This paper investigates a recursive formulation of auto-regressive multi-fidelity Gaussian process regression in the challenging setting of noisy and non-nested high- and low-fidelity data. We propose a decoupled optimization strategy based on the expectation-maximization algorithm, which exploits the structure of the recursive model. In particular, we derive closed-form update formulas when the scaling factor is modeled as a parametric linear predictor. This approach is compared with the fully coupled likelihood maximization of the classical non-recursive formulation introduced by Kennedy and O'Hagan. A series of benchmark experiments, covering applications of increasing complexity, highlights the performance of both approaches. The results demonstrate that the proposed recursive strategy significantly reduces training time, especially when large low-fidelity datasets are available, while maintaining competitive predictive accuracy and uncertainty estimation.

We generalize certain totient functions using elementary symmetric polynomials and derive explicit product forms for the totient functions involving the second elementary symmetric sum. This work follows from the work of Toth [The Ramanujan Journal, 2022] where the totient function was generalized using the first and the kth elementary symmetric polynomial. We also provide some observations on the behavior of the totient function with an arbitrary jth elementary symmetric polynomial. We then outline a method for solving a certain the restricted linear congruence problem with a greatest common divisor constraint on a quadratic form, illustrated by a concrete example. Most importantly, we demonstrate the equivalence between obtaining product forms for generalized totient functions, counting zeros of specific polynomials over finite fields, and resolving a broad class of restricted linear congruence problems .

We investigate the constraints and phenomenology of a three Higgs doublet model (3HDM) with a $\Z2\times\Z2$ symmetry, featuring two inert scalar doublets that give rise to a two-component dark matter (DM) scenario. We analyze the model's vacuum structure, exploring the competition between different symmetry-breaking minima, and subject it to comprehensive theoretical and current experimental constraints. Our analysis reveals previously unexplored regions of parameter space with viable dark matter candidates. Notably, we identify scenarios where both DM particles contribute comparably to the observed relic density, offering distinctive experimental signatures that could guide future searches.

We theoretically investigate the shape dependence and microscopic mechanism of the magnetoelectric (ME) effect, including both nonmagnetic (Edelstein-type) and magnetic origins, in a V-shaped one-dimensional chain model. Our goal is to establish a symmetry-based framework linking local geometry to ME responses. Numerical calculations based on the Kubo formula reveal that the nonmagnetic-driven ME response is maximized at an apex angle of $θ\approx 0.6π$. To clarify its origin, we derive a low-energy effective Hamiltonian in the $s$-orbital subspace and demonstrate that the polarity induced by the V-shaped geometry manifests as an effective spin--orbit interaction. An analytical derivation of the Green's function shows that the geometric effect can be described as a $T$-matrix contribution associated with local symmetry breaking. This formulation provides a unified description of geometry-induced responses in terms of a scattering framework. Using a multipole-basis representation, we identify symmetry-based selection rules for the ME tensor and show that the coupling between the effective spin--orbit interaction and the orbital angular momentum generated across the apex plays an essential role. The resulting angular dependence, $\sinθ\sin{θ/2}$, peaks at $θ= 2\tan^{-1}\sqrt{2} \approx 0.608π$, in good agreement with the numerical results. We also analyze a ferromagnetic V-shaped model including the Zeeman interaction and show that the magnetic-driven ME response originates from the spin magnetization induced by the coupling between the electric-field--driven charge-potential gradient and the Zeeman term. These results reveal distinct ME mechanisms depending on the presence or absence of time-reversal symmetry and provide a microscopic framework for geometry-induced multipole phenomena.

This paper investigates the parameter identification for multi-participant autoregressive exogenous input (ARX) systems while protecting the system input and output. To do so, the discrete Gaussian noise in the standard Cheon-Kim-Kim-Song (CKKS) cryptosystem is replaced with a truncated one. By using the CKKS cryptosystem with the truncated discrete Gaussian noise and the key-switching technique, a proxy re-encryption scheme is developed. Based on this scheme, a secure parameter identification algorithm is proposed for multi-participant ARX systems. By rigorously proving that the statistical distance between the discrete Gaussian noise and the truncated one is negligible, the polynomial-time reduction between the standard Ring-Learning with Errors (RLWE) problem and the RLWE problem with the truncated discrete Gaussian noise is established. This result ensures the indistinguishability under chosen-plaintext attacks (IND-CPA) security of the algorithm. By giving a lower bound condition on the size of the plaintext space, the computational overflow in encryption is avoided. Based on this condition, the mean square convergence and convergence rate of the algorithm are given. The trade-off between the security level and the convergence of the algorithm is presented. Finally, a numerical example is given to verify the effectiveness of the algorithm.

Recent theoretical and experimental advances have identified cobaltates with a high-spin $d^7$ electronic configuration as promising hosts for spin-orbit entangled $J_{eff} = 1/2$ magnetism that can support bond-dependent exchange interactions. In two-dimensional triangular lattices, the coexistence of such exchange frustration along with geometric frustration gives rise to a rich landscape of competing magnetic phases, establishing monolayer triangular $d^7$ cobaltates as a compelling platform for frustrated magnetism. Here we investigate a representative triangular-lattice monolayer cobaltate CoBr$_2$, where first-principles density functional theory (DFT) calculations reveal a dominant nearest-neighbor $t_{2g}$-$e_g$ hopping channel that enhances the ferromagnetic Kitaev-type exchange interactions. In contrast, the nearest-neighbor Heisenberg term is highly sensitive to a direct $t_{2g}$-$t_{2g}$ hopping path and electronic correlations. The magnetic exchange parameters are evaluated using the hopping amplitudes obtained from DFT calculations within an exact diagonalization framework. We construct the first and third nearest neighbor Heisenberg exchange dependent $J_1$-$J_3$ magnetic phase diagram in the physically relevant regime and identify multiple competing ground states, including ferromagnetic, stripy, spiral, and $120^{\circ}$ antiferromagnetic orders. The Luttinger-Tisza analysis further predicts a Z$_2$ vortex crystal phase, while exact diagonalization reveals a bond-nematic phase stabilized by the longer-range couplings. Going beyond the conventional bond-independent XXZ picture typically applied to Co$^{2+}$ systems, our results on monolayer CoBr$_2$ establish d$^7$ cobalt dihalides as a promising platform to explore the interplay of long-range Heisenberg and bond-dependent exchange interactions that can stabilize diverse magnetic ground states on a triangular lattice.

We derive an analog of the Lellouch-Lüscher (LL) relation for few-body bosonic systems, linking few-body scattering loss rates to the energies and widths of the corresponding harmonically trapped few-body states. Three-body numerical simulations show that the LL relation applies across a broad range of interaction strengths and energies and allows the determination of scattering rates within a single partial wave. Our work establishes a robust theoretical framework for understanding the role of the finite volume effect in few-body observables in optical lattice and tweezer experiments, enabling precise determination of multi-body scattering rates.

Deep learning has emerged as a key tool for designing nanophotonic structures that manipulate light at sub-wavelength scales. We investigate how to inversely design plasmonic nanostructures using conditional generative adversarial networks. Although a conventional approach of measuring the optical properties of a given nanostructure is conceptually straightforward, inverse design remains difficult because the existence and uniqueness of an acceptable design cannot be guaranteed. Furthermore, the dimensionality of the design space is often large, and simulation-based methods become quickly intractable. Deep learning methods are well-suited to tackle this problem because they can handle effectively high-dimensional input data. We train a conditional generative adversarial network model and use it for inverse design of plasmonic nanostructures based on their extinction cross section spectra. Our main result shows that adding label projection and a novel embedding network to the conditional generative adversarial network model, improves performance in terms of error estimates and convergence speed for the training algorithm. The mean absolute error is reduced by an order of magnitude in the best case, and the training algorithm converges more than three times faster on average. This is shown for two network architectures, a simpler one using a fully connected neural network architecture, and a more complex one using convolutional layers. We pre-train a convolutional neural network and use it as surrogate model to evaluate the performance of our inverse design model. The surrogate model evaluates the extinction cross sections of the design predictions, and we show that our modifications lead to equally good or better predictions of the original design compared to a baseline model. This provides an important step towards more efficient and precise inverse design methods for optical elements.

In "Faces of a Hyperplane Arrangement Enumerated by Ideal Dimension, with Applications to Plane, Plaids, and Shi," Zaslavsky showed how to compute the number $r_\ell(\mathcal{A})$ of regions of a real hyperplane arrangement $\mathcal{A}$ with a given level, refining his well known enumeration of regions and relatively bounded regions. We restate this theorem in terms of a construction called the centralization of $\mathcal{A}$, give a bijective proof, and then apply it in two ways to answer questions concerning the concept of level. Firstly, a consequence of this enumeration is that $r_\ell(\mathcal{A})$ depends only on the intersection poset $\mathcal{L}(\mathcal{A})$, such that both $r_\ell$ and centralization can be defined in the more general setting of geometric semilattices. In this context we derive a very general expression for the characteristic polynomial of a geometric semilattice with several interesting corollaries. Secondly, recent investigations into the phenomenon of level have made little use of Zaslavsky's level-counting theorem, but it can be applied to obtain or generalize many of their results. In particular we show how exponential generating function identities (arXiv:2410.10198, arXiv:2411.02971) and an expression giving the characteristic polynomial in terms of $r_\ell$ (arXiv:2411.03756) can be derived for deformations of the braid arrangement.

The $Λ$CDM model, while successful on large cosmological scales, faces challenges on small scales. A promising solution posits that dark matter (DM) exhibits strong self-interaction, enhanced through the narrow resonance or Sommerfeld effects. We demonstrate that the ``super-resonance" phenomenon, combining these effects, significantly amplifies the DM self-scattering cross section, enabling strong self-interactions for DM candidates in the $\mathcal{O}(100)$ GeV mass range. This mechanism also enhances the DM annihilation cross section, causing early kinetic decoupling that renders the standard Boltzmann equation inadequate. By implementing coupled Boltzmann equations, we achieve precise calculations of the relic density for super-resonant DM, aligning with observational constraints.

Radiation-mediated shocks (RMSs) occurring below the photosphere in a gamma-ray burst (GRB) jet could play a crucial role in shaping the prompt emission. In this paper, we study the time-resolved signal expected from such early shocks. We model an internal collision using a 1D special relativistic hydrodynamical simulation, and we follow the photon distributions in the resulting forward and reverse shocks as well as in the common downstream region to well above the photosphere using a designated RMS simulation code. We compute the light curve and time-resolved spectrum of the resulting single pulse taking into account the emission at different optical depths and angles to the line of sight. For the specific case considered, we find a light curve consisting of a short pulse lasting $\sim 0.1~$s for an assumed redshift of $z = 1$, which could constitute a whole short GRB or be a building block within a highly variable longer GRB light curve. The efficiency is large, with $\approx 23$% of the total burst energy being radiated. The spectrum has a complex shape at very early times, after which it settles into a more generic shape with a smooth curvature below the peak energy, $E_p$, and a clear high-energy power law that cuts off at $\sim 5~$MeV in the observer frame. The spectrum becomes narrower and softer at late times with $E_p$ steadily decreasing during the pulse decay from $E_p \approx 250~$keV to $E_p \approx 100~$keV. The low-energy index, $α$, decreases during the bright part of the pulse from $α\approx -0.5$ to $α\approx -1$, although the low-energy part is better fit with a broken power law when the signal-to-noise ratio is high. The high-energy power law is generated by the reverse shock at low optical depths ($τ< 30$) and has an index that decreases from $β\approx -2$ to $β\approx -2.4$.

The problem of super-resolution, roughly speaking, is to reconstruct an unknown signal to high accuracy, given (potentially noisy) information about its low-degree Fourier coefficients. Prior results on super-resolution have imposed strong modeling assumptions on the signal, typically requiring that it is a linear combination of spatially separated point sources. In this work we analyze a very general version of the super-resolution problem by considering completely general non-negative signals (equivalently, distributions) over the $d$-dimensional torus $[0,1)^d$; we do not assume any spatial separation between point sources, or even that the distribution is a finite linear combination of point sources. The question naturally arises: what can be said about super-resolution in such a general setting? - As a warm-up, we first give a set of results for reconstructing distributions under the Wasserstein distance. We establish essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $κ$ of the noise for which accurate reconstruction is possible: we show that for $d$-dimensional distributions, estimates of $\approx \exp(d)$ many Fourier coefficients are both necessary and sufficient for accurate Wasserstein reconstruction. - As our main result, we define a new notion of "heavy hitter" reconstruction for distributions, which essentially amounts to achieving high-accuracy reconstruction of all "sufficiently dense" regions of the distribution. We give essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $κ$ of the noise for which accurate reconstruction is possible under this notion. Our results show that (in sharp contrast with Wasserstein reconstruction) accurate estimates of only $\approx \exp(\sqrt{d})$ many Fourier coefficients are both necessary and sufficient for heavy hitter reconstruction.

We have developed a new version of the high-performance Jülich universal quantum computer simulator (JUQCS-50) that leverages key features of the GH200 superchips as used in the JUPITER supercomputer, enabling simulations of a 50-qubit universal quantum computer for the first time. JUQCS-50 achieves this through three key innovations: (1) extending usable memory beyond GPU limits via high-bandwidth CPU-GPU interconnects and LPDDR5 memory; (2) adaptive data encoding to reduce memory footprint with acceptable trade-offs in precision and compute effort; and (3) an on-the-fly network traffic optimizer. These advances result in a 16.6-fold speedup over the previous 48-qubit record on the K computer

We give an introduction to renormalisation, focusing first on a pedagogical description of fundamental concepts of the procedure and its features, then we introduce the renormalisation group and its equations. We discuss then the case of gauge theories such as QCD summarising the current state of the art. We introduce the renormalisation scale setting problem in QCD and we give an illustration of the possible optimisation procedures currently in use.

We construct $AdS_3\times Y_7$ solutions of type IIB supergravity, where $Y_7$ is a smooth $S^5$ bundle over a spindle $Σ(n_N,n_S)$, which are dual to $\mathcal{N}=(0,2)$ SCFTs in $d=2$. The solutions are constructed using the $D=5$ STU $U(1)^3$ gauged supergravity theory coupled to a hyperscalar charged under $U(1)_B$. We investigate spindle solutions with two new features: first, we allow $(n_N,n_S)$ to be non-coprime integers, including orbifolds of the round $S^2$, which can lead to non-unique, inequivalent uplifts, distinguished by the hyperscalar spectra, for given magnetic flux through the spindle. Second, we also allow the hyperscalar to vanish at the poles leading to solutions carrying non-vanishing $U(1)_B$ flux. The new hyperscalar $AdS_3$ solutions can naturally arise as the endpoint of RG flows, triggered by relevant hyperscalar deformations of the $AdS_3$ solutions of the STU model.

Detecting multimodality in empirical distributions is a fundamental problem in statistics and data analysis, with applications ranging from clustering to the study of complex systems. In practice, however, assessing departures from unimodality in a consistent and comparable way remains challenging. Widely used methods such as Hartigan and Hartigan's Dip Test illustrate these difficulties, as the interpretation of their statistics depends strongly on sample size, requires calibration to determine significance, and, for large samples, exhibit increasing sensitivity, leading to rejection of unimodality for arbitrarily small deviations from the null. We introduce Z-Dip, a standardized measure of multimodality that addresses these limitations. By treating the Dip statistic as a random variable under the null hypothesis of unimodality and standardizing its observed value, the proposed approach yields scores that are directly comparable across datasets of different sizes. Using simulation-based calibration, we derive a universal decision threshold that closely reproduces classical Dip Test decisions without requiring sample-size-specific adjustments. Extensive validation on simulated data and on more than 88,000 empirical opinion distributions shows near-perfect agreement with the classical Dip Test while providing a more interpretable and comparable measure of modality. Finally, we propose a downsampling-based correction that mitigates residual sensitivity in extremely large samples. Open-source software and reference tables are provided to facilitate practical adoption.

In this work, we demonstrate the complete process of using space-based gravitational wave detectors to measure properties of the stochastic gravitational wave background arising from a first-order electroweak phase transition. Based on frequency-domain simulations of the Taiji mission, including instrumental noise and astrophysical foregrounds, we perform parameter inference using both the Fisher information matrix and Bayesian Markov Chain Monte Carlo sampling. We show how the reconstructed spectrum constrains the macroscopic parameters of the phase transition, and further how these constraints map onto the underlying particle-physics parameters in a singlet-extended Standard Model. Our results demonstrate that the Higgs cubic and quartic self-couplings can be significantly constrained using gravitational wave observations, despite limitations arising from parameter degeneracy.

We analyse the behaviour of the linear phase of the Rayleigh-Taylor instability (RTI) in the presence of a foam. Such a problem may be relevant, for example, to some inertial confinement fusion (ICF) scenarios such as foams within the capsule or lining the inner hohlraum wall. The foam displays 3 different phases: by order of increasing stress, it is first elastic, then plastic, and then fractures. Only the elastic and plastic phases can be subject to a linear analysis of the instability. The growth rate is analytically computed in these 2 phases, in terms of the micro-structure of the foam. In the first, elastic, phase, the RTI can be stabilized for some wavelengths. In this elastic phase, a homogenous foam model overestimates the growth because it ignores the elastic nature of the foam. Although this result is derived for a simplified foam model, it is likely valid for most of them. Besides the ICF context considered here, our results could be relevant for many fields of science.

Magnetic bimerons, similar to skyrmions, are topologically nontrivial spin textures characterized by topological charge $Q$. Most studies so far have focused on low-$Q$ solitons ($|Q| \leq 1$), such as skyrmions, bimerons, and vortices. Here, we present the first calculations of the lifetimes of {ring-like} high-$Q$ bimerons and demonstrate that they are fundamentally more stable than high-$Q$ skyrmions over a wide range of temperature. To obtain realistic results, our chosen system is an experimentally feasible van der Waals interface, Fe$_3$GeTe$_2$/Cr$_2$Ge$_2$Te$_6$. We show that the lifetimes of high-$Q$ (anti)bimerons can exceed the lifetime of those with $|Q|=1$ by 3 orders of magnitude. Remarkably, this trend remains valid even when extrapolated to room temperature (RT), as the lifetimes are dominated by entropy rather than energy barriers. This contrasts with high-$Q$ skyrmions, whose lifetimes fall with $|Q|$ near RT. We attribute this fundamental difference between skyrmions and bimerons to their distinct magnetic texture symmetries, which lead to different entropy-dominated lifetimes.

In the presence of a background magnetic field, axions or axion-like particles (ALPs) can be resonantly converted to photons when their mass is nearly equal to the effective photon mass. In this paper, we propose a novel method to constrain the parameter space of ALPs by investigating the resulting imprints of axion-photon conversion in the cosmic microwave background (CMB) observations. We show that a helical magnetic field existing prior to the CMB epoch can generate an excess population of photons carrying net circular polarization due to the axion-photon conversion mechanism. Consequently, current measurements of the angular power spectrum of circular polarization ($V$-mode) in the CMB can be used to constrain the parameter space of ALP mass and its coupling to photons. In the optimistic scenario of a maximally helical magnetic field with strength $\sim {\rm nG}$, we find that CLASS observations at $40 \, {\rm GHz}$ can probe the previously unconstrained regions of axion-photon coupling corresponding to ALP masses in the range $10^{-10}-10^{-8} \, {\rm eV}$.

This work is concerned with the coordination gain in integrated sensing and communication (ISAC) systems under a compress-and-estimate (CE) framework, wherein inference performance is leveraged as the key metric. To enable tractable transceiver design and resource optimization, we characterize inference performance via an error probability bound as a monotonic function of the discriminant gain (DG). This raises the natural question of whether maximizing DG, rather than minimizing mean squared error (MSE), can yield better inference performance. Closed-form solutions for DG-optimal and MSE-optimal transceiver designs are derived, revealing water-filling-type structures and explicit sensing and communication (S\&C) tradeoff. Numerical experiments confirm that DG-optimal design achieves more power-efficient transmission, especially in the low signal-to-noise ratio (SNR) regime, by selectively allocating power to informative features and thus saving transmit power for sensing.

We study the geodesic structure and observable properties of asymptotically flat regular black holes sourced by a phantom scalar field characterized by a scalar charge $A$. This parameter removes the central singularity and continuously deforms the Schwarzschild geometry. The equations of motion for test particles and photons are derived, and the resulting null geodesics are analyzed, including the deflection of light, gravitational time delay, and redshift, in order to constrain $A$ using classical Solar System tests. These observations impose stringent limits on the scalar charge, confirming that $A$ must remain extremely small in the weak-field regime to ensure full consistency with general relativity. In the strong-field regime, we compute the Lyapunov exponent $λ$ associated with the photon sphere and establish its exact relations with the critical impact parameter $\mathcal{B}_u$ and the angular size of the shadow $α_{\mathrm{sh}}$, given by $\mathcal{B}_u = 1/|λ|$ and $α_{\mathrm{sh}} = 1/(r_{0}|λ|)$. These correspondences reveal that the dynamical instability of null circular orbits governs the optical appearance of the black hole. Our results show that increasing $A$ reduces the instability of photon trajectories and enlarges the angular size of the shadow, indicating that the regularization scale leaves a distinct observational imprint on the geometry of regular black holes. In addition, constraints derived from Event Horizon Telescope observations of M87* and Sgr A* further restrict the allowed range of the scalar charge, reinforcing the consistency of the model with current astrophysical observations.

Large Language Models (LLMs) are increasingly deployed in complex multi-agent applications that rely on external function calls. This workload creates severe performance challenges for the KV Cache: spatial contention leads to the eviction of critical agents' caches and temporal underutilization leaves the cache of agents stalled on long-running function calls idling in GPU memory. We present TokenCake, a KV-Cache-centric serving framework that bridges this gap by co-optimizing scheduling and memory management through an agent-aware design. TokenCake's Temporal Scheduler employs an event-driven, opportunistic policy to proactively offload idle KV Caches during function calls and uses predictive uploading to hide data transfer latency. TokenCake's Spatial Scheduler uses dynamic memory partitioning, guided by a hybrid priority metric combining graph structure and runtime state, to reserve GPU memory for critical-path agents. Our evaluation on representative multi-agent benchmarks shows that TokenCake reduces end-to-end latency by over 47.06% and improves effective GPU memory utilization by up to 16.9% compared to vLLM.

Human mobility, enabled by diverse transportation modes, is fundamental to urban functionality. Studying these movements across scales-from microscopic to macroscopic-yields valuable insights into urban dynamics. Local adaptation and (self-)organization in such systems are expected to result in dynamical behaviors that are represented by stationary trajectories of an appropriate effective action. In this study we develop a Lagrangian dynamical model for movement processes, using local population functions as the coordinate variables. An efficient gradient descent algorithm is introduced to estimate the optimal Lagrangian parameters minimizing a local error function of the dynamical process. We show that even a quadratic Lagrangian, incorporating dissipation, effectively captures the dynamics of synthetic and empirical movement data. The inferred models reveal that inertia and dissipation are of comparable importance, while interactions and randomness in the movements induce significant qualitative changes in model parameters. Our results provide an interpretable and generative model for human mobility, with potential applications in movement prediction.

We propose a new version of the Frank-Wolfe method, called the (L0, L1)-Frank-Wolfe algorithm, developed for optimization problems with (L0, L1)-smooth objectives. We establish that this algorithm achieves superior theoretical convergence rates compared to the classical Frank-Wolfe method. In addition, we introduce a novel adaptive procedure, termed the Adaptive (L0, L1)-Frank-Wolfe algorithm, which dynamically adjusts the smoothness parameters to further improve performance and stability. Comprehensive numerical experiments confirm the theoretical results and demonstrate the clear practical advantages of both proposed algorithms over existing Frank-Wolfe variants.

Nanoporous metals are extensively investigated as platforms for applications in plasmonics. They present high surface areas and strong local electric fields that can be tuned at different energies, playing with the choice of the metals and the morphology of the porous layers. Until recently, research in the field of plasmonics has primarily focused on porous metals composed of a single element, with limited attention given to the impact of alloy composition. The investigation of bi-metallic systems has only just begun to emerge in the literature. In particular, combining two or more different plasmonic metals, it could be possible to explore the interactions between two metals excited at specific energies. This involves plasmonic coupling, electron transfer, band hybridization at the interface, electromagnetic field interactions, and possibly thermal and electronic energy transfer depending on separation, size, and materials involved. The analysis of bi-metal systems can also be interesting in biomolecule detection, such as in the case of Surface Enhanced Raman Scattering (SERS). Here we report, for the first time, a detailed study (comprising morphological analyses, numerical modelling, and optical spectroscopies) on bi-metal nanoporous platforms prepared with a dry-synthesis method enabling the easy and controllable fabrication of bilayers combining different metals such as Au, Ag, and Cu.

In this work, we introduce a novel computational framework for solving the two-dimensional Hele-Shaw free boundary problem with surface tension. The moving boundary is represented by point clouds, eliminating the need for a global parameterization. Our approach leverages Generalized Moving Least Squares (GMLS) to construct local geometric charts, enabling high-order approximations of geometric quantities such as curvature directly from the point cloud data. This local parameterization is systematically employed to discretize the governing boundary integral equation, including an analytical formula of the singular integrals. We provide a rigorous convergence analysis for the proposed spatial discretization, establishing consistency and stability under certain conditions. The resulting error bound is derived in terms of the size of the uniformly sampled point cloud data on the moving boundary, the smoothness of the boundary, and the order of the numerical quadrature rule. Numerical experiments confirm the theoretical findings, demonstrating high-order spatial convergence and the expected temporal convergence rates. The method's effectiveness is further illustrated through simulations of complex initial shapes, including interfaces driven by anisotropic surface tension, which correctly evolve towards circular equilibrium states under the influence of surface tension, highlighting the versatility of the method for complex geometry-dependent interface dynamics.

The Pontryagin-Thom theorem gives an isomorphism between the cobordism group of framed $n$-dimensional manifolds, $ω_n$, and the $n^{th}$ stable homotopy group of the sphere spectrum, $π_n(\mathbb{S})$. The equivariant analogue of this theorem, gives an isomorphism between the equivariant cobordism group of $V$-framed $G$-manifolds, $ω_V^G$, and the $V^{th}$ equivariant stable homotopy group of the $G$-sphere spectrum, $π_V^G(\mathbb{S})$, for a finite group $G$ and a $G$-representation, $V$. In this paper, we explicitly identify the images of each element of $ω_1^{C_2}$ and $ω_σ^{C_2}$ in $π_1^{C_2}(\mathbb{S})$ and $π_σ^{C_2}(\mathbb{S})$ under the equivariant Pontryagin-Thom isomorphism.

The imposition of tariffs by President Trump during his second term had far-reaching consequences for global markets, including Australia. This study investigates how both the announcement and subsequent implementation of these tariffs, specifically on 02-Apr-2025, affected the Australian stock market, focusing on the S\&P/ASX 200 index over the period from 21-Jan-2025 to 25-Jul-2025. To accurately capture the significance and behavior of market fluctuations, the exploratory data analysis (EDA) techniques are applied. Furthermore, the impact of tariffs on stock performance is evaluated using machine learning-based regression models. A comparative assessment of these models is conducted to determine their predictive accuracy and robustness in capturing tariff-related market responses.

A character table $X$ for a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is the square matrix of values associated to a basis of the lattice of virtual $\mathcal{F}$-stable ordinary characters of $S$. We investigate a conjecture of the second author which equates the determinant of $X \overline{X}$ (the square of the volume of this lattice) with the product of the orders of $S$-centralisers of fully $\mathcal{F}$-centralised $\mathcal{F}$-class representatives. This statement is exactly column orthogonality for the character table of $S$ when $\mathcal{F}=\mathcal{F}_S(S)$. We prove the conjecture when $\mathcal{F}=\mathcal{F}_S(G)$ is realised by some finite group $G$ with Sylow $p$-subgroup $S$, and for all simple fusion systems when $|S| \le p^4$. We also put forward a potential strategy for the general case, which would exploit properties of the characteristic idempotent of $\mathcal{F}$.