We investigate the dynamical behavior of compositions of affine transformations over finite fields from a linear algebraic perspective. For a prime power q and integers n>m≥0, we consider functions of the form F:Fqn→Fqn defined by F(x)=T(x+Btf(Ax)), where T:Fqn→Fqn, A:Fqn→Fqm, and B:Fqn→Fqn−m are full-rank linear transformations satisfying the orthogonality condition ABt=0, and f:Fqm→Fqn−m is an arbitrary function. This construction, introduced by Gravel and Panario (2019), extends any function f into a permutation over Fqn and has applications in cryptographic schemes such as Feistel networks.Our main contribution is a complete characterization of cycle structures—the decomposition of the permutation into disjoint cycles—in terms of the spectral properties of T. When T is the identity matrix, we prove that the functional graph contains only cycles of length 1 and p (where p=char(Fq)), with the number of fixed points given by qn−m⋅|f−1(0)|. For general invertible T and constant functions f(x)=y, we establish that all cycle lengths are multiples of a minimal integer d determined by the column space of Td−I. Using the Jordan canonical form of T and Möbius inversion techniques, we derive explicit formulas for the number of elements in cycles of each length, showing that the possible cycle lengths have the form lcm(S,d,pm) where S is a subset of eigenvalue orders and m is bounded by Jordan block sizes.We extend these results to compositions of multiple linear extensions with different functions fj, proving that the cycle structure depends on the sum ∑j=1ℓfj and establishing formulas for cycle lengths in terms of greatest common divisors with the number of iterations. Our results reveal an intricate connection between the algebraic structure of linear transformations and the combinatorial properties of the resulting dynamical system.
Motivated by the classification of one-to-one rational functions of low degree in terms of their equivalence classes, we determine all many-to-one (including one-to-one) rational functions of degree two or three on the projective line explicitly in terms of their coefficients. Furthermore, we study the linear-fractional equivalence classes and the value sets of these rational functions. As an application, we characterize two classes of many-to-one quadranomials using their coefficients. These one-to-one quadranomials unify and generalize many results in the literature.
Multi-modal trajectory forecasting requires balancing predictive diversity and accuracy under a constrained sampling budget. Generative models with mode-structured priors provide strong global expressiveness; however, their instance-agnostic sampling strategies often distribute computation across trajectory modes that are weakly supported by the observed history. This paper presents a hierarchical generative framework that enables instance-adaptive sampling without modifying the underlying generative architecture. The proposed approach introduces a lightweight mode router that estimates an observation-conditioned categorical distribution over trajectory modes. This distribution is used to construct an uncertainty-adaptive feasible mode set via top-(p) selection, where the number of retained modes varies with the uncertainty of the predicted distribution. The retained probability mass is then renormalized into mode-specific sampling weights for downstream trajectory generation. By explicitly allocating the sampling budget at inference time, the proposed sampler concentrates samples on modes with higher posterior support while preserving multi-modal coverage. Experimental results show that the proposed sampler improves trajectory alignment under a fixed sampling budget. The advantage is particularly clear in the constrained Best-of-1 setting, where the prediction is limited to a single final trajectory; our method consistently outperforms MGF across ETH-UCY, SDD, and nuScenes, with additional nuScenes results showing similar gains for a DiffusionDrive-style predictor. The framework provides a practical mechanism for controlling the inference-time trade-off between diversity and accuracy, making it suitable for both real-time trajectory forecasting and decision-oriented trajectory generation in autonomous systems.
Physical activity offers numerous health and wellbeing benefits, yet people experiencing gender-based violence, housing precarity, systemic oppression, and other forms of trauma are at heightened risk of physical inactivity. Trauma- and violence-informed physical activity (TVIPA) programming is a promising approach to support physical activity amongst these populations. This study examined the effects of a multi-site, community-based TVIPA intervention for women who have experienced interpersonal or structural violence. Using a feminist participatory action research approach, interventions were co-developed with community partners and implemented across four sites in Canada, prioritizing communities with limited access to structured physical activity programs. Each site offered a series of 7 six-week physical activity programs (e.g., dance, swimming, yoga) designed to foster safety, empowerment, and connection. Immediate effects were assessed using pre- and post-session mood ratings, and longer-term effects were captured through pre- and post-program measures of subjective mental and physical wellbeing across the six-week series. Of the 168 enrolled participants, 143 contributed repeated self-reported mood ratings before and after physical activity sessions. Participants reported moderately positive mood before sessions and further elevated positive mood afterward (M change = 0.94/5, SD = 0.94, d = 1.00), with the greatest improvements occurring on days when they began in a less positive mood. Across the six-week program, greater attendance was associated with improvements in mental health (b = 0.06, p < .001), but with lower physical health scores (b = −0.06, p < .001). Findings underscore the potential of community-driven, trauma- and violence-informed physical activity to enhance wellbeing among women affected by violence and structural inequities.
Long Range (LoRa) networks have emerged as a vital solution catering to applications requiring coverage over relatively long distances in the world of Internet of Things (IoT), where connectivity and efficient data transmission are paramount. LoRa’s utilization of the unlicensed Industrial, Scientific, and Medical (ISM) band not only underscores its cost-effectiveness but also positions it favorably against licensed technologies in terms of deployment cost. Consequently, LoRa has been used as the underlying communication technology for applications across various industrial IoT scenarios. Despite its immense promise in reshaping IoT connectivity, LoRa does have some shortcomings and challenges that the research community has yet to address to unleash its full potential. These limitations have triggered substantial attention from diverse entities, including research institutions, organizations, and industry stakeholders. This paper reviews the literature aimed at improving the capacity and scalability of LoRa networks specifically at the Medium Access Control (MAC) and data link layers. Unlike other surveys, this study focuses on these layers because they play a pivotal role in managing collision rates, which significantly impact network scalability. The paper suggests a comprehensive review of the literature, organizing it based on key limitations that could hinder the network’s ability to meet its performance objectives, including scalability, Packet Delivery Ratio (PDR), and energy efficiency. In addition, the paper provides a summary of these research efforts and offers insight into potential directions for future research in this area.
