In this paper, we study the eigenvalues of the matrices $T_n(a)+γE_{n,1,1}$ where $T_n(a)$ is the Toeplitz matrix with generating symbol $a(t)=t-t^{-1}$, $E_{n,1,1}$ is the $n\times n$ matrix whose upper left component is $1$ and the other components are zero, and $γ$ is a fixed complex number such that $0<|γ|<1$. As $n\to\infty$, the eigenvalues of these matrices are asymptotically distributed as the function $2 i \sin(x)$, $x\in[0,2π]$. Our main result is an asymptotic formula for every eigenvalue with a residue of the order $O(1/n^3)$.

Stacking multiple SiSiGe channels in advanced logic devices faces severe thermal budget accumulation, which degrades interfaces via Ge-Si interdiffusion and strain relaxation.This strategy lowers the Ge diffusion coefficient to 5.6-7% of its value at 650C (Arrhenius estimate), suppressing interdiffusion and preserving pseudomorphic strain. The 4 + 4 channel stack exhibits clear XRD satellite peaks, fully coherent strain state (reciprocal space mapping), sharp interfaces (1.5-2.6 nm transition width) and low RMS roughness (0.08 nm). Quantitative analysis from bottom to top reveals that prolonged high-temperature exposure broadens bottom interfaces and dilutes Ge concentration (from 20% to 18.5%), while the top stack maintains design targets. This work provides a process-physics understanding of thermal budget effects in multi-channel superlattices and establishes a high-quality material foundation for advanced logic devices beyond 2 nm node.

Standard cardiovascular risk calculators, including the Framingham Risk Score and the ACC/AHA Pooled Cohort Equations, estimate the conditional probability P(CHD | SysBP = s) rather than the interventional quantity P(CHD | do(SysBP = s)). When confounding is present, this distinction has direct clinical consequences: observational estimates may systematically overstate the absolute benefit of antihypertensive treatment. We applied Pearl's do-calculus to the Framingham Heart Study Offspring Cohort (n = 4,240; primary analysis on 3,776 complete cases; 574 ten-year coronary heart disease events). A structurally corrected directed acyclic graph (DAG) was specified and evaluated using conditional independence testing. The average causal effect (ACE) of a 20 mmHg systolic blood pressure reduction was estimated by g-computation with bootstrap confidence intervals, corroborated by propensity score matching and inverse probability weighting. G-computation yielded an ACE of 3.40 percent absolute risk reduction (95 percent CI: 2.64 to 4.14), compared with a naive observational estimate of 4.14 percent, corresponding to an approximate 21.8 percent relative overestimation. Conditional average treatment effects were estimated using R-Learner and T-Learner metalearners. These findings suggest that observational cardiovascular risk tools may overestimate the absolute benefit of blood pressure reduction, with implications for clinical risk stratification and prescribing thresholds.

We study threefolds of general type constructed as $\mathbb{Z}_2^s$-covers of weighted projective spaces with a particular focus on their invariants, deformation theory, and the behavior of the $m$-canonical map. For the invariants, we write the ratios of the volume to the topological and holomorphic Euler characteristics as functions of the ratios of the degree of the branch divisors with respect to the total degree. From this expression, we obtain their asymptotic behavior, bounds, and a counterexample to a conjecture made by Bruce Hunt about the non-existence of smooth threefolds in a forbidden zone. From the perspective of deformation theory, we extend the criterion for such covers to be general in their moduli to the case when the weighted projective threefold has isolated singularities and the cover is non-flat, i.e., the pushforward of the structure sheaf splits as a direct sum of reflexive sheaves as opposed to line bundles. As an application, we present new numerical criteria for constructing components of the moduli spaces of stable threefolds and give concrete examples illustrating their application. Finally, we introduce techniques from Fourier transforms on finite groups to completely classify when a $\mathbb{Z}_2^s$-cover is a flat pluricanonical map. For $s \geq 2$, there are $32$ deformation types. We also show that there exist non-flat canonical and bicanonical $\mathbb{Z}_2^s$-covers for arbitrarily large values of $s$.

This paper reports a Ricci-Gauss-Bonnet (RGB) dark energy model within the framework of Hořava-Lifshitz cosmology and presents a scalar-field reconstruction of the effective dark energy sector. In a spatially flat FRW background with a power-law scale factor, we derive analytical expressions for cosmological parameters, scalar field kinetic term, and the reconstructed potential. The reconstructed EoS parameter exhibits smooth transition toward a cosmological-constant-like regime at late times for suitable choices of the model parameters. The classical stability of the model is analyzed through the squared sound speed, and stable regions of the parameter space are identified. Finally, the generalized second law of thermodynamics is investigated at the apparent horizon, and it is shown that the total entropy variation remains non-negative in this model. From these results it can be concluded that the model provides a theoretically consistent description of late-time acceleration, with physical viability maintained within a specific range of the model parameters.

Balmer-dominated shocks in Type Ia supernova remnants offer powerful probes into collisionless shock physics and hints towards supernova progenitor environments. Prior studies focused on the hydrogen Balmer lines, which manifest as a superposition of broad and narrow emission lines. Using integral-field spectroscopy with MUSE, we discovered broad and narrow helium emission lines from Balmer-dominated filaments of three Type Ia supernovae remnants in the Large Magellanic Cloud: SNR 0509-67.5, SNR 0519-69.0 and N103B. We detect broad and narrow He~\textsc{i} 5876~Å~,7065~Å emission in SNR 0519 and N103B and He \textsc{ii} 8236~Å in SNR 0519. In SNR 0509 we detect narrow He~\textsc{i} 5015~Å, 6678~Å, 7065~Å and 7281~Å, with only 7065~Å~ exhibiting a broad component. The detection of narrow He\,\textsc{ii} challenges existing shock models, where such emission is not expected, and may indicate either incomplete ion-ion equilibration behind the shock or an origin in shock precursors. For SNR 0509 and N103B, the neutral He/H line ratios indicate enhanced helium abundances, whereas SNR 0519 is consistent with the primordial He/H value. We therefore propose helium emission in Balmer-dominated shocks as a new diagnostic of shock physics and Type Ia supernova circumstellar environments. Although our modeling is primarily a proof of concept, it demonstrates the possibility to infer the total He-to-H abundance ratio, with dominant uncertainties arising from the assumed initial ionization fractions. Despite the uncertainties, we demonstrate that narrow helium lines can serve as effective probes of circumstellar conditions and progenitor evolution when analysed alongside reliable constraints on the preshock neutral H/He abundance ratio.

These notes provide a pedagogical introduction to the role of transversality theory in the analysis of statistical degeneracies within the framework of distributional statistical models. The classical question of when a statistical model is well-behaved - in the sense of being identifiable, having non-singular Fisher information, and admitting robust estimation - is reformulated as a question about the geometry of a kernel-induced feature map. Statistical pathologies correspond to geometric degeneracies of this map, and transversality theory provides a precise language for understanding when and why such degeneracies are non-generic. The exposition is organised in three parts. Part I surveys the statistical phenomena that motivate the geometric treatment: representation failure, non-identifiability, moment indeterminacy, singular information, nuisance parameters, and the Behrens-Fisher problem. Part II develops the necessary geometric toolkit - smooth maps, Sard's theorem, transversality, jets, stratifications, and the parametric transversality theorem - at a level accessible to students with a background in analysis and linear algebra but no prior exposure to differential topology. Part~III returns to the statistical problems of Part~I and shows how each one admits a unified geometric interpretation as a transversality condition on the feature map. These notes are a pedagogical companion to the research paper Labouriau (2026) "Transversality and Geometric Regularisation in Distributional Statistical Models" (arXiv:2605.04536 [math.ST]), expanding its arguments with motivating examples, geometric intuition, and exercises aimed at advanced Master's and PhD students with a background in mathematical statistics and measure theory. They are designed to support seminars or reading groups.

Bernstein modes (BM) in graphene are finite-wavevector magnetoplasmons excited by contact near fields, whereas ordinary cyclotron resonance (CR) probes $q\approx0$. We derive the BM peak absorption in the quasiclassical ballistic regime and show that it factorizes into a launch spectrum, Bernstein-mode splitting, turning-point enhancement, and residual dielectric-response factor. At fixed excitation frequency, BM overtones ($n\ge2$) are sampled, to leading order, at the same momentum $q\simeqω/v_F$. Smooth launch and screening factors therefore cancel in inter-harmonic peak ratios, yielding $I_n/I_m\simeq m/n$, modified by linewidth corrections and one residual response ratio for each harmonic pair. In smooth-launcher synthetic tests, noisy full-$q$ spectra recover the residual ratio within errors: moderate launcher/dielectric misspecification within this benchmark family shifts it by only $\sim\!1$--$2\%$, whereas linewidth assumptions shift it by $\sim\!10$--$30\%$. The same factorization connects low-power amplitudes to nonlinear saturation. If BM harmonics share the same cooling region and bolometric readout, the low-power slope times onset intensity is harmonic independent, while BM and CR power sweeps obey distinct normalized saturation curves with linewidth scalings $Γ^{-1/2}$ and $Γ^{-1}$.

In this paper, we derive new commutator estimates in the Triebel-Lizorkin spaces by employing Bony's para-product decomposition, the Nikol'skij representation, and the Fefferman-Stein vector-valued maximal function. These estimates are then applied to develop a general theory for transport equations. Although analogous results are already available in the setting of Besov spaces, the methods developed there do not carry over directly to the Triebel-Lizorkin case. Our approach works for Triebel-Lizorkin spaces and, as a byproduct, also yields the corresponding results in Besov spaces. All proofs are presented in a unified manner that applies to both scales of function spaces, thereby extending and sharpening previous results on transport equations in these frameworks. Furthermore, the general theory we obtain is widely applicable to evolution equations, including incompressible and compressible ideal fluid flows, shallow water waves, and related models. As an illustration, we consider the two-component Euler-Poincaré system. Using the theoretical framework developed herein, we establish its local well-posedness and a blow-up criterion in both sub-critical and critical Triebel-Lizorkin spaces.

We present spatially resolved Keck/LRIS spectroscopy of three clumpy star-forming galaxies at $z\sim0.1$, comparing outflow properties traced by H$α$ and Mg II emission with those probed by Mg II and Na I D absorption. Outflow velocities measured using Mg II absorption ($\langle v_{\rm out} \rangle = -560 \pm 30$~\kms) are consistently higher than those traced by H$α$ emission ($\langle v_{\rm out} \rangle = -124 \pm 3$~\kms) across $\sim$5 kpc$^{2}$ regions. Despite this offset, the correlation between $v_{\rm out}$ and galaxy properties, such as SFR and $Σ_{\rm SFR}$, show similar slopes for both tracers, with Mg II absorption systematically offset by $\sim 0.4$ dex. In two galaxies, Mg II emission is also detected, yielding velocities consistent with H$α$. In one galaxy we also detect outflows in Na I D absorption and find similar velocities as Mg II in absorption, which leads to a $\sim$0.4 dex higher Na I D outflow velocities compared to those measured in emission. Our spatially resolved results are consistent with those found for galactic-scale measurements, implying the outflow relationships are similar from the sales of $\sim$1-2 kpc to global measurements. Combined with literature measurements, these results suggest that the offset in velocities is driven not by ionisation state, but rather by the systematics associated to how absorption and emission measures trace the gas density.

We study Liouville-type results for the stationary Navier--Stokes equations in $\mathbb{R}^3$. We prove that any $\dot{H}^1(\mathbb{R}^3)$ solution is trivial under an integrability condition imposed only on the radial component of the velocity, namely $u_ρ(x) \in L^p(\mathbb{R}^3)$ with $3/2 < p \leq 3$. We also establish a uniqueness result in a variable-exponent setting, where an $L^6$-type condition is required only on a bounded region, while the exponent approaches the critical value $3$ at infinity. Our analysis reveals that the rigidity of the stationary Navier--Stokes system can be driven by localized and radial integrability properties, rather than uniform global conditions.

High-order adaptive time-stepping algorithms are of significant practical value and theoretical interest for accelerating long-time fluid-flow simulations and resolving complex dynamical behaviors. While several high-order implicit-explicit schemes have been proposed in the literature, their long-time stability properties remain largely unexplored. We develop a family of long-time stable implicit-explicit Runge-Kutta (IERK) methods, up to fourth-order temporal accuracy, for the two-dimensional incompressible Navier-Stokes equations in vorticity-stream function formulation. By combining a convolution-type Hölder inequality with a damping-type multistage Grönwall inequality, we establish a unified analytical framework that proves long-time stability in both the $L^2$ and $H^1$ norms. A key component of the analysis is a mathematical-induction argument that ensures stage-wise boundedness of the vorticity in the $H^δ$ norm for some $δ>0$. To the best of our knowledge, this is the first work to establish large-time stability results for high-order IERK algorithms for the two-dimensional incompressible Navier-Stokes equations. Our IERK schemes employ stiffly accurate diagonally implicit Runge-Kutta approximations for the linear diffusive term together with explicit Runge-Kutta approximations for the nonlinear advection term. By exploiting the specific structure of the Navier-Stokes model, we derive a reduced set of order conditions-requiring only 5 and 11 conditions for the third- and fourth-order methods, respectively, in contrast to the classical 6 and 18-allowing the construction of a parameterized family of efficient schemes. These IERK methods are particularly well suited for adaptive time-stepping, as they permit significantly enlarged step sizes in actual computations.

Federated learning (FL) is attractive for cloud-edge intrusion detection because it enables collaborative training over distributed telemetry without centralizing raw logs. In production security analytics pipelines, however, only a subset of clients participates in each round, and heterogeneous bandwidth, stragglers, and dropouts can cause the server to rely on stale client information. This paper studies client participation as a timeliness-aware systems problem using Age of Information (AoI). We compare three lightweight policies for federated intrusion detection: AoI-first, utility-first, and a hybrid AoI+utility rule with a tunable trade-off parameter. Across a CIC-IDS2017 DDoS/PortScan mini subset, NSL-KDD, ToN-IoT, and a synthetic drift benchmark under clean, poisoning, and poisoning-plus-robust-aggregation settings, AoI-aware selection reduces average AoI by about 39--41% and peak AoI by about 70% relative to random sampling while keeping the per-round communication budget fixed. The hybrid policy usually preserves Macro-F1/AUC and provides an interpretable knob for balancing freshness, detection quality, and robustness, although it is not uniformly Pareto-dominant once false positive rate is included. Robustness is evaluated by combining AoI-guided selection with trimmed-mean aggregation under label-flip poisoning; the selection policy itself is not intended as a standalone Byzantine defense. The main practical message is that cloud-edge, privacy-preserving intrusion analytics can improve timeliness through a lightweight scheduling layer without changing the underlying FL participation budget.

Phase-coherent optical frequency transfer is essential for optical clock networking, relativistic geodesy, and distributed precision metrology. However, realizing coherent optical networks spanning thousands of kilometers in standard single-mode fiber (SMF) generally requires densely distributed amplifiers or repeater stations together with complex operational control, while long-term instability remains limited by thermally driven residual phase fluctuations. Here we show that hollow-core fiber (HCF) can simultaneously improve transfer instability and relax the reach limitation of long-span optical frequency transfer. Compared with SMF, HCF exhibits lower fiber-induced phase noise and shorter propagation delay, supporting improved short-term instability, while its much lower thermal sensitivity supports nearly one-order-of-magnitude better long-term instability. In addition, for long-haul HCF links, no observable stimulated Brillouin scattering induced saturation is found up to the maximum available injected power of 34 dBm, whereas the threshold of an equal-length SMF link remains only a few dBm. Together with the lower attenuation achievable in modern HCF, this enables ultra-long single-span optical frequency transfer. Using a 152 km HCF link with an average attenuation of 0.18 dB/km, we demonstrate single-span optical frequency transfer, achieving a fractional frequency instability of 7.3 x 10^-21 at 10,000 s and a fractional uncertainty of 1.8 x 10^-20. These results establish HCF as a transmission medium that simultaneously improves instability and extends single-span reach, opening a practical route toward future intercontinental optical frequency networks with ultrahigh precision.

Large language model (LLM) serving is now limited by the key-value (KV) cache. During decode, each new token rereads prior KV state, so attention becomes a bandwidth- and capacity-heavy memory task. HBM-PIM helps by moving attention closer to memory, but current stack organizations still waste resources. In practice, only hot KV blocks benefit from near-memory compute. Weights, activations, and cold KV mainly need dense storage and GPU-visible bandwidth. A uniform HBM-PIM stack makes all layers pay for PIM logic, while a dedicated-PIM design such as AttAcc recovers capacity but shrinks the HBM bandwidth left for GPU-side work. We propose TokenStack, a vertically heterogeneous HBM-PIM architecture for KV-centric LLM serving that leverages HBM4's logic-die substrate. TokenStack separates each stack into dense capacity layers and PIM-enabled compute layers, then uses the logic base die as a stack-local control point that manages cross-layer movement without host-side overhead. The base-die controller handles cross-layer DMA, layered address translation, attention-side gather/broadcast coordination, and inline quantization during migration. On top of this hardware, TokenStack uses topology-aware KV placement, workload-aware eviction, and bounded replication to keep hot KV near PIM compute while moving colder state to dense layers. Using production-derived traces across four models, completed multi-QPS runs show that TokenStack increases geometric-mean token throughput by 1.62x and SLO-compliant serving capacity by 1.70x over AttAcc, and reduces per-token energy by 30-47%.

It is known that the weighted $L^2$ projection operator exhibits approximation properties different from those of the classical $L^2$ projection, in the sense that the $L^2$ error of the weighted $L^2$ projection of an $H^1$ function generally cannot be bounded by the $H^1$ semi-norm of the function. In this paper, we establish sharper $L^2$ error estimates for the weighted $L^2$ projection of an $H^1$ function under general weight distributions. These new estimates show that the $L^2$ errors of the weighted $L^2$ projection can be controlled by the $H^1$ semi-norm of the function, except when the weight distribution is highly irregular, such as those resembling a ``checkerboard" pattern. These results can be applied to more refined analyses of domain decomposition methods and multigrid methods for certain partial differential equations with large jump coefficients.

The ground-state properties of neutron-rich nuclear clusters in the inner crust of neutron stars are investigated within the Wigner-Seitz approximation using a relativistic mean-field framework. The radial Dirac equations are solved with an asymmetric finite-difference scheme, by which the hermiticity is preserved and spurious states are eliminated. Calculations are performed for representative Wigner-Seitz cells employing TM1-based interactions with different symmetry-energy slope parameters $L$, as well as a parametrization with a larger nucleon effective mass. It is found that the binding energy per nucleon decreases systematically with increasing $L$, while a larger effective mass leads to further reduction, particularly at higher densities. Quantum shell effects, which are absent in the Thomas-Fermi approximation, give rise to oscillatory density distributions and modify neutron properties. Within the Wigner-Seitz cell, the resulting neutron root-mean-square radius and chemical potential are shown to be sensitive to both $L$ and the effective nucleon mass, underscoring their important roles in determining the microscopic structure of the neutron-star inner crust.

We show that the Bell-state phase of a polarization-entangled photon pair controls the biexciton eigenstate distribution produced by entangled two-photon absorption (ETPA) in monolayer WSe$_2$. In a frequency-nondegenerate ladder scheme, two independent valley pathways ($K$ and $K'$) share no intermediate state, so the biphoton phase sets the relative amplitude between them. Within the valley-symmetric limit this phase factorizes from the material response, and the resulting selection rule partitions the excitation among biexciton eigenstates according to the Bell-state phase $φ$. The symmetric Bell state ($φ= 0$) selectively drives bright eigenstates, while the antisymmetric state ($φ= π$) drives the exchange-dark eigenstate. No classical polarization source reproduces this $φ$-dependent eigenstate distribution. Including valley dephasing and intervalley scattering at 4~K, the phase-scan visibility exceeds $0.97$ for broadband SPDC ($T_e \sim 100$~fs) with high source purity.

We study the behavior of the elastic polymer, a model of a directed polymer in a continuous Gaussian random environment that is independent in time and correlated in space, as the dimension of the environment is taken to infinity. We give an explicit asymptotic formula for the free energy, which is given in terms of the distribution of the inner product of two sampled configurations, which we also obtain an implicit formula for. From this, we provide an explicit characterization of both the low- and high-temperature phases of this model in terms of the spatial correlation function of the environment. We find asymptotics for the wandering exponent when the spatial correlation function is either an exponential or a power-law decay. Our results show that when the correlations are either suitably weak or short ranged, the model is asymptotically diffusive. On the other hand, for suitably strong long ranged correlations, the model is asymptotically superdiffusive. Moreover, we show that this transition coincides exactly with another transition where the model goes from being one-step replica symmetry breaking to full-step replica symmetry breaking. This rigorously confirms many of the findings of Mezard and Parisi [53] in the physics literature.

Tensor parallelism (TP) in large-scale LLM inference and training introduces frequent collective operations that dominate inter-GPU communication. While in-switch computing, exemplified by NVLink SHARP (NVLS), accelerates collective operations by reducing redundant data transfer, its communication-centric design philosophy introduces the mismatch between its communication mode and the memory semantic requirement of LLM's computation kernel. Such a mismatch isolates the compute and communication phases, resulting in underutilized resources and limited overlap in multi-GPU systems. To address the limitation, we propose CAIS, the first Compute-Aware In-Switch computing framework that aligns communication modes with computation's memory semantics requirement. CAIS consists of three integral techniques: (1) compute-aware ISA and microarchitecture extension to enable compute-aware in-switch computing. (2) merging-aware TB (Thread Block) coordination to improve the temporal alignment for efficient request merging. (3) graph-level dataflow optimizer to achieve a tight cross-kernel overlap. Evaluations on LLM workloads show that CAIS achieves 1.38$\times$ average end-to-end training speedup over the SOTA NVLS-enabled solution, and 1.61$\times$ over T3, the SOTA compute-communicate overlap solutions but do not leverage NVLS, demonstrating its effectiveness in accelerating TP on multi-GPU systems.

The extremely metal-poor nature of the C-19 stream indicates that its progenitor was a primordial stellar system born in the very early Universe. Current observations show that it has a small metallicity dispersion (0.18 at the 95% confidence level), which is the signature of a globular cluster origin, while at the same time displaying an unusually large velocity dispersion ($\sim10$ km/s) typical of dwarf galaxies. To reconcile this conflicting observational evidence, previous simulations have focused on potential interactions with dark matter subhalos, which can efficiently make a cluster stream dynamically hot. In this work, we explore internal dynamical processes in star cluster formation, focusing on initial conditions shaped by gas expulsion and a top-heavy initial mass function. We find that the large observed velocity dispersion and broad stream morphology can be reproduced by a cluster that underwent severe gas expulsion and expansion during its birth phase, which is potentially a typical formation scenario of extremely metal-poor star clusters. A top-heavy IMF and binaries can also increase the velocity dispersion. The formation of C-19 may involve a combination of these effects.

We study the question of explicitly constructing variety-evasive subspace families, a pseudorandom primitive introduced by Guo (Computational Complexity 2024) that generalizes both hitting sets and lossless rank condensers. Roughly speaking, a variety-evasive subspace family $\mathcal{H}$ is a collection of subspaces such that for every algebraic variety $V$ in a fixed family $\mathcal{F}$, there is some subspace $W \in \mathcal{H}$ that is in general position with respect to $V$. We give an explicit construction of a subspace families that evade all degree-$d$ varieties in an $n$-dimensional affine or projective space. Our construction improves on the size of the variety-evasive subspace families constructed by Guo and, for varieties of degree $n^{1 + Ω(1)}$, comes within a polynomial factor of Guo's lower bound on the size of any such variety-evasive subspace family. Our variety-evasive subspace families rely on an improved construction of hitting sets for Chow forms of algebraic varieties.

Ga$_2$O$_3$/SiC heterointegration is attractive for ultra-wide-bandgap power electronics, but interfacial thermal boundary conductance (TBC) remains a major heat-removal bottleneck. Direct experimental access to intrinsic atomistic interfacial transport remains limited, particularly for ideally synthesized materials with defect-free interfacial contact. First-principles simulations are too expensive at relevant length and time scales, while empirical Molecular Dynamics (MD) potentials often lack transferability across oxide and carbide bonding environments. We develop a unified feedforward neural network potential and validate it against density-functional data, bulk phonon dispersions, and anisotropic thermal-conductivity trends in both $β$-Ga$_2$O$_3$ and SiC. Nonequilibrium simulations show that TBC decreases with transport length, increases with temperature, and is consistently higher for Ga$_2$O$_3$$(\bar{2}01)$/SiC(0001) than for Ga$_2$O$_3$(100)/SiC(0001). These trends are explained by attenuation of long-mean-free-path carriers, enhanced incoherent and anharmonic interfacial exchange within broadly unchanged spectral channels, and stronger bonding and vibrational coupling at the $(\bar{2}01)$ interface. The results show how a single transferable feedforward neural network potential can enable large-scale transport prediction and physics-grounded mechanistic understanding of thermal boundary conductance. Code for NEP training and simulation workflows is available at the project repository https://github.com/knowhow07/TBC_Ga2O3_SiC.git

The rigorous stability analysis of high-order implicit-explicit multistep (IEMS) methods for nonlinear parabolic equations by using discrete energy arguments is a long standing open issue due to their non-A-stable property. A novel semi-generating function approach combined with the global discrete energy analysis is suggested to the stability and convergence analysis of general IEMS methods for nonlinear parabolic equations. Inspired from the Grenander-Szegö theorem for the Toeplitz matrix, the semi-generating function approach is used to handle the three groups of discrete coefficients via three complex rational polynomials on the unit circle. A unified theoretical framework is then presented to establish the unconditional stability of IEMS methods if the minimum eigenvalue of composite convolution kernels for the implicit part is properly large and the spectral norm bound of composite convolution kernels for the explicit part is properly small. An indicator, called implicit-explicit controllability intensity, is then introduced to evaluate the degree of controllability of implicit part over explicit part. Some of existing IEMS methods, up to the fifth-order time accuracy, are revisited and compared by computing the associated implicit-explicit controllability intensities such that one can choose certain IEMS method or proper parameter to maintain the unconditional stability for a specific nonlinear parabolic model. We also propose a new parameterized class of IEMS methods, up to the eighth-order time accuracy, which satisfy the priori settings of our theory and have a large value of the implicit-explicit controllability intensity by choosing proper parameter so that they would be well suited for a wide class of nonlinear parabolic problems.

We study the algorithmic tractability of finding large independent sets in dense random hypergraphs. In the sparse regime, much of the natural algorithms can be formulated within either the local or the low-degree polynomial (LDP) framework, and a rich literature has subsequently identified nearly sharp algorithmic thresholds within these classes by exploiting their stability. In the dense setting, however, the algorithmic paradigms are fundamentally different: they are online and thus need not be stable. Perhaps more crucially, even for the classical Erdős-Rényi random graph $G(n,p)$, LDPs are conjectured to fail in the 'easy' regime accessible to online algorithms, thereby challenging their viability for dense models. Our focus is on two models: (i) finding large independent sets in dense $r$-uniform Erdős-Rényi hypergraphs, and (ii) the more challenging problem of finding large $γ$-balanced independent sets in dense $r$-uniform $r$-partite hypergraphs, where the $i$-th coordinate of $γ\in\mathbb{Q}^r$ specifies the proportion of vertices from $V_i$ in the independent set. For both models, we pinpoint the size of the largest independent set and design online algorithms that achieve a multiplicative approximation factor of $r^{1/(r-1)}$ in the uniform and $(\max_i γ_i)^{-1/(r-1)}$ in the $r$-partite model. Furthermore, we establish matching algorithmic lower bounds, showing that these computational gaps are sharp: no online algorithms can breach these gaps.

Tunneling ionization in static or slowly varying electric fields is a cornerstone of strong-field physics and provides the entry point for semiclassical descriptions of above-threshold ionization and high-harmonic generation. In conventional quantum mechanics, the Perelomov--Popov--Terent'ev (PPT) theory and its Ammosov--Delone--Krainov (ADK) form yield an ionization rate whose defining feature is an exponential dependence governed by an under-barrier (imaginary-time) action. Here we develop an analytical ADK-like tunneling model within \emph{space-fractional} quantum mechanics, where the quadratic kinetic energy is replaced by the Riesz fractional Laplacian of order $1<α\le2$. Working in a static electric field in the length gauge, we derive a closed-form tunneling exponent for a triangular exit barrier. The fractional kinetic operator deforms the conventional $I_p^{3/2}$ scaling to $I_p^{1+1/α}$ and introduces a characteristic $\sin(π/α)$ factor encoding the complex-phase structure associated with nonlocal dispersion. We position this benchmark relative to prior tunneling studies in fractional quantum mechanics (primarily scattering through model barriers and fractal potentials) and provide a validation protocol for testing the exponent in time-dependent simulations of the fractional Schrödinger equation under a constant field. The result establishes a transparent reference for static-field ionization in nonlocal quantum dynamics and a baseline for strong-field approaches extensions.

Constacyclic codes over finite fields are of theoretical importance as they are closely related to a number of areas of mathematics such as algebra, algebraic geometry, graph theory, combinatorial designs and number theory. However, the study of constacyclic codes in this context remains limited compared to classical cyclic codes. This paper provides two infinite families of $λ$-constacyclic codes over $\mathbb{F}_{q^2}$ that support infinite families of 3-designs, which generalize the results in [IEEE Trans. Inf. Theory 69(4): 2341-2354, 2023]. The parameters and weight distributions are determined completely. Besides, we study their subfield subcodes and applications on constructing entanglement-assisted quantum error-correcting codes (EAQECCs) and locally recoverable codes (LRCs). It is worthy to mention that two classes of maximal entanglement EAQECCs with a negative or a high positive net rate are derived. Moreover, two classes of distance-optimal and dimension-optimal LRCs are also obtained.

The stoichiometric antiferromagnetic insulator EuTiO$_3$ is proximate to a ferroelectric phase. Whereas cation substitution has been used as a tuning parameter to introduce charge carriers and manipulate the magnetism, the effects of oxygen-vacancy doping have been less explored. Here we report a detailed study of the charge transport and magnetic properties of metallic, oxygen-vacancy-doped EuTiO$_{3-δ}$. Using CaH$_2$ as an oxygen getter to achieve a higher carrier concentration than previously reported, we find that the phase diagram of the oxygen-vacancy-doped system is distinct from that obtained via cation doping. In particular, we uncover a change from antiferromagnetic to ferromagnetic order in the metallic state, with a maximum Curie temperature of TC $\approx$ 11 K at the highest carrier concentration of n $\approx$ 10$^{21}$ cm$^{-3}$. These findings are supported by density functional theory calculations, which indicate a significant change in the nearest-neighbor magnetic exchange constant with increasing electron doping. We also present x-ray diffuse scattering and complementary first-principles results that reveal that, similar to the prominent incipient ferroelectric perovskite SrTiO$_3$, the data for EuTiO$_3$ are consistent with thermal diffuse scattering and with the absence of quasi-elastic contributions. Finally, we report specific heat measurements that confirm the magnetic transition temperatures deduced from magnetization measurements and corroborate the lattice dynamics picture inferred from the diffuse scattering data.

We develop and analyze a class of matrix-valued spherical-convolution kernels stemming from scaled zonal functions on $\mathbb{S}^2,$ the unit sphere embedded in $\mathbb{R}^3$. The construct of these kernels utilizes the Legendre differential equation and requires less stringent regularity conditions on the original zonal kernels. The induced integral operators are simple Fourier-Legendre multipliers that not only deliver optimal Sobolev error estimates (in terms of the scaling parameter) but also yield natural Helmholtz-Hodge decompositions on the $L_2$-tangential vector fields on $\mathbb{S}^2$. Via discretization of the underlying convolution integrals, we harvest a family of vector-valued quasi-interpolants that accomplish our approximation goal in the divergence/curl-free vector field. The quasi-interpolation algorithm is robust against noisy data. The implementation process is adaptive to human-improvision, involving neither evaluating the convolution integrals nor solving systems of linear equations. The computational efficiency and executory robustness of the quasi-interpolation algorithm stand in sharp contrast to the existing kernel-based vector field interpolation method.

Periodically driven quantum systems, known as Floquet systems, provide a versatile platform for engineering novel topological phases absent in static settings. However, dynamically characterizing these non-equilibrium topological invariants remains a challenge. Here, we develop a Floquet perturbation theory in the extended Hilbert space to analytically describe the center-of-mass (CoM) dynamics of a wave packet. When applied to the driven Su-Schrieffer-Heeger model, our theory reveals that the CoM exhibits multi-frequency Zitterbewegung oscillations, whose spectral composition and phase are directly tied to the system's Floquet band structure. Crucially, we find that band inversions at topological phase transitions imprint distinct signatures in the CoM dynamics, including the emergence of low-frequency modes and phase shifts of the oscillatory trajectory. These dynamical signatures offer a practical protocol for detecting Floquet topological invariants, which we demonstrate for both high-frequency and strongly driven regimes. Our work establishes CoM dynamics as a simple and experimentally accessible probe for exploring topological phase transitions in Floquet systems.