Papers

In this paper, the 3D scheme of the fuel code ALCYONE is applied to a database consisting of more than 50 base irradiations and ramp tests performed on rods with UO 2 , MOX or Cr-doped UO 2 fuels and Zy4 or M5__ cladding tubes with burn-ups up to 70 GWd/tU. The ability of the 3D scheme to predict the behaviour of a single fuel pellet -cladding element situated at the maximum Linear Heat Rate (LHR) during ramp testing is demonstrated by comparing the following experimental and calculated data: residual clad diameter after base irradiation and ramp test, height of inter-pellet and mid-pellet ridges after base irradiation and ramp test, dish filling after ramp test. In the second part of the paper, 3D simulations that catch the stress -strain localization at the triple point are presented. It is shown that the stress-strain concentration occurs only at the end of the power transient and is quickly relaxed by pellet and clad creep.

This reply addresses a recent comment concerning the proof of the Nernst theorem. I clarify how a Carnot engine can consistently operate at $T=0$ through a continuous deformation of a cycle operating at $T>0$. By examining the limit where heat exchange with the cold reservoir vanishes, I show that the Nernst theorem ensures that the concept of temperature remains physically consistent at the absolute zero limit.

A ball spinning about a vertical axis is pushed sideways by the Magnus force, so its ground track curves. We show that when the lift and drag coefficients are constant, the horizontal part of this problem is exactly solvable in elementary functions. The step that makes this work is to write the horizontal velocity as a complex number and to use path length rather than time as the independent variable. Both aerodynamic forces are quadratic in speed, so the speed cancels and what remains is a linear equation with constant coefficients in which gravity does not appear. Three exact results follow from it. The heading turns at a fixed rate per unit path length, so a half turn always costs the same distance whatever the launch conditions and whatever the drag. The horizontal speed decays exponentially in the turn angle at a rate fixed by the drag-to-lift ratio, which makes the horizontal hodograph a logarithmic spiral. The radius of curvature of the ground track equals the lift length times the cosine of the flight-path angle. The whole flight problem also collapses to one scalar differential equation controlled by two dimensionless numbers. We use this to settle a concrete question. Two identical balls thrown in opposite horizontal directions with opposite spin can meet head-on when drag is absent, on a one-parameter family we give in closed form, and they fail to meet once drag acts. Every analytic claim is checked against machine-precision numerics, and the code and data are openly available.

We report the first calculation of the lowest-known energy configuration for $N=100\,000$ classical point charges confined to a disk and interacting via the $1/r$ Coulomb potential - a system size never before achieved for the Thomson problem in a disk, more than doubling the previous record of $N=40\,886$ reported by Amore and Zarate. An adaptive defect-targeting subdomain optimization strategy yields an approximately linear growth of the wall-clock time per cycle with $N$; for $N=100\,000$ a single $24$-core CPU workstation without GPU acceleration required $\approx31$ hours. The energy $E_{\min}=7.80466624157\times10^{9}$ deviates from the standard empirical asymptotic expansion by only $2.525\times10^{-7}$. We demonstrate that this remarkable agreement provides a stringent test of the rigorous next-order asymptotic theory for Riesz interactions, while the residual deviation is consistent with an $O(N^{4/3})$ boundary-layer correction whose absence from the standard fitting basis biases the empirical $N^{3/2}$ coefficient away from its exact crystalline value, thereby tying the deviation to the two-dimensional crystallization conjecture. Bond-orientational order analysis reveals a polycrystalline bulk threaded by radial grain boundaries, establishing a new benchmark for two-dimensional Coulomb systems.

This study uses wall-resolved large-eddy simulations to investigate bypass transition in a flat-plate boundary layer subjected to free-stream turbulence. Two thermal configurations are considered: an adiabatic wall and a uniformly cooled wall. The plate geometry includes an elliptic leading edge and is designed for direct experimental reproduction, providing a framework for future numerical-experimental comparisons. The resolved leading edge allows direct examination of the early stages of bypass transition. Both simulations recover the classical sequence of receptivity, vortex tilting, lift-up, streak amplification, secondary instability and turbulent-spot growth. The wall-normal transport term is identified as a precursor of streak formation, while shear sheltering is quantified and linked to the frequency-dependent penetration of free-stream disturbances into the boundary layer. Wall cooling does not modify the bypass-transition mechanisms nor the onset location of transition. Instead, it generates thermal streaks alongside the velocity streaks and shifts the latter slightly closer to the wall, consistent with optimal-perturbation predictions. The velocity streaks retain similar amplitudes, growth rates and spanwise spacings in both thermal conditions. A conditional analysis reveals a pronounced asymmetry between high- and low-velocity streaks. Although breakdown is systematically observed within low-velocity streaks, the strongest pre-transitional evolution occurs within the high-velocity streak population, which undergoes significant amplification and a progressive displacement towards the wall before the onset of intermittency. These observations suggest that high-velocity streaks may actively contribute to the reorganisation of the streak field leading to secondary instability and breakdown.

The continuity equation serves as a fundamental principle for mass transport in continuous media. While its mathematical structure mirrors that of probability transport and the Liouville equation, an informational interpretation of macroscopic fluid flow is less commonly explored in engineering contexts. This paper treats fluid density as a spatial probability density function, modeling macroscopic motion as the continuous transport of uncertainty. I derive the temporal evolution of Shannon entropy under general flow conditions, establishing how entropy generation depends on macroscopic compressibility and microscopic diffusion. Microscopic diffusion is shown to act as a strictly positive entropy source governed exactly by local Fisher Information. Furthermore, the framework yields an explicit algebraic scaling law characterizing the equilibrium thickness of compressive mixing layers at a local Peclet number of unity. The theoretical model is computationally validated through finite-difference simulations of one-dimensional canonical flows and advection-diffusion within a three-dimensional, spatially varying Arnold-Beltrami-Childress (ABC) flow, yielding a mean relative error of $0.001$. The framework provides an analytical perspective on thermodynamic irreversibility, with potential applications to turbulence modeling, thermal entropy generation in heat exchangers, in-cylinder mixing in internal combustion engines, and aerodynamic flows.

Classical electrodynamics including Lorentz-invariant classical electromagnetic zero-point radiation leads to a ground state and resonant excited states for a charged particle in a Coulomb potential. These resonant states correspond to integer values of the action variables analogous to those appearing in the Bohr theory of the hydrogen atom. Thus there is a purely classical electromagnetic derivation of the Bohr energy levels. According to classical electromagnetic theory, it is not that there are states where the charged particle does not radiate, but rather in these states there is a resonant balance between radiation emission and absorption The work on classical zero-point radiation reported here is a continuation of the analysis published in 1975, but with the addition of the ideas of relativity and resonance between the charged-particle orbit and classical zero-point radiation.

The intricate procedure of mass renormalization in quantum field theory has a simpler precursor in classical electromagnetism. There, the renormalized mass of a charged body includes both its bare mass and its electromagnetic mass. For a body moving at constant low velocity, the electromagnetic mass can be calculated either from the total momentum of the electromagnetic field or from the total energy of the electromagnetic field in the body's rest frame. For a spherically symmetric charge distribution, these methods appear to disagree by a factor of 4/3. This 4/3 problem has a long history. Here, we see that the concept of field velocity can be used to quickly resolve the 4/3 problem. The two methods for finding the electromagnetic mass agree after noting that the average velocity of energy flow in the electromagnetic field is 4/3 the velocity of the body. The energy in the electromagnetic field moves faster than the charged body because, in addition to moving with the body, the field energy must make its way from the back of the body (where it is being emitted) to the front (where it is being absorbed).

We propose a computationally inexpensive algorithm to determine whether an opaque spherical body eclipses a luminous one, using only classical geometrical tools. We apply this method to the Sun-Earth-Moon system and predict solar eclipses, including their type, beginning, and duration, using the three-body problem. For predictions over a one-year period, the resulting errors are in the order of minutes.

We report improved global-minimum configurations for the classical Thomson problem of $N=60$, $61$, $92$, and $99$ repulsive Coulomb charges confined to a disk. By combining the quenched molecular dynamics (QMD) method with the fixed-border heuristic introduced by Amore and Zarate, we systematically obtain configurations with energies $E_{\mathrm{QMD}}(60)=2159.3584240930$, $E_{\mathrm{QMD}}(61)=2237.19264190$, $E_{\mathrm{QMD}}(92)=5358.35353314$, and $E_{\mathrm{QMD}}(99)=6254.83029083$, which improve upon the previously best-known values. For $N=60$, the Voronoi diagram of our configuration differs from the one reported earlier. The symmetries for $N=61$ $(C_{2})$ and $N=99$ $(D_{1})$ are confirmed and are consistent with the known symmetry patterns for these configurations. For $N=92$, whereas the previously reported Voronoi diagram has lower symmetry, our solution exhibits a clear $D_{1}$ (axial) symmetry of defects. The results are highly reproducible across multiple independent runs, providing strong evidence that these configurations are robust global-minimum candidates.

Multiple pendulum with branching is proposed as a convenient platform to study energy transfer between different modes. The antiphase oscillation mode is localized to the "child" links, which makes it easy to prepare initial conditions without exciting the antiphase mode. A manageable expression for the energy transfer is derived theoretically and evaluated with experimental data.

Near-field propagation between finite apertures can support multiple spatial channels, while far-field transmission is effectively single mode and follows the Friis transmission formula. This letter establishes a direct connection between these two regimes through the mutual shadow area between the transmitting and receiving apertures. The mutual shadow determines the asymptotic number of spatial degrees of freedom and, in the paraxial regime, leads to a simple beamforming distance that provides a characteristic scale for the transition from multimode to effectively single-mode propagation. The channel singular-value sum rule further shows that, below this distance, increasing separation primarily reduces the number of supported spatial channels while their average normalized strength remains approximately constant. Beyond the transition, the remaining channel strength decreases according to the conventional free-space quadratic power law, recovering the Friis transmission formula. Numerical results for circular apertures demonstrate the transition, the scaling with aperture size, and the dependence on the transmitter and receiver apertures. The mutual-shadow formulation also extends the description beyond the paraxial regime.

We consider the integrable dynamics of a planar Kepler billiard in the plane bounded by a branch of a hyperbola focused at the Kepler center. As a sequel to our previous work \cite{JZ2}, we use the existence of the foci-caustic circle to identify an elliptic curve on which the dynamics is linearized and we analyze Cayley's condition on $n$-periodic orbits in this setting. By taking various limits also we discuss the dynamics of planar Kepler billiards with boundary being a focused parabolic arc, as well as a straight line. In this last case, our approach offers a different way to deduce the elliptic curve as in \cite{Felder}.

Acoustic lenses have been introduced recently to compensate for the phase distortions induced by the propagation across a human skull for ultrasonic deep-brain stimulation in humans. In this study, we present bifocal lenses that compensate for human skull aberrations and allow simultaneous targeting of multiple structures deep in the brain. We investigated the impact of phase unwrapping in the design of the lenses and how this process improves the distribution of pressure produced in N=5 human skulls for two different spatial arrangements of the targets. The results show that unwrapping the phase computed during the design increases the fidelity of the pressure field generated across the human skulls. The spatial precision is on average improved by 73%, and out of target energy deposition is on average reduced by 58%. The results presented in this study highlight the importance of phase unwrapping to optimize the safety and efficacy of future transcranial ultrasound stimulations targeting multiple regions.

The topology of a bounded three-dimensional domain can strongly affect electromagnetic fields. In a topologically complex domain, a curl-free field may not have a globally single-valued scalar potential and a divergence-free field may also fail to have a global vector potential. These topological effects lead naturally to harmonic vector fields in the Helmholtz decomposition. This paper gives a geometric and constructive study of such fields using vector analysis, circulation integrals, cutting surfaces, scalar Laplace problems, Stokes' theorem, and Green's identity. The first Betti number is interpreted through independent handle-type circulations, whereas the second Betti number counts enclosed voids. Direct proofs are given for the dimensions of the Neumann and Dirichlet harmonic-field spaces. The analysis is also extended to anisotropic lossless media. The results provide a simple topological interpretation of harmonic fields and physical DC modes in bounded electromagnetic resonators.

Accurate prediction of transition-metal hydride (TM-H) bond dissociation energies (BDEs) remains challenging because of strong electron correlation, relativistic effects, and nuclear quantum contributions. Here, we assess the performance of neural-network variational Monte Carlo (NN-VMC) based on the Psiformer ansatz for systems with increasing complexity (LiH, OH, TiH and NiH), and compare it against CCSDT(Q)/CBS as well as available theoretical and experimental data. Throughout this study, both NN-VMC and ab initio calculations employ a common ccECP Hamiltonian to enable tractable and consistent comparisons. To reduce finite-training errors, we introduce complementary zero-variance and infinite-step extrapolation schemes. For LiH and OH systems, NN-VMC yields total energies that differ within sub-milli Hartree compared to CBS extrapolated ab initio results and the BDE differences remain under 2$\sigma$. In case of Ti and TiH, the variational NN-VMC energies at the end of training are already consistent with CCSD(T)/CBS, while post-training extrapolation systematically closes the gap towards the CCSDT(Q)/CBS results. In contrast, NiH provides a stringent test of wavefunction expressivity, where increasing the number of determinants in NN-wavefunction ansatz improves the recovered correlation energy. The comparison of NiH BDE reveals that, while the existing theoretical predictions cluster into distinct high and low BDE groups, the broad scatter in available experimental data prevents a definitive assessment of the most accurate theoretical approach. This work demonstrates that NN-VMC with ccECPs Hamiltonian provides a competitive framework for quantitative prediction of main-group and early TM-H energetics, while identifying late TM-H as an important benchmark for future developments in neural-network wavefunctions and electronic structure theory.

To enhance the performance of vibration energy harvesters, it is essential to maximise the electromechanical coupling coefficient k${}^2$ of piezoelectric devices. This enables sufficient harvested power and tuning capability of the resonant frequency through electrical methods. While much literature treats the optimisation of piezoelectric cantilevers, the optimisation range is usually limited by the transverse coupling coefficient k31${}^2$ of the material. This work introduces an innovative solution to extend the optimisation range and increase the coupling coefficient of piezoelectric cantilevers. This is achieved by minimising lateral strain in the beam using lateral bars to maximise the equivalent material coupling coefficient. The theoretical basis of this innovation is demonstrated through the exploitation of the constitutive equations of piezoelectric materials. The interest of the addition of lateral bars to increase the coupling coefficient is demonstrated and studied with simulations based on the finite elements method. Finally, a proof of concept is realised using a aluminum cantilever prototype integrating a lead-free KNaNbO3 (KNN) piezoelectric material. It is tested under vibration at 0.1 m/s${}^2$ with variable resistive loads. The results show that the coupling coefficient increases by 30% and the relative frequency bandwidth by 32% with resistive tuning of the resonant frequency, by fixing eight steel bars to the cantilever. The designed prototype is a highly performant leadfree vibration energy harvester. It produces a maximum power of 49.9 $\mu$W at resonance, and its normalised power density is equal to 16.5 mW/G${}^2$/cm${}^3$. Its relative frequency bandwidth is equal to 3.1%.

Using large-scale, three-dimensional discrete element simulations, we investigate the kinematic properties of granular chute flows of cohesionless grains, together with the evolution of the coordination number and the fraction of sliding contacts. Our results reveal that, under certain conditions, increasing the grain–grain friction coefficient leads to a decrease in the global dissipation of the system. This finding highlights that the overall dissipation is not solely governed by the energy lost during sliding contacts, but also by the probability that such contacts occur, a probability that decreases markedly as the friction coefficient increases. Together, these effects demonstrate the subtle and nontrivial interplay between grain-scale frictional interactions and macroscopic flow behavior in dense and cohesionless granular materials.

We study theoretically the peeling behavior of adhesives. Adopting a fracture mechanics approach, we derive the equation of motion of the adhesion front propagating at the interface between the adhesive and the substrate from which the peel strength is inferred. The originality of our approach lies in the description of the interplay during peeling between the stretching and the bending modes of deformation of the adhesive that is described as a F{\”o}ppl-Von Karman's thin film. Considering first a straight adhesion front, we retrieve the most salient feature of homogeneous adhesives, namely a peeling angle dependent peel strength driven by bending at large angles and by stretching at low angles. We also derive the shape of the adhesive that can be described using a single bending length scale derived from our model. We then investigate the impact of adhesion heterogeneities. We evidence that the deformations of the adhesion front are governed by a non-local interface elasticity the strength of which decreases with the peeling angle. This phenomenon reflects the transition between a stretching dominated peeling at low angle to a bending driven peeling at large angles that is captured in our model. This transition impacts the stability of adhesive fronts that relaxe more slowly from perturbations and gives rise to a stronger toughening effect in presence of a disorder distribution of adhesion energy at low peeling angles. Overall, this study sheds light on the central role played the elastic deformations of adhesives on their peeling behavior. The proposed framework unfolds the complex interplay between the deformation of adhesives and the peeling driving force that may be leveraged to engineer heterogeneous adhesives with enhanced properties. It also provides rich insights on the mechanisms underlying the emergence of non-local elasticity in interface problems.

The mechanics and friction of contact interfaces involving soft materials like gel, rubber or human skin are of both fundamental and applied interest. Recent insights have been made into this field thanks to in situ observations of the contact interface. However, current soft-material-oriented tribometers enable only few degrees of freedom for the actuation of the contact, far from covering the richness of the loading conditions relevant to real tribological contacts. Here, we introduce an apparatus dedicated to the study of the contact mechanics and friction of soft materials which, in addition to in situ / in operando optical monitoring of the interface, enables simultaneous actuation along five degrees of freedom: three translations and two rotations. While all three translations feature large velocity/large stroke motion, the one responsible for normal contact loading can also apply high frequency/small amplitude vibrations. The contact's dynamical response is monitored using both a 6-axes force/torque sensor and a 6-axes displacement/rotation sensor. We first describe the structure of the apparatus, its implementation, alignment, calibration and resolutions. We then illustrate its capabilities through a series of experiments on elastomer contacts. Our apparatus will be useful to investigate the mechanics of a wide range of soft interfaces submitted to rich, tribology-relevant kinematic or dynamic stimuli.

Hamiltonian and Lagrangian mechanics are powerful frameworks for analyzing physical systems. Previous work has extended these formalisms to ecological systems, such as the predator-prey Lotka-Volterra (LV) system. In this Article, we derive a canonical Lagrangian for the two-dimensional LV model directly from its Hamiltonian representation. We find that the two-dimensional LV system admits a standard canonical Lagrangian formulation with one degree of freedom and a non-quadratic kinetic structure. This formulation admits a mechanical interpretation in terms of a particle moving in a potential well, where the non-standard kinetic structure produces a position-dependent damping term that can instead act as "revving." The derivation provides a direct connection between predator-prey dynamics and a canonical formulation of mechanical dynamics. As a verification of the construction, we apply Noether's procedure to the explicitly time-independent derived Lagrangian and reveal that the well-known Hamiltonian of the LV system is the corresponding conserved quantity. We also uncover a subtle redundancy associated with the choice of canonical momentum and its identification with the original population variables.

Generalization of the inductance to the case of non-quasistatic electromagnetic field oscillations appears to be fruitful when considering wireless power transfer and RF metamaterials consisting of thin wire loop meta-atoms. When dealing with large systems of interacting loops carrying currents, efficiency and precision of calculation in the presence of retardation is crucial. In this work, we derive a series expansion of such generalized inductance and propose a way for its efficient numerical approximation. Illustrative examples are provided both for inductance convergence of a pair of two loops and extinction efficiency for scattering by metamaterial samples.

Modern design optimization frameworks aim first and foremost for models with the most accurate predictions without balancing computational overhead. It remains a reason why scaled architecture and multidisciplinary design optimization problems are difficult to address, even with sample-efficient Bayesian optimizers. In this paper, a metric quantifying the computational energy footprint is introduced within a Bayesian optimization framework to guide the parameter setting of a model towards configurations that balance both performance and frugality. The computer experiments highlighted existing tradeoffs between optimum convergence and the underlying energy footprint, and sometimes resulted in both a better-found optimum and lower energy consumption.

We experimentally demonstrate second- and third-order exceptional bound states in the continuum (EP-BICs), formed by the merging of two and three symmetry-protected BICs at an exceptional point (EP). Our passive reciprocal acoustic platform consists of symmetry-protected BIC cavities coupled through an acoustic waveguide and enables independent control of intrinsic loss, radiative loss, and near-field coupling between the cavities. A nonuniformly distributed intrinsic loss provides the non-Hermiticity required for EP formation while preserving decoupling from the waveguide radiation channel. The EP-BICs are probed using both near- and far-field excitation. In the near field, local intracavity excitation directly accesses the symmetry-protected BICs and reveals their spectral evolution. For far-field excitation, we intentionally break the protecting geometrical symmetry, converting the BICs into quasi-BICs and making them accessible through transmission measurements. Extending the system to three cavities with graded intrinsic loss realizes a third-order EP-BIC and yields a larger spectral response to the implemented coupling perturbation. These results establish a passive reciprocal route to higher-order exceptional degeneracies with independent control over intrinsic loss, near-field coupling, and radiative access.

We demonstrate that low loss radio frequency plasmon-like surface electromagnetic waves may propagate along rough interfaces between air and highly conductive media, such as water or metal. Unlike the well-known Zenneck wave solutions, these surface waves have mode index larger than 1, so that their propagation properties are generally similar to the properties of surface plasmons in the visible frequency range. The wave impedance of these modes is derived, which demonstrates the need to use special plasmonic inducers or exciters for efficient generation of these novel surface electromagnetic modes. Propagation characteristics of these radio frequency plasmons were explored both experimentally and theoretically in the cases of water-air and water-ice-air interfaces, and the two-dimensional character of this propagation has been revealed. Other important applications of radio frequency plasmons were demonstrated, such as communication along metal infrastructure immersed in water, communication along a lake floor, underwater plasmonic radar, as well as plasmonic communications underground and in the dense jungles.

Linear response is cheap to measure; the bistability boundaries it organizes are not. For a passive network with one localized cubic, the driving-point receptance $G$ fixes the period-one cusp set at fundamental-harmonic order: cusps lie on a fixed phase contour of $G$, a tangency of that contour under parameter variation creates a pair, and its curvature separates a gap opening from an isolated loop. For a two-mode absorber the linear prediction locates a benchmark birth coupling to $0.3\%$, and to $0.03\%$ once a third-harmonic correction of scale $|G(3\Omega)/G(\Omega)|$ is included.

We propose an all-reflective two-mirror projection system for extreme ultraviolet (EUV) lithography operating at exposure wavelengths of $13.5$~nm (Mo/Si) and $11.2$~nm (Ru/Be), delivering a fourfold ($4\times$) demagnification of the periodic mask pattern at a numerical aperture approaching unity ($\mathrm{NA}_{\max} \approx 0.993$). In contrast to conventional EUV projection objectives that incorporate 6–10 aspheric mirrors with an overall optical throughput of less than $15\%$, the proposed design redirects each accepted discrete spatial diffraction order scattered by the mask onto the wafer via a dedicated pair of planar mirror facets. The number of reflections is strictly fixed at two for all accepted orders, retaining $50$–$60\%$ of the power leaving the mask in each accepted order. We derive a spatial geometry providing rigorous optical path length equalization across all diffraction orders, thereby removing order-dependent propagation phase shifts. Individually optimized 30-bilayer Bragg multilayer coatings are designed for each facet using the transfer matrix method combined with global evolutionary optimization algorithms. The architecture is generalized to a three-dimensional vector formulation with a two-dimensionally periodic mask. Utilizing inverse lithography technology, Fourier parameterization, and a differentiable electromagnetic modal waveguide solver, we solve the synthesis problem for binary absorber masks (La absorber on a Ru/Be/Sr multilayer mirror). We demonstrate simulated aerial images of sub-10-nm features on the wafer (isolated peaks with a full width at half maximum (FWHM) of approximately $5.4$~nm and line pairs with a critical dimension of $6$~nm) and find that the two peaks remain resolved for the tested wafer defocus values from $0$ to $5$~nm along the $z$-axis.

We study the electromagnetic radiation and wave-optical imaging of an oscillating electric dipole orbiting a Kerr black hole. We derive the effective 4-current associated with a pointlike oscillating electric dipole in curved spacetime, and use black hole perturbation theory to compute the resulting radiation field at future null infinity, from first principles. We then develop a wave-optical imaging framework for a moving electromagnetic source in curved spacetime. We obtain images of an orbiting electric dipole, displaying relativistic beaming, gravitational lensing, and Einstein rings. We study polarization-dependent scattering by comparing the images produced by spinning dipoles with opposite helicities, finding a displacement that roughly decreases with the inverse of the radiation frequency, as expected for a beyond-geometric-optics effect. Our results provide a first-principles benchmark for beyond-geometric-optics descriptions of electromagnetic radiation in Kerr spacetime.

Optimizing slender elastic structures for renewable energy applications requires non-classical continuum formulations capable of accounting for spatial micro-interactions without sacrificing analytical tractability. Here, we extend beam vibration mechanics by replacing standard spatial derivatives with the Dunkl differential operator. This modification introduces a reflection-coupled mathematical structure that accounts for spatial parity effects across the beam domain. We formulate the governing dynamic equations into a generalized eigenvalue problem and derive exact analytical expressions for modal characteristics under standard boundary conditions. The classical limit confirms exact convergence to classical Euler-Bernoulli formulations. Parametric analyses reveal that the Dunkl parameter acts as a reflection-induced modulation parameter, significantly shifting natural frequencies and altering the modal characteristics of higher modes. These results provide an analytical baseline for dynamic optimization in lightweight structural components.

We present an analytical, statistical solution to the binary-binary (2+2) outcome of the chaotic non-hierarchical four-body problem. The solution is based on the density-of-states formulation pioneered by J. J. Monaghan. The method skips the computationally expensive integration of the equations of motion, and instead samples the outcome from the chaotic phase-space, subject to conservation of energy, momentum, and angular momentum. From the joint distribution, we extract marginal distributions of several key parameters using Monte-Carlo integration, and numerically verify them by comparing to an identical ensemble of scattering experiments produced by the FEWBODY code. From the comparison, we identify a regime not represented by the density-of-states formulation: the hard binary regime, where one binary is much harder then the four-body energy scale, and the system acts as an effective three-body system. We hypothesize that at this regime the system's probability distribution spreads over an effective reduced three-body chaotic phase space. The process of transfer from four-body to three-body phase-space is still not understood, and represents the next natural extension of density-of-states methods.