was high (85.3%, 872/1022). ‘Access’ antibiotics constituted 62.7% (750/1196) of prescriptions. Single-dose prescriptions for surgical prophylaxis accounted for 17.6% (15/85). This study demonstrates progress in stewardship, particularly among paediatric inpatients. Ongoing monitoring of broad-spectrum antibiotic use and adherence to single-dose surgical prophylaxis guidelines are essential priorities. Continued Global-PPS surveillance is crucial to track trends and guide future AMS interventions.

Building energy management (BEM) tasks require processing and learning from a variety of time-series data. Existing solutions rely on bespoke task- and data-specific models to perform these tasks, limiting their broader applicability. Inspired by the transformative success of Large Language Models (LLMs), Time-Series Foundation Models (TSFMs), trained on diverse datasets, have the potential to change this. Were TSFMs to achieve a level of generalizability across tasks and contexts akin to LLMs, they could fundamentally address the scalability challenges pervasive in BEM. To understand where they stand today, we evaluate TSFMs across four dimensions: (1) generalizability in zero-shot univariate forecasting, (2) forecasting with covariates for thermal behavior modeling, (3) zero-shot representation learning for classification tasks, and (4) robustness to performance metrics and varying operational conditions. Our results reveal that TSFMs exhibit limited generalizability, performing only marginally better than statistical models on unseen datasets and modalities for univariate forecasting. Similarly, inclusion of covariates in TSFMs does not yield performance improvements, and their performance remains inferior to conventional models that utilize covariates. While TSFMs generate effective zero-shot representations for downstream classification tasks, they may remain inferior to statistical models in forecasting when statistical models perform test-time fitting. Moreover, TSFMs’ forecasting performance is sensitive to evaluation metrics, and they struggle in more complex building environments compared to statistical models. These findings underscore the need for targeted advancements in TSFM design, particularly their handling of covariates and incorporating context and temporal dynamics into prediction mechanisms, to develop more adaptable and scalable solutions for BEM.