Abstract: Multicellular individuality depends upon constraints that limit the establishment and evolutionary success of genetically foreign cellular lineages. Naturally transmissible cancers expose the permeability of these constraints because malignant cells can propagate between unrelated organisms. Although ecological barriers to transmission, allorecognition and host–parasite genotype-by-genotype interactions are established evolutionary concepts, their integration into a stage-resolved account of somatically derived cellular parasitism remains incomplete. This review critically synthesises evidence from transmissible cancers in dogs, Tasmanian devils, bivalves and freshwater catfish alongside genetically regulated cellular recognition in colonial invertebrates. Historical experimental transmission in soft-shell clams, recent environmental-survival studies, comparative tumour genomics and contemporary host population-genetic responses provide complementary evidence. A Genetic Firewall Framework is proposed as a restricted application of established host–parasite compatibility theory to clonally transmissible malignant cells. It distinguishes exposure, viable-cell entry, recognition-mediated elimination, post-entry niche compatibility and onward transmission. An explicit recipient-genotype-by-clone-genotype establishment model is linked to host fitness and malignant-lineage frequency dynamics. The primary empirical prediction concerns reproducible, stage-specific compatibility effects, with susceptibility-rank reversal representing a particularly discriminating but non-essential interaction pattern. A falsification-oriented programme centred on Mya arenaria separates inherited resistance from tumour-intrinsic attenuation, environmental confounding and phylogenetic pseudoreplication. The framework neither assumes homologous recognition systems across taxa nor claims an established universal antitumour mechanism. Its value lies in determining when genetic compatibility explains cellular parasitism and when simpler ecological or physiological explanations prevail.

Abstract: Pancreatic ductal adenocarcinoma (PDAC) is one of the most aggressive human cancers and is characterized by a high frequency of activating KRAS mutations, most commonly KRASG12D. Yet oncogenic KRAS alone is insufficient for malignant progression, indicating that additional cellular and environmental factors shape disease evolution. Obesity is a major modifiable risk factor for PDAC and creates a chronic metabolic environment marked by excess nutrient and lipid availability, inflammation, and altered redox homeostasis. In acinar cells, these conditions can increase metabolic and mitochondrial stress, promoting reactive oxygen species (ROS) generation and molecular and organellar damage. Cellular responses to such damage, however, do not necessarily lead to either complete recovery or cell death. Antioxidant defenses, autophagy, mitochondrial quality control, proteostasis, and DNA repair can contain or process damage, allowing cells to remain viable despite persistent or incompletely resolved injury. Thus, cellular fitness under chronic stress may depend not only on the amount of damage generated, but also on the capacity to tolerate damage while maintaining sufficient function. Here, we conceptualize the relationship between damage burden and cellular fitness as a dynamic redox damage tolerance threshold. Transient metabolic and redox stress may allow acinar cells undergoing acinar-to-ductal metaplasia (ADM) to return toward homeostasis, whereas sustained metabolic stress, as occurs in obesity, in the presence of oncogenic KRAS may enable ADM cells to remain functional despite residual damage. Such persistence may increase the opportunity for additional alterations and progression toward pancreatic intraepithelial neoplasia (PanIN). This framework places cellular damage tolerance, rather than damage generation alone, at the center of early PDAC development and highlights the mechanisms that determine this threshold as potential therapeutic vulnerabilities.

Abstract: Charcoal in sub-Saharan Africa is commonly described as a fuel that villagers produce for urban consumers while they themselves cook with collected wood. We tested this view in three purposively selected villages around Kolwezi, Democratic Republic of the Congo, asking whether residual fuelwood use is associated mainly with household attributes or with the village. We surveyed 235 households and fitted nested Firth logistic regressions, adding village after household attributes. Charcoal was the main cooking fuel of 97.4% of households and, by recall, had already been so for 92.8% five years earlier. Fuelwood remained a main or secondary fuel for 24.7% of households, from 2.0% in Kawama to 51.0% in Mushima. Household attributes discriminated poorly between users and non-users (Tjur R² = 0.09), whereas village alone discriminated better (Tjur R² = 0.28). Charcoal producers had higher odds of fuelwood use before adjustment (odds ratio 2.86, 95% CI 1.56–5.31) but not within villages (Mantel–Haenszel odds ratio 0.86, 95% CI 0.38–1.92), and no clear association was detected with income or education. Villagers, producers included, are thus mainly charcoal users. Energy and forest policies should count these villages among charcoal consumers, and the village conditions behind residual fuelwood use warrant testing.

Abstract: Semantic entropy, the entropy of the meanings among a language model's sampled answers, is estimated from noisy pairwise judgments. Its minimax theory assumes a judge that recognizes every meaning equally well. We show that a meaning-dependent sensitivity changes the minimax rate even when it is known exactly and the judge recognizes every meaning at least as well as a homogeneous one. Two answers with meaning \( c \) are declared equivalent with probability \( q+\Delta w_c \), two with different meanings with probability \( q \), where \( q \) is bounded away from 0 and 1 and the number \( K\ge3 \) of meanings is fixed. For almost every sensitivity profile \( w \), the entropy is uniformly consistently estimable exactly when \( N:=n^{2K-1}\Delta^{4K-4}\to\infty \): for \( \Delta=n^{-\beta} \) the threshold falls from \( \beta<K/(2K-2) \) for a homogeneous judge to \( \beta<(2K-1)/(4K-4) \), that of such a judge on \( 2K-1 \) meanings, and from 3/4 to 5/8 for three meanings. On every fixed polynomial window \( n\Delta^2\le n^{-\kappa} \) the minimax squared risk is of order \( \min\{1,N^{-1/(2K-1)}\} \). Star statistics computable in \( O(n^2K) \) time and a projection attain this rate through a weighted entropy modulus with optimal exponent \( 1/(2K-1) \). The lower bound uses a moment-preserving path near the interior point \( \theta\propto1/w \), along which the judge's channel moves with the distribution.

Abstract: Background: Point-of-care (POC) diagnostics could improve timely diagnosis of skin neglected tropical diseases (skin NTDs) in settings with limited access to laboratory services. However, evidence on their diagnostic performance, operational feasibility and readiness for decentralized implementation remains fragmented. Methods: We systematically searched PubMed, Web of Science, Google Scholar, and grey literature sources for studies evaluating POC diagnostic technologies for skin NTDs. Diagnostic performance, evidence maturity and operational characteristics were synthesized narratively because of heterogeneity across technologies, populations, reference standards, and reported outcomes. Diagnostic-accuracy studies were assessed using the Quality Assessment of Diagnostic Accuracy Studies-2 (QUADAS-2). Results: A short summary of the article’s main findings. Conclusions: POC diagnostics for skin NTDs show promising technological development but uneven evidence of maturity. Molecular technologies have generally progressed further in analytical development than in clinical and field validation, while simpler rapid tests have greater field experience but variable performance. Future research should prioritize prospective clinical and implementation-focused evaluation of complete diagnostic workflows in representative endemic settings.

Abstract: In this paper a turbineless hybrid electric-hydrogen propulsion system was built and tested under static conditions. Two electric ducted fans (EDFs) supplied the baseline thrust and combustion airflow. An alkaline aluminum-water reactor generated hydrogen on demand for combustion downstream of the fans. Rod immersion depth controlled hydrogen production, which was measured at each condition with a hydrogen-calibrated mass-flow meter. Aluminum consumption was also measured to determine the effective hydrogen yield. Tests covered ten immersion depths, with 76 independent paired EDF-only and hybrid runs per condition. At the highest hydrogen flow, thrust rose from 8.49 ± 0.15 to 9.94 ± 0.21 N, an increase of 17.1%. Hydrogen production increased approximately linearly with exposed aluminum area (R² = 0.997), while the marginal thrust gain decreased. The momentum-based combustion-to-jet conversion indicator ranged from 7.3 to 9.1% and was highest at intermediate hydrogen flows. Chamber pressure and fan supply power were measured in a separate session with paired runs without and with combustion. Measured thrust was 1.7 to 3.5% below the ideal nozzle prediction from chamber pressure. Combustion increased fan supply power from 1.49 kW to between 1.60 and 2.02 kW. As hydrogen flow increased, jet kinetic power fell from 40% to 19% of the combined electrical and hydrogen chemical exergy input.

Abstract: Cisplatin-induced hearing loss (CIHL) is a debilitating side effect in cancer patients undergoing chemotherapy. So far, epigenetic studies in the inner ear have largely focused on developmental biology to regenerate hair cells. In post-mitotic hair cells, on the other hand, DNA methylation regulates transcriptomic response to external stimuli. Adult mammalian hair cells are locked in the G0 phase, and passive demethylation via DNA replication does not occur under normal physiological conditions. However, external stimuli such as cisplatin can aberrantly upregulate DNMT methyltransferases (DNMTs), resulting in promoter methylation and silencing protective genes. From this angle, DNMTs modulation is an attractive therapeutic strategy to protect hearing during chemotherapy. Recent studies suggest that non-nucleoside inhibitors such as RG108 and natural polyphenolic compounds downregulate DNMTs, and at the same time, protect auditory cells. These compounds function via the inhibition of the DNMT catalytic site but can also act through downregulating DNMT1 and DNMT3A/B at mRNA and protein levels. Polyphenols are also antioxidants, decreasing reactive oxygen species, thereby increasing cell survival. Furthermore, it is suggested that upregulating TET-mediated active demethylation can be another angle to modify DNA methylation in cancer and CIHL models. However, functional DNA methylation should be distinguished from non-functional, passenger DNA methylation to elucidate the compounds’ otoprotective effects. In addition, the candidate compounds should not interfere with cisplatin’s antitumor effect. In this review, we investigate the relationship between DNA methylation and CIHL in the post-mitotic inner ear and evaluate the therapeutic potential of candidate DNMT inhibitors.

Abstract: Three-dimensional Gaussian splatting is the latest computer-graphics technique to be adopted for medical visualisation, following volume rendering and cinematic rendering. Unlike its predecessors, which render images directly from the acquired volumetric data, Gaussian splatting is a reconstruction: a cloud of anisotropic Gaussian primitives is optimised against a finite set of rendered two-dimensional views, after which the source volume plays no further part. This distinction has practical consequences that are not evident from the exterior images typically used to demonstrate the technique. In this narrative review we describe what we term the view-space limitation: because training cameras orbit an anatomical structure from outside, the interior is never observed, and no clipping-plane position recovers intracardiac anatomy from a cardiac splat. We argue that this is a structural consequence of camera distribution rather than an implementation defect, that a splat is an inferred representation whose geometric and densitometric fidelity must be measured rather than assumed, and that until such validation is reported, splat reconstructions should be reserved for communication, education, and qualitative review rather than primary quantitative interpretation. We also argue that the case in favour is stronger than usually made: a splat of a patient’s heart renders at interactive frame rates in a consumer smartphone browser and can be shared by a link, extending photorealistic anatomy to referring clinicians, multidisciplinary meetings, trainees, patients, and health systems without workstation-class hardware. We propose a framework for matching visualisation technique to clinical setting and outline validation priorities, including accuracy against ground truth, projection-domain sparse-view formulations, clippable variants, and dynamic extensions for gated cardiac data, that should precede further application papers.

Abstract: Vehicular safety communications rely on authenticated basic safety messages (BSMs); however, cryptographic authentication cannot prevent compromised insiders from disseminating correctly signed yet malicious information. Existing misbehavior detection systems often rely on individual-message or isolated kinematic analysis, limiting their ability to identify plausible falsifications, identity-based attacks, and coordinated multi-vehicle attacks. In this paper, we propose TRUST-V, a prior-guided spatio-temporal graph learning framework for cross-layer V2X misbehavior detection. TRUST-V first applies physical plausibility, freshness, and transmission-rate checks to remove readily identifiable violations and derive vehicle-level behavioral evidence (Stage 1). It then constructs dynamic interaction graphs in which spatial, mobility, and communication consistency priors explicitly guide graph attention while temporal dependencies are captured through recurrent modeling (Stage 2). Independent multi-label heads enable unified detection of attack classes related to false BSMs, denial-of-service, Sybil-based identity churning, coordinated multi-vehicle misbehavior, replay, identity cloning, and the overlapping behaviors of these attacks. We further develop a high-resolution Network Simulator 3 (NS-3)-generated dataset comprising 830,706 BSM records spanning benign, individual, stealthy, identity-based, compound, and coordinated attack scenarios. Evaluations demonstrate that Stage 1 removes 39.73% of the raw BSM workload while the Stage 2 graph-based analysis identifies most relational behaviors. Experimental results show an overall micro-F1 score of 93.16%, with F1-scores of 93.20%, 97.44%, 93.19%, 88.30%, 98.90%, and 96.94% for False-BSM, DoS, Sybil, Coordinated, Replay, and Identity-Clone attacks, respectively. On a Raspberry Pi 5, the complete online pipeline requires approximately 8.63 ms per 100-ms observation window under the evaluated workload, demonstrating the feasibility of combining physical knowledge with spatio-temporal relational learning for low-latency V2X misbehavior detection.

Abstract: Nuclear fusion is a pathway toward clean, virtually limitless energy. Its practical use requires the world’s greatest engineering skills alongside the best mathematicians, physicists, and the most capa- ble technology from the extremes of what is possible on Earth. This literature review evaluates a varied selection of reactor designs, including tokamaks, stellarators, inertial confinement systems, and Helion En- ergy’s hybrid Field Reversed Configuration (FRC). The main focus is directed at recent high-temperature superconductor (HTS) advancements, detailing the micron level layering of rare earth barium copper oxide (REBCO/YBCO) thin-film tapes, which are utilized to achieve the magnetic fields required for high-density plasma stabilization in these designs except for inertial confinement. This paper analyzes new methods and technology designed to mitigate helium embrittlement within fusion reactor walls, directly comparing the use of iron silicate (Fe₂SiO₄) atomic sponges against self venting nanocomposite vascular vein-like systems. The path to fusion is not exclusively in tech, engineering and physics. The second half of the fusion industry is the bureaucracy and national interests of many countries across the world, both as a collective group and in nations working internally. The changing US regulatory frame- works introduced by the U.S, specifically the ADVANCE Act of 2024, will be evaluated to see the DOE’s roadmap to US energy dominance, which is synthesized alongside the logistical, chemical, structural and cost considerations of potential alternative fuels, such as the aneutronic Deuterium–Helium-3 as fuel. By examining breakthroughs in physics, materials science, metallurgy, the law, alongside private and public sector designs, this review shows the technical path governing next generation, grid-scale fusion use.

Abstract: Management consulting sells structured judgment: research, analysis, synthesis and recommendation, delivered under time pressure by teams organised as a pyramid of junior analysts and senior partners. Generative and agentic artificial intelligence (AI) now performs a growing share of the pyramid's base — desk research, data analysis, benchmarking, first-draft synthesis — and is beginning to perform parts of its middle, including option generation and scenario construction. This conceptual paper asks what happens to strategic advice when the technology moves from assisting consultants to acting alongside them. Drawing on the automation–augmentation paradox, research on the changing sources of competitive advantage in AI, and the dynamic capabilities tradition, we develop a five-level maturity framework for agentic AI in consulting and strategic advice: (1) Assisted, (2) Augmented Analysis, (3) Generative Partnership, (4) Agentic Workflows, and (5) Governed Autonomous Operations. For each level we specify the AI capabilities in use, the changing human role, the principal risks, and the governance required before a firm should advance to the next level. We then propose four governance principles for AI-augmented advice — named human accountability, an explicit decision-rights ladder, evidence and audit requirements, and a protected right to override — and derive three propositions for empirical research. The framework is addressed to consulting firm leaders redesigning their leverage model, to clients learning how to buy and challenge AI-augmented advice, and to business schools whose graduates will enter a profession whose entry-level tasks are being automated first.

Abstract: The predictive organization previously identified in SPARC galaxy rotation-curve discrepancies [1] is tested here across a substantially larger and more distant population. We use 1,387 galaxies drawn from four independently acquired PROBES source surveys. SPARC is used only to help establish the stellar-disk normalization and an empirical radial gas-velocity reconstruction. For each galaxy we define the signed squared-velocity discrepancy, ΔV²(r) = Vobs²(r) − Vbar²(r), characterize its radial structure, and test whether population relationships learned from other galaxies predict the withheld galaxy. Strict leave-one-galaxy-out prediction gives a velocity RMSE of 47.34 km s⁻¹, compared with 50.86 km s⁻¹ for a population-average residual control, a 6.92% reduction across 75,722 held-out radial predictions. Each of the four constituent surveys independently shows positive error reduction. A leave-one-survey-out test, training on three surveys and predicting the fourth, gives a pooled 7.20% gain; the SCII survey is statistically unresolved. In survey-stratified grouped withholding, the gain remains 6.81% even when approximately half of the 1,387 galaxies are excluded from model construction. The composite sample has a median distance of 88.7 Mpc, approximately 5.2 times that of the original SPARC population. These results show that the same galaxy-level predictive architecture identified in SPARC can be recovered independently in a much larger, more distant, and less homogeneous four-survey population, even though its baryonic baseline must be reconstructed rather than supplied directly. Fixed SPARC coefficients expectedly do not transfer unchanged to PROBES, and direct near-to-far prediction is not established due to survey heterogeneity. Against a training-fitted radial-acceleration relation (RAR), the PRIM (Predictive Residual Informational Mapping) to PROBES is competitive overall but not uniformly superior; its physical origin remains unresolved.

Abstract: We propose that the quantum vacuum is a dynamic superfluid that is dragged by the rotating Sun, forming a large-scale vortex. The Sun and the vortex constitute an open system in which total angular momentum is conserved. A single partial slip parameter, fixed solely from the anomalous precession of Mercury, determines the coupling between solar rotation and vacuum entrainment. Over the age of the Solar System, the Sun has progressively entrained outer vacuum layers. The effective circulation of the vortex therefore grows linearly with distance, producing a constant orbital velocity. This naturally yields a distance-independent inward acceleration in the outer Solar System, consistent with the Pioneer anomaly observed between twenty and seventy astronomical units, and, at larger scales, flat galactic rotation curves without invoking dark matter. The model connects, with no additional free parameters, Solar System tests of gravity to galactic dynamics and provides a falsifiable, unified origin for three phenomena that have been treated as separate for more than a century.

Abstract: Oral ulcerative and inflammatory mucosal diseases, exemplified by recurrent aphthous stomatitis (RAS) and cancer-therapy-induced oral mucositis (OM), remain challenging to treat locally because salivary clearance, enzymatic activity, mechanical movement and limited residence time constrain conventional topical drug delivery. This review examines mucoadhesive biomaterial platforms as functional delivery systems and asks whether their material properties—including retention, interfacial adhesion and controlled release—contribute therapeutic value beyond that attributable to the incorporated bioactive payload. Evidence for mucoadhesive tablets, buccal films and patches, hydrogels, polymeric nanocarriers, dissolving microneedles and electrospun systems is appraised according to an evidence hierarchy that distinguishes randomised clinical evidence from animal and in vitro findings. Mucoadhesive tablets have been associated with reductions in RAS ulcer diameter at days 5 and 7 in a small pooled analysis, while licorice-loaded dissolving and bioadhesive patches provide repeated clinical evidence for phytochemical delivery through mucoadhesive devices. Hydrogel and nanocarrier systems show preclinical benefits in wound healing and local delivery, including substantial dose sparing of triamcinolone acetonide in a microneedle platform, but remain without controlled clinical translation in the indications reviewed. Where evidence permits, retention and release are considered as material properties distinct from the biological responses they enable; however, existing studies rarely isolate the therapeutic contribution of the carrier from that of the payload. Priorities for translation include standardised mucoadhesion characterisation, physiologically relevant human oral mucosa models, botanical and phytochemical standardisation, and controlled trials comparing identical active agents in conventional versus mucoadhesive vehicles.

Abstract: This study presents a full year of continuous field monitoring of a balcony-mounted photovoltaic system with battery energy storage (PV-BESS) operating in zero feed-in mode near Parma, Italy. The 1.38 kWp array (60° tilt) was coupled to a 6 kW hybrid inverter and a 2.56 kWh LiFePO₄ battery to shift daytime solar surplus toward the evening demand peak. The system achieved an average annual Self-Sufficiency Ratio (SSR) of 32.3% (55.8% in summer), while zero feed-in curtailment removed 16.4% of the theoretical yield. An unexpected finding, termed the Appliance Paradox, reveals that in deep winter the inverter's standby consumption exceeded the scarce PV yield, turning winter SSR negative (−10.6%). A novel Energy Advantage Index (EAI) is introduced as a PV-BESS design figure of merit, predicting the optimal deactivation window and, prospectively, sizing inverter pairings to avoid the paradox. Economically, a CAPEX of about €1,100 yields a 4.6-year payback and a ten-year ROI of about 137%, improving to 4.3 years and 154% under EAI-driven seasonal control, while cutting the effective cost of electricity by 32% and avoiding 2.35 t CO₂eq over ten years. Balcony PV-BESS systems can meaningfully advance residential energy autonomy, provided inverter sizing and seasonal operation are managed proactively.

Abstract: Bluetongue is an economically important arboviral disease affecting domestic and wild ruminants health and well-being, caused by bluetongue virus (BTV) and transmitted by Culicoides midges. More than 30 BTV serotypes may co-circulate regionally, while current vaccines face limitations related to safety and incomplete cross-protection. Here, reverse vaccinology and immunoinformatics were used to design four serotype-specific and one multiserotype multi-epitope vaccine candidates targeting BTV-1, BTV-3, BTV-4 and BTV-8. Cytotoxic T-lymphocyte (CTL) and helper T-lymphocyte (HTL) epitopes were predicted from VP2, VP5, VP7 and NS1, and linear B-cell epitopes from VP2 and VP5. Twenty-five CTL, 24 HTL and 15 B-cell epitopes met the computational antigenicity, non-allergenicity and non-toxicity screening criteria; HTL epitopes were also predicted to induce interferon-γ. Five constructs comprising 13-25 epitopes (252-471 amino acids) were fused to bovine β-defensin 2 as a putative adjuvant. Molecular docking with bovine Toll-like receptor 4 ectodomain yielded predicted binding energies ranging from -19.6 to -10.7 kcal/mol. Molecular dynamics simulations (100 ns), immune simulation, codon optimization and in silico cloning further characterized the candidates. These computational findings support the prioritization of these constructs for experimental evaluation as potential serotype-specific and multiserotype BTV vaccine candidates.

Abstract: This study investigated behavioural responses of the black sea urchin Arbacia lixula to underwater anthropogenic noise, artificial light at night (ALAN), lunar phase, and seawater temperature under semi-controlled conditions, over eight weeks. The degree of mobility, the rate of dislocation, as well as spatial distribution of the sea urchins within the tank of locomotion after approximately three weeks of exposure, followed by a return towards baseline activity levels by the final week, suggesting a temporary behavioural suppression followed by acclimation. Continuous sound treatments produced greater movement under several tested frequencies, whereas pulse-burst exposure produced fewer apparent effects, with the clearest response occurring at 10,000 Hz after prolonged exposure. Spatial patterns should be interpreted cautiously since the light and sound gradients across the quadrants were not instrumentally measured and the experimental tank restricted the available movement space. Within the observed range, seawater temperature and lunar phase were not clearly associated with movement responses. No spawning was observed during the trials, indicating that short-term exposure to these stimuli did not trigger spawning under the present experimental conditions. Overall, Arbacia lixula showed behavioural plasticity in response to an array of anthropogenic stimuli, but the effects appeared transient and should be interpreted in light of the exploratory statistical design and calibration limitations.

Abstract: Aeration diffusers in wastewater treatment plants are designed under a two factor protocol: deliver the actual oxygen requirement at the lowest practical energy. This paper argues that the protocol is incomplete, because the same design choices that improve oxygen transfer and energy, namely lower dissolved oxygen setpoints, finer bubbles and deeper submergence, also change nitrous oxide (N₂O) emission and bioaerosol release. A structured review of the literature found the four factors studied separately or in pairs, but no protocol that brings all four into diffuser design. The paper assembles the governing equations and links them through the field oxygen transfer efficiency, which fixes the air flow rate, then proposes an integrated protocol with a composite objective subject to the oxygen demand constraint. An illustrative example for a 1,000 m³ tank, using assumed dissolved N₂O concentrations, shows that the option preferred by the conventional protocol, fine bubble aeration at a setpoint of 1.0 mg/L, has the lowest energy use (422 kWh/d) but the highest total carbon footprint (1,743 kg CO₂e/d), because the added N₂O outweighs the energy saving. The ranking reverses once the dissolved N₂O ratio between the two setpoints exceeds about 1.23, an increase of more than about 23%, and published simulations suggest that threshold is likely to be exceeded. The ranking of fine against coarse bubbles depends on an uncalibrated bubble size exponent, so the bioaerosol results are reported as a parametric family in that exponent rather than as predictions. A staged measurement programme is set out to calibrate and verify the protocol.

Abstract: We study envelopes of families of chords generated by arithmetic maps on the unit circle, which we call arithmetic envelopes. For the doubling and tripling maps these are the classical cardioid and nephroid, which are also caustics of a circular mirror; more generally, the map \( x\mapsto kx \bmod 1 \) produces an epicycloid with k-1 cusps. We then change the coordinate system on the circle by labelling the odd integers with dyadic rationals, which identifies them with the Prüfer 2-group, and we recall its group law from earlier work and describe it by a simple rule on binary digits. In these coordinates multiplication by 3 becomes a piecewise linear circle homeomorphism, conjugate to the rotation by \( \log_2 3 \), and its chords leave a region in the centre of the circle free. We prove that this region, observed by Henkel and called the convexoid, is convex and bounded by a segment of a diameter and by arcs of two epicycloids with cusps at 1/5 and 1/7. The pedal curve of the same family is heart-shaped and lies on the algebraic curves \( x=\rho\,T_5(\rho) \) and \( x=\rho\,T_7(\rho) \) with \( \rho^2=x^2+y^2 \), where T_n denotes the n-th Chebyshev polynomial. We compute these and further envelopes explicitly with Gr\"obner bases and resultants, and we include code for all computations; an appendix discusses the tropicalization of the resulting curves. We close with questions on how such pictures might help to visualize problems such as Furstenberg's \( \times2 \) \( \times3 \) conjecture.

Abstract: Why does a magnetic field change the direction of a charged particle's motion but never its kinetic energy? The textbook answer is that the magnetic force is perpendicular to the velocity, so . In this paper we take that answer as a starting point and ask what can be said in general about how much a magnetic field can turn a particle. First, we show that the forces that do no work on any motion are exactly those of the form , and that among forces linear in the velocity these are precisely the magnetic-type forces. Second, the curvature of every trajectory in an arbitrary static field is . As a consequence, the angle between any two velocities along the path is bounded by the field integral . Combined with the theorems of Fenchel and Fáry–Milnor, this gives minimum field integrals for closed and knotted orbits. Third, in any static unidirectional field, whatever its spatial profile, the velocity stays on a cone around the field direction, so the direction can change by at most twice the initial pitch angle, independent of the field strength. Fourth, for a magnetic barrier , transmission, exit angle and specular reflection depend only on the running flux per unit length , not on the profile. All results hold relativistically with and are verified by symbolic computation and numerical integration in random fields.