In this work, within the framework of the Glauber-Sitenko approximation, an analysis of the differential cross section for deuteron breakup into a proton in the reaction H(d,p)X is presented. The study is carried out using various parameterizations of the deuteron wave function, including the single-Gaussian parametrization, the multi-Gaussian K2 parametrization, and models based on the Av18 and NijmI nucleon-nucleon potentials. Special attention is given to the effects of small longitudinal components of the transferred momentum (Qz < 0.5 GeV/c) and the transverse momentum of the proton-neutron pair (p_perp < 0.5 GeV/c) in the anti-laboratory reference frame. The results are compared with experimental data, particularly in the region of longitudinal momenta p\_3 = 0.25-0.5 GeV/c, where quark effects are expected to manifest. Preliminary estimates show a decrease in the cross section with increasing transverse momentum, as well as a relatively small shift (and growth) of the cross-section maximum due to the inclusion of the longitudinal component Qz.

We study interaction-induced bound states in a system of ultracold bosons loaded into the states with orbital angular momentum in a one-dimensional staggered lattice of rings. We consider the hard-core limit and strong nearest-neighbour interactions such that two particles in next neighbouring sites are bound. Focusing on the manifold of such bound states, we have derived the corresponding effective model for doublons. With orbital angular momentum $l=1$, the original physical system is described as a Creutz ladder by using the circulations as a synthetic dimension, and the effective model obtained consists of two Su-Schrieffer-Heeger (SSH) chains and two Bose-Hubbard chains. Therefore, the system can exhibit topologically protected edge states. In a structure that alternates $l=1$ and $l=0$ states, the original system can be mapped to a diamond chain. In this case, the effective doublon model corresponds to a Creutz ladder with extra vertical hoppings between legs and can be mapped to two SSH chains if all the couplings in the original system are equal. Tuning spatially the amplitude of the couplings destroys the inversion symmetry of these SSH chains, but enables the appearance of multiple flat bands.

In addition to their broad physics reach enabled by their high energies and precision, future multi-TeV muon colliders will also be the world's most intense sources of neutrinos. This offers the opportunity to search for new non-standard neutrino interactions, possible by installing a dedicated forward neutrino detector in the straight sections of the collision ring, which is then used to measure reactions initiated by neutrinos from the decaying beam muons. In this paper, we show that these searches can exceed current and upcoming bounds on non-standard neutrino interactions from low-energy precision experiments and the LHC. This is achieved by the large flux of high-energetic neutrinos, the precise knowledge of the neutrino flavor composition on each side of the interaction point and the chirality of the neutrinos. We further discuss the technical requirements of the proposed forward neutrino detector, \FASERmuC, to maximally exploit this physics potential.

Quantum metrology has recently emerged as a powerful approach for dark matter (DM) searches, particularly using nonclassical bosonic states in microwave cavities that are sensitive to weak signals. Nonclassical cat states - macroscopic superpositions of coherent states featuring sub-Planck interference structures - offer promising advantages for high-precision measurements. However, their practical utility in DM search remains unexplored. Here, we report the first experimental application of four-component cat states within a high-quality superconducting microwave cavity to search for dark photons, a potential DM candidate. We demonstrate an 8.1-fold enhancement in the signal photon rate and constrain the dark photon kinetic mixing angle to an unprecedented $ε< 7.32 \times 10^{-16}$ near 6.44~GHz (26.6~$μ$eV). By employing a parametric sideband drive to actively tune the cavity frequency, we achieve dark photon searches and background subtraction across multiple frequency bins, yielding a sensitivity at the $10^{-16}$ level within a 100~kHz bandwidth. Our cat-assisted DM (CaD) search and frequency-scanning techniques demonstrate substantial improvements over previous results, promising potential implications in quantum-enhanced searches for new physics.

We study two complementary methodologies for calibrating implied volatility surfaces: analytical approximations and data-driven models based on rough path theory. On the analytical side, we revisit a second-order asymptotic expansion for the Heston model, and we propose a new, VIX-based calibration scheme for the rough Bergomi model. Both methods yield highly accurate and computationally efficient calibration formulas when the underlying dynamics are well specified. In parallel, we develop a signature-based approach in which volatility is represented as a linear functional of the truncated signature of a primary stochastic process, providing a flexible and model-agnostic alternative. Our numerical experiments compare the two approaches across both Markovian and non-Markovian settings. In the Heston case, signature-based models achieve a level of accuracy comparable to analytical expansions. In the rough Bergomi setting, using a fractional Brownian motion as the primary process, the signature approach continues to perform strongly and in some cases improves upon the Markovian specification, reflecting its ability to capture more complex temporal dependencies. Overall, the results illustrate that analytical methods are highly effective when the model is correctly specified, while signature-based methods offer a robust and flexible framework for calibration across a wider range of volatility dynamics.

This paper investigates the qualitative behavior of a system of ordinary differential equations (ODEs) defined by a matrix operator derived from the algebraic structure of the Alpha Group. The system depends on a rotational parameter that continuously deforms the underlying geometry of the phase space. Using a fourth-order Runge-Kutta numerical scheme, we analyze the evolution of trajectories and identify the presence of critical parameter values at which the system undergoes qualitative transitions. In particular, we observe the emergence of critical dynamical regions associated with changes in the interaction between dynamically defined subspaces. As the rotation parameter varies from $0$ to $π/2$, the system transitions from a regime with Euclidean-type geometric behavior to a non-Euclidean configuration induced by the Alpha Group structure. These transitions correspond to changes in stability and global phase space organization, including the formation of invariant structures and attractor-like behavior at infinity. The results suggest that the underlying matrix operator acts as a generator of structured transformations governing the system dynamics. This work provides a computational and qualitative framework for studying parameter-dependent dynamical systems with evolving geometric structure.

We explore how recent advances in the manipulation of single-ion wave packets open new avenues for detecting weak magnetic fields sourced by ultralight dark matter. A trapped ion in a ``Schrödinger cat'' state can be prepared with its spin and motional degrees of freedom entangled and be used as a matter-wave interferometer that is sensitive to the Aharonov-Bohm-like phase shift accumulated by the ion over its trajectory. The result of the spin-motion entanglement is a parametrically-enhanced sensitivity to weak magnetic fields as compared with an un-entangled ion in a trap. Taking into account the relevant boundary conditions, we demonstrate that a single trapped ion can probe unexplored regions of kinetically-mixed dark-photon dark matter parameter space in the $10^{-15}~\text{eV} \lesssim m_{A'} \lesssim 10^{-14}$~eV mass window. We also show how such a table-top quantum device will also serve as a complementary probe of axion-like particle dark matter in the same mass window.

We study the interactions mediated by a symmetric superconducting quantum interference device (SQUID), their renormalizations, and the applicability of the anharmonic oscillator (bosonic) model for a coupled phase qubit. The latter dwells in its metastable well holding a number of anharmonic energy states. The coupling SQUID can switch between the single- and two-photon interactions in situ. We find that, in the bosonic two-photon regime, the cross-Kerr coupling never vanishes as it dresses due to asymmetry in the qubit potential and nonlinearity of the coupler. Our quantitative results also depend on the bosonic approximation. We approach determining its limits by finding the minimum number of coherent energy states required for a dressing. For that, we lay out diagrams of the dressing virtual processes that climb the qubit ladder as high as possible. Near the two-photon resonance with a coupled resonator, we systematically calculate other relevant renormalizations due to nonresonant interactions. We provide verifiable predictions for the coupling rates. Modified systems can be applied for two-photon detection and for quantum-nondemolition readout of a qubit with an asymmetrical potential.

We investigate out-of-equilibrium entanglement dynamics in a generalization of the so-called $QSSEP$ model, which is a free-fermion chain with stochastic in space and time hopping amplitudes. In our setup, the noisy amplitudes are spatially-modulated satisfying a $ν$-site translation invariance but retaining their randomness in time. For each noise realization, the dynamics preserves Gaussianity, which allows to obtain noise-averaged entanglement-related quantities. The statistics of the steady-state correlators satisfy nontrivial relationships that are of topological nature. They reflect the Haar invariance under multiplication with structured momentum-dependent random $SU(ν)$ matrices. We discuss in detail the case with $ν=1$ and $ν=2$. For $ν=1$, i.e., spatially homogeneous noise we show that the entanglement dynamics is describable by a stochastic generalization of the quasiparticle picture. Precisely, entanglement is propagated by pairs of quasiparticles. The entanglement content of the pairs is the same as for the deterministic chain. However, the trajectories of the quasiparticles are random walks, giving rise to diffusive entanglement growth.

We address the problem of reliable data transmission within a finite time horizon $T$ over a binary erasure channel with unknown erasure probability. We consider a feedback model wherein the transmitter can query the receiver infrequently and obtain the empirical erasure rate experienced by the latter. We aim to minimize a regret quantity, i.e. how much worse a strategy performs compared to an oracle who knows the probability of erasure, while operating at the same block error rate. A learning vs. exploitation dilemma manifests in this scenario -- specifically, we need to balance between (i) learning the erasure probability with reasonable accuracy and (ii) utilizing the channel to transmit as many information bits as possible. We propose two strategies: (i) a two-phase approach using rate estimation followed by transmission that achieves an $O({T}^{\frac 23})$ regret using only one query, and (ii) a windowing strategy using geometrically-increasing window sizes that achieves an $O({\sqrt{T}})$ regret using $O(\log(T))$ queries.

The superconducting transition temperature ($T_{\mathrm{c}}$) of trilayer or quadruple-layer cuprates typically surpasses that of single-layer or bilayer systems. This observation is often interpreted within the ``composite picture", where strong proximity effect between inner CuO$_2$ planes (IPs) and outer CuO$_2$ planes (OPs) is crucial. Albeit intriguing, a straightforward scrutinization of this composite picture is still lacking. In this study, using angle-resolved photoemission spectroscopy to investigate (Cu,C)Ba$_2$Ca$_3$Cu$_4$O$_{11+δ}$ (CuC-1234) with a high $T_{\mathrm{c}}$ of 110~K, we found that the OPs are not superconducting at the $T_{\mathrm{c}}$ of the material. Instead, the large pairing strength and phase coherence concurrently emerge at the underdoped IPs, suggesting that the high $T_{\mathrm{c}}$ is primarily driven by these underdoped IPs. Given that the $T_{\mathrm{c}}$ of CuC-1234 is comparable to other trilayer or quadruple-layer cuprates, our findings suggest that the conventional ``composite picture" is not universally required for achieving high $T_{\mathrm{c}}$. More importantly, we demonstrate that CuO$_2$ planes free of apical oxygen can support superconductivity up to 110~K even at a doping level of 0.07 holes per Cu, a level that lies deep in the underdoped regime of single- and bilayer cuprates. These findings provide new insights into the origin of high $T_{\mathrm{c}}$ in multilayer cuprates.

Initialization of mechanical modes in the quantum ground state is crucial for their use in quantum information and quantum sensing protocols. In quantum processors, impurity of the modes' initial state affects the infidelity of subsequent quantum algorithms. In quantum sensors, excitations out of the ground state contribute to the noise of the detector, and their prevalence puts a bound on rare events that deposit energy into the mechanical modes. In this work, we measure the excited-state populations of GHz-frequency modes in a high-overtone bulk acoustic wave resonator (HBAR). We find that the population of the first excited state can be as low as $P_p$=(1.2$\pm$5.5)$\times10^{-5}$, corresponding to an effective temperature of 25.2 mK, which are upper bounds limited by imperfections in the measurement process. These results compare favorably to the lowest populations measured in superconducting circuits. Finally, we use the measured populations to constrain the amplitude of high-frequency gravitational waves, the kinetic mixing strength of ultra-light dark matter, and non-linear modifications of the Schrödinger equation describing wavefunction collapse mechanisms. Our work establishes HBARs as a versatile resource for quantum state initialization and studies of fundamental physics.

Signal detection in high dimensions is a critical challenge in data science. While standard methods based on random matrix theory provide sharp detection thresholds for finite-rank perturbations, such as the known Baik-Ben Arous-Péché (BBP) transition, they are often insufficient for realistic data exhibiting nearly continuous (extensive-rank) signal distributions that merge with the noise bulk. In this regime, typically associated with real-world scenarios such as images for computer vision tasks, the signal does not manifest as a clear outlier but as a deformation of the spectral density's geometry. We use the functional renormalisation group (FRG) framework to probe these subtle spectral deformations. Treating the empirical spectrum as an effective field theory, we define a scale-dependent "canonical dimension" that acts as a sensitive order parameter for the spectral geometry. We show that this dimension undergoes a sharp crossover, interpreted as a "dimensional phase transition", at signal-to-noise ratios significantly lower than the standard BBP threshold. This dimensional instability is shown to correlate with a spontaneous symmetry breaking in the effective potential and a deviation of eigenvector statistics from the universal Porter-Thomas distribution, confirming the consistency of the method. Such behaviour aligns with recent theoretical results on the "extensive spike model", where signal information persists inside the noise bulk before any spectral gap opens. We validate our approach on realistic datasets, demonstrating that the FRG flow consistently detects the onset of this bulk deformation. Finally, we explore a formalisation of this methodology for analysing nearly continuous spectra, proposing a heuristic criterion for signal detection and a method to estimate the number of independent noise components based on the stability of these canonical dimensions.

We propose and analyze an energy-stable fully discrete parametric approximation for Willmore flow of hypersurfaces in two and three space dimensions. We allow for the presence of spontaneous curvature effects and for open surfaces with boundary. The presented scheme is based on a new geometric partial differential equation (PDE) that combines an evolution equation for the mean curvature with a separate equation that prescribes the tangential velocity. The mean curvature is used to determine the normal velocity within the gradient flow structure, thus guaranteeing an unconditional energy stability for the discrete solution upon suitable discretization. We introduce a novel weak formulation for this geometric PDE, in which different types of boundary conditions can be naturally enforced. We further discretize the weak formulation to obtain a fully discrete parametric finite element method, for which well-posedness can be rigorously shown. Moreover, the constructed scheme admits an unconditional stability estimate in terms of the discrete energy. Extensive numerical experiments are reported to showcase the accuracy and robustness of the proposed method for computing Willmore flow of both curves in $\mathbb{R}^2$ and surfaces in $\mathbb{R}^3$.

Confidence sequences are collections of confidence regions that simultaneously cover the true parameter for every sample size at a prescribed confidence level. Tightening these sequences is of practical interest and can be achieved by incorporating prior information through the method of mixture martingales. However, confidence sequences built from informative priors are vulnerable to misspecification and may become vacuous when the prior is poorly chosen. We study this trade-off for Gaussian observations with known variance. By combining the method of mixtures with a global informative prior whose tails are polynomial or exponential and the extended Ville's inequality, we construct confidence sequences that are sharper than their non-informative counterparts whenever the prior is well specified, yet remain bounded under arbitrary misspecification. The theory is illustrated with several classical priors.

In minimum trace (MinT) forecast reconciliation, the covariance matrix of the base forecasts errors plays a crucial role. Typically, this matrix is estimated and then treated as known. This can lead to underestimation of the variance of the predictive distribution. To address the problem, we propose a Bayesian reconciliation model that accounts for the uncertainty in the estimation of the covariance matrix. By adopting an Inverse-Wishart prior and assuming Gaussian residuals, the reconciled predictive distribution follows a multivariate t-distribution, obtained in closed-form, rather than a multivariate Gaussian distribution. We evaluate our method on three tourism-related datasets, including a new publicly available dataset. Empirical results show that our approach consistently improves prediction intervals compared to MinT reconciliation.

In this paper, we study the problem of model-checking quantum pushdown systems from a computational complexity point of view. We arrive at the following equally important, interesting new results: We first extend the notions of the {\it probabilistic pushdown systems} and {\it Markov chains} to their quantum counterparts, i.e., {\em quantum pushdown system (qPDS)} and {\em quantum Markov chains}, and prove a necessary and sufficient condition for a qPDS to be well formed, also presenting a method to extend the local transition function of a well-formed qPDS to a unitary local time evolution operator. Next, we investigate the question of whether it is necessary to define a quantum analogue of {\it probabilistic computational tree logic} to describe the probabilistic and branching-time properties of the {\it quantum Markov chain}. We study its model-checking question and show that model-checking of {\it stateless quantum pushdown systems (qBPA)} against {\it probabilistic computational tree logic (PCTL)} is generally undecidable, i.e., there exists no algorithm for model-checking {\it stateless quantum pushdown systems (qBPA)} against {\it probabilistic computational tree logic}. We then study in which case there exists an algorithm for model-checking {\it stateless quantum pushdown systems} and show that the problem of model-checking {\it stateless quantum pushdown systems (qBPA)} against {\it bounded probabilistic computational tree logic} (bPCTL) is decidable, and further show that this problem is in $\mathit{NP}$-hard. Our reduction is from the {\it bounded Post Correspondence Problem} for the first time, a well-known $\mathit{NP}$-complete problem.

We explore the use of flexible GMRES as an outer wrapper for sketched GMRES. Building on a new bound for the residual of FGMRES in terms of the residual of the preconditioner, we derive a practical randomized solver that requires very little parameter tuning, while still being efficient and robust in the sense of generating non-increasing residual norms.

Dark-field X-ray imaging visualizes structural inhomogeneities through small-angle scattering, but existing directional methods are confined to the micrometer scale. While recent advances have extended dark-field capabilities to nanoscale transmission X-ray microscopy, directional scattering retrieval - critical for characterizing anisotropic nanostructures - has remained inaccessible for imaging resolutions in the sub-micrometer scale. Here, we demonstrate the first directional dark-field setup for nanoimaging, achieving orientation mapping of scattering features below the spatial resolution limit. Our method is experimentally simple to implement with existing transmission X-ray microscopy setups. We validate its performance by successfully resolving sub-resolution test structure orientations, cross-correlating orientational changes within hierarchical nanoporous materials, and mapping the directional arrangement of hydroxyapatite nanocrystals 30 - 70 nm within human tooth enamel. By utilizing shadow regions in the optical configuration, we further extend the detectable scattering vector range, demonstrating a pathway toward size-selective dark-field imaging. This advancement enables the quantitative structural characterization of anisotropic nanomaterials, which are critical to biomineralization, advanced materials, and nanotechnology applications.

For an integer $t\geq 4$, a $t$-sun is a graph obtained from a $t$-vertex cycle $C$ by adding a degree-one neighbor for each vertex of $C$. Trotignon asked whether every triangle-free graph of sufficiently large chromatic number has an induced subgraph that is a $t$-sun for some $t\geq 4$. This remains open, but we show that every triangle-free graph of chromatic number at least $48$ has an induced subgraph that is either a $t$-sun for some $t\geq 5$, or a $4$-sun with a single degree-one vertex deleted. In fact, we prove that for all $\ell\geq 5$, there exists $c=c(\ell)\in \mathbb{N}$ such that every triangle-free graph of chromatic number at least $c$ has an induced subgraph that is either a $t$-sun for some $t\geq \ell$, or a $4$-sun with a single degree-one vertex deleted.

In general, the rates of infection and removal (whether through recovery or death) are nonlinear functions of the number of infected and susceptible individuals. One of the simplest models for the spread of infectious diseases is the SIR model, which categorizes individuals as susceptible, infectious, recovered or deceased. In this model, the infection rate, governing the transition from susceptible to infected individuals, is given by a linear function of both susceptible and infected populations. Similarly, the removal rate, representing the transition from infected to removed individuals, is a linear function of the number of infected individuals. While nonlinear infection and removal rates have been extensively studied in deterministic epidemiological models, analytic results for stochastic dynamics with general nonlinear rates remain limited. This work presents an analytic expression for the number of infected individuals considering nonlinear infection and removal rates. In particular, we examine how the number of infected individuals varies as cases emerge and obtain the expression accounting for the number of infected individuals at each moment. This work paves the way for new quantitative approaches to understanding infection dynamics.

The rapid turn-on of a strong, resonant, continuous wave laser field may trigger the formation of a transient oscillation akin to a train of damped solitons, before the vapor-field system relaxes into a stationary state. We study this transient dynamic on theoretical models of a rubidium vapor. We also consider doubly resonant V-systems, for which the transients take the form of trains of damped simultons. We compute the propagating field(s) by solving the Maxwell-Bloch equations, taking homogeneous broadening, Doppler broadening and the full hyperfine structure of the atoms into account. We also compare the actual fields to the stationary dnoidal fields predicted by the Maxwell-Bloch equations in conditions of self-induced transparency. A similar dynamics is expected to occur in any atomic vapor at the turn-on of a strong resonant continuous wave field provided the turn-on is sufficiently fast compared to relaxation.

Quantum geometry may enable the development of quantum phases ranging from superconductivity to correlated topological states. One powerful probe of quantum geometry is the nonlinear Hall response which detects Berry curvature dipole in systems with time-reversal invariance and broken inversion symmetry. With broken time-reversal symmetry, this response is also associated with quantum metric dipole. Here we investigate nonlinear thermal and thermoelectric responses, which provide a wealth of new information about quantum geometry. In particular, we uncover a web of connections between these quantities that parallel the standard Wiedemann-Franz and Mott relations. Implications for the studies of a variety of topological systems, including Weyl-Kondo semimetals and Bernal bilayer graphene, are discussed.

The concept of statistical depth extends the notions of the median and quantiles to other statistical models. These procedures aim to formalize the idea of identifying deeply embedded fits to a model that are less influenced by contamination. In the multivariate case, Tukey's median was a groundbreaking concept for multivariate location estimation, and its counterpart for scatter matrices has recently attracted considerable interest. The breakdown point and the maximum asymptotic bias are key concepts used to summarize an estimator's behavior under contamination. We explicitly obtain the maximum bias curve, contamination sensitivity and breakdown point of the deepest scatter matrices. In the multivariate and regression setting we analyse recently introduced error bounds that provide a unified framework for studying both the statistical convergence rate and robustness of Tukey's median, depth-based scatter matrices and multivariate regression estimators. We observe that slight variations in these inequalities allow us to visualize the maximum bias behavior of the deepest estimators. We also point out that all the halfspace depths under consideration can be obtained from a unifying concept called residual smallness depth. A numerical study is performed to compare the finite sample bias performance of several robust estimators in the multivariate setting.

We propose an approach for studying the spin liquid phase of the Hubbard model on the triangular lattice by combining the Baym-Kadanoff formalism with the slave rotor parton construction. This method enables the computation of a series of two-body Feynman diagrams for the Luttinger-Ward (LW) functional using a one-loop truncation. This approach enables us to study the U(1) quantum spin liquid phase characterized by a spinon Fermi surface and to derive the Green's functions for spinons, chargons, and electrons. Our findings extend beyond the standard mean-field approximation by accounting for the effects of gauge field fluctuations. The spatial components of the U(1) gauge field are equivalently represented by interactions that incorporate corrections from the spinon-chargon two-particle random phase approximation. This framework effectively captures the long-range correlations inherent to the U(1) quantum spin liquid and combines non-perturbative quantum field theory with the projective construction, providing new insights into the study of quantum spin liquids and other strongly correlated electron systems. We demonstrate that our approach correctly computes the low-temperature linear temperature dependence of the specific heat in the U(1) spin liquid, in agreement with the behavior expected for a Fermi surface. Moreover, this approach reproduces the resonant peaks in the Mott gap, as observed in cobalt atoms on single-layer 1T-TaSe2.

We present a framework for mechanochemical self-organization in active solids where elasticity is reciprocally coupled to Hopf oscillators. Our model reveals a rich landscape of symmetry-protected phases, identified through amplitude equations and group-theoretic analysis. We uncover a universal transition to compression-driven oscillation death (COD), providing a physical basis for localized signaling dampening in biological tissues that resolves inconsistencies in previous models. Our work demonstrates that complex self-organization in active solids can be classified purely through symmetry arguments.

Graph neural network simulators (GNS) have emerged as a computationally efficient tool for simulating granular flows. Previous efforts have been limited to simplified homogeneous geometries characterized only by the friction angle, which does not reflect the complexity of realistic slopes encountered in engineering practice. This study introduces a differentiable GNS framework designed for multi-layered slope systems comprising both forward and inverse modeling components. The forward component relies on a fine-tuned GNS that incorporates both friction angle and cohesion. Its performance is demonstrated through column collapse and multi-layered slope runout simulations, where the GNS replicates multi-material flow dynamics while achieving significant computational speedup over the Material Point Method (MPM). The inverse modeling component leverages the trained GNS, reverse-mode automatic differentiation, and L-BFGS-B optimization to infer material properties from a target runout geometry. Its performance is demonstrated by back-calculating the material strengths that led to failure-induced runout in a dam system composed of multiple materials. Results are obtained within minutes and show good agreement with the target strength values. The framework introduced in this study provides an efficient approach for forward runout assessments and inverse strength back-calculation in realistic slope systems.

We consider a family of potential quasi-lisse affine vertex algebras $L_{k_m}(D_4)$ at levels $k_m =-6 + \frac{4}{2m+1}$. In the case $m=0$, the irreducible $L_{k_0}(D_4)$--modules were classified in arXiv:1205.3003, and it was proved in arXiv:1610.05865 that $L_{k_0}(D_4)$ is a quasi-lisse vertex algebra. We conjecture that $L_{k_m}(D_4)$ is quasi-lisse for every $m \in {\mathbb{Z}}_{>0}$, and that it contains a unique irreducible ordinary module. In this article we prove this conjecture for $m=1$, by using mostly computational methods. We show that the maximal ideal in the universal affine vertex algebra $V^{k_1}(D_4)$ is generated by three singular vectors of conformal weight six. The explicit formulas were obtained using software. Then we apply Zhu's theory and classify all irreducible $L_{k_1}(D_4)$--modules. It turns out that $L_{k_1}(D_4)$ has $405$ irreducible modules in the category $\mathcal O$, but a unique irreducible ordinary module. Finally, we prove that $L_{k_1}(D_4)$ is quasi-lisse by showing that its associated variety is contained in the nilpotent cone of $D_4$. We also prove that the associated variety $X_{L_{k_1}(D_4)}$ is $\overline{\mathbb O}_{sreg}$, the Zariski closure of the subregular nilpotent orbit in $D_4$.

Self-interacting dark matter (SIDM) theories predict that dark matter halos experience core-collapse in late-stage evolution, a process where the halo's inner region rapidly increases in density and decreases in size. This process can be modeled by treating the dark matter as a gravothermal fluid, and solving the fluid equations to predict the density profile evolution. This model is incomplete without calibration to N-body simulations, through a constant factor $β$ included in the thermal conductivity for the long-mean-free-path limit. The value of $β$ employed in the gravothermal fluid formalism has varied between studies, with no clear universal value in the literature. In this work, we use the N-body code Arepo to conduct a series of isolated core-collapse simulations across a range of scattering cross-sections, halo concentrations, and halo masses to calibrate the heat transfer parameter $β$. We find that $β$ is independent of cross-section, halo concentration, and halo mass for velocity independent elastic scattering cross-sections. We present a model for an effective $β$ as a function of a dimensionless cross-section, to describe halo evolution in the long mean free path limit, and show that it accurately captures halo evolution as long as the cross section is not too large. This effective model facilitates comparisons between simulations and the gravothermal model, and enables fast predictions of the dark matter density profile at any given time without running N-body simulations.

We construct an algorithm that, given a pair of homomorphisms between polycyclic-by-finite groups, determines whether their Reidemeister number is finite, and if so returns a set of representatives of the twisted conjugacy classes. Moreover, we show how this algorithm can be applied to compute double cosets and orbits of affine actions.