New binning schemes for the Dalitz plots of $D \rightarrow K^0_{\mathrm S}\pi^+\pi^-$ and $D \rightarrow K^0_{\mathrm S} K^+ K^-$ decays are presented. These are determined using a new figure of merit for optimisation that better represents the sensitivity to $\gamma$ than previous metrics. Further augmentation comes from including consideration of degeneracy with other physics parameters determined simultaneously and a more accurate treatment of relevant backgrounds. The overall expected improvement in the precision of $\gamma$ measurements using $B^\pm\to DK^\pm$ decays, followed by $D \rightarrow K^0_{\mathrm S}\pi^+\pi^-$, is estimated at around 5$\%$. In addition, the first dedicated optimisation is performed for the observables relevant to charm mixing, $x_{CP}$ and $\Delta x$, in $D \rightarrow K^0_{\mathrm S}\pi^+\pi^-$ decays. This procedure accounts for both statistical sensitivity and the migration of events between bins due to detector resolution. The resulting binning scheme leads to an estimated gain of approximately 20$\%$ in statistical sensitivity, while maintaining the bias due to bin migration at a level comparable to the currently used scheme.

Comparing causal relationships across populations is essential in many scientific domains. This paper studies the problem of inferring a difference graph between two environments and proposes a causal discovery method for linear structural causal models based on equality tests of regression coefficients. We show that invariance of regression coefficients is governed by graphical conditions that go beyond standard d-separation. Therefore, we introduce diff-separation, a graphical criterion that characterizes when a conditioning set blocks all paths capable of inducing differences in regression coefficients across environments. Building on this criterion, we introduce a corresponding diff-faithfulness assumption, linking graphical diff-separation statements to equality constraints on regression coefficients. Finally, we propose LDiffPC, a PC-style algorithm that uses equality tests of regression coefficients to recover the differences from multi-environment data.

This paper studies an inverse problem in nonlinear optical encryption. We examine chosen-plaintext attacks (CPA) on a nonlinear optical encryption strategy that integrates double random phase encryption (DRPE) into a nonlinear optical propagation model to enhance the security of the combined system. We first demonstrate that the system's phase information can be decoded from carefully designed differential CPA data. We then demonstrate that the strength of the optical device's nonlinearity can also be recovered from CPA data, indicating that including this parameter as an additional security key does not enhance protection against CPA attacks, although numerical simulations show that strong nonlinearity still poses significant challenges for CPA attacks. Finally, we provide a stability analysis to demonstrate that small errors in decoded security keys result in only small errors in the decrypted text, even though the encryption process is nonlinear.

We employ a model-independent Effective Field Theory (EFT) to analyze the possibility of a strong First-Order Phase Transition (FOPT) in extensions Beyond the Standard Model (BSM). We find that sizable deviations in the Higgs cubic and quartic interactions that are still allowed experimentally could lead to a strong FOPT, whereas the Higgs interactions to the top quark yield a weak FOPT. We also study the Gravitational Wave (GW) power spectra corresponding to the strong FOPT and find that they could be detectable in future experiments. In particular, we find that deformations of the Higgs quartic coupling have the dominant impact on the FOPT, with a GW signal that could be probed by a number of future experiments, such as LISA, BBO, and DECIGO. We also study the magnetic field produced by the corresponding FOPT and find that it could explain the primordial magnetic field puzzle. We find that for the size of deformations that could induce a strong FOPT, a scale of NP can be as low as $\sim 4\text{--}5~\text{TeV}$ for deformations in the Higgs cubic coupling, and $\sim 9\text{--}11~\text{TeV}$ for deformations in the Higgs quartic coupling. This highlights the synergy between collider searches and GW experiments in probing the Higgs couplings, specifically the Higgs quartic coupling.

The BICEP series of experiments has been observing CMB polarisation from the South Pole for over 20 years, with the goal of constraining inflationary gravitational waves. The upcoming data release, using data taken through 2024, is forecasted to constrain the tensor-to-scalar ratio $r$ at the level of $\sigma(r) \sim 0.005$ (including delensing), with the longer-term goal of reaching $\sigma(r) \sim 0.001$ by 2034. As the survey sensitivity increases, it is crucial to control instrumental systematics to unprecedented levels, and our goal is to limit dominant sources of systematics to 20% of $\sigma(r)$ or lower. Achieving this requires careful instrumental characterisation and dedicated end-to-end studies to evaluate the impact of each systematic on cosmological parameters. We first present the BICEP calibration program, which characterises the optical, spectral, and polarisation response of the receivers. We then describe the analysis strategies implemented to identify and mitigate systematic contamination, and we detail simulations used to evaluate the impact of residual effects. We focus in particular on beam systematics, the dominant source of systematics for BICEP receivers. Finally, we report preliminary estimates of the expected level of systematic contamination for upcoming BICEP results, and we discuss approaches to evaluate and mitigate instrumental systematics for future surveys.

We study inverse boundary value problems for the nonlinear Monge--Ampère equation \[ \det D^2u=a(x,u,\nabla u) \] in a bounded domain. We introduce a nonlinear Cauchy data set and investigate the recovery of the nonlinearity from boundary measurements. Linearizing around a strictly convex background solution, we establish local well-posedness, smooth dependence on boundary data, and a higher-order linearization framework. We show that the first variation of the nonlinear Cauchy data is governed by a linear elliptic operator whose principal coefficient is the cofactor matrix of the background Hessian. An Alessandrini-type identity then yields a reduction principle from the nonlinear inverse problem to an anisotropic Calderón-type inverse problem. As a consequence, under suitable uniqueness assumptions for the associated linear problem, the nonlinear Cauchy data determine the first-order derivatives of the nonlinearity along the background jet. Higher-order linearization identities provide recovery of higher derivatives and, under suitable density assumptions, determine the full Taylor expansion of the nonlinearity along the background solution. We also discuss applications to semilinear equations and nonlinearities arising in optimal transport.

Energetic particle (EP) transport stiffness plays a crucial role in determining EP confinement in tokamak plasmas. This work investigates the impact of nonlinear thermal plasma dynamics on EP transport using global gyrokinetic simulations with the TRIMEG code. Simulations are performed for the ITPA toroidal Alfv'en eigenmode (TAE) benchmark, and extended to beta-induced (BAE) and reversed shear Alfven eigenmodes (RSAE). Two models are compared: a fully nonlinear treatment of all species and a reduced model with only nonlinear EP. For TAE simulations, while linear properties of the instability are identical in both cases, significant differences arise in the nonlinear regime. Including thermal plasma nonlinearity reduces the saturation level following an overshoot phase and modifies the radial mode structure, including mode broadening and poloidal harmonic splitting. These changes significantly affect EP transport. Specifically, the dependence of EP flux on the EP drive becomes weaker when nonlinear thermal plasma dynamics are included. The saturation level changes from an approximately quadratic scaling to a weaker, nearly linear dependence, reducing the scaling of EP flux with the EP gradient from quartic to quadratic. Zonal flow generation is observed but plays a minor role in regulating the instability. Without accounting for thermal nonlinearity, the EP flux can be overestimated by an order of magnitude. Simulations of BAEs and RSAEs demonstrate similar effects of thermal plasma nonlinearity on mitigating the saturation level. These results demonstrate that nonlinear thermal plasma effects provide an important feedback mechanism that reduces EP transport stiffness and regulates Alfvenic mode saturation, which is essential for predictive modeling of EP confinement in burning plasmas such as ITER.

Our aim is to investigate the existence of local maximizers for the singular Trudinger-Moser functional $$ F_{\alpha}(u):= \int_{\Omega} \frac{e^{\alpha u^2}-1}{|x|^{a}} \mathrm{d}x,\;\;\; u \in W_0^{1,2}(\Omega), $$ restricted to the manifold $ \Sigma=:\left\{u\in W_0^{1,2}(\Omega):\|\nabla u\|_2=1\right\},$ where $a\in [0, 2)$, $\alpha\ge 0$ and $\Omega$ denotes a smooth bounded domain $\mathbb{R}^2$ containing the origin. Adimurthi and Sandeep (Nonlinear. Differ. Equ. Appl. \textbf{13}, 2007) showed the following singular Trudinger-Moser type estimate \begin{equation}\nonumber \sup_{u \in W_0^{1,2}(\Omega), \, \|\nabla u\|_{L^2} \le 1} F_\alpha(u)<\infty\;\;\;\mbox{iff}\;\;\alpha \leq \alpha_a:=2\pi(2-a). \end{equation} In particular, the functional $F_{\alpha}$ is bounded on $ \Sigma$ whenever $\alpha \le \alpha_a$. In addition, Csató and Roy (Calc. Var. Partial Differ. Equ., \textbf{54}, 2015) were able to ensure the existence of maximizers for $F_{\alpha}$ on $\Sigma$ when $\alpha\le \alpha_a$. In the supercritical regime $\alpha > \alpha_a$, the functional $F_{\alpha}$ becomes unbounded on $\Sigma$. Nevertheless, we prove that $F_{\alpha}$ still possesses local maximizers on $\Sigma$ beyond the critical threshold, at least for $\alpha > \alpha_a$ sufficiently close to $\alpha_a$. Our approach relies on a variational analysis near the set of maximizers associated with the critical parameter $\alpha_a$, together with a suitable local compactness argument. Our result improves and complements related findings due to Struwe (Ann. Inst. Henri Poincaré, Anal. Non Linéaire, \textbf{5}, 1988).

The baryonic Tully-Fisher relation (BTFR), a well-established galaxy scaling relation linking the dynamical mass of rotation-supported galaxies through their maximum circular velocity to the baryonic luminous mass, has emerged over the decades as a fundamental scaling relation and as a robust calibrated distance indicator, thereby providing a robust benchmark to test galaxy formation and evolution theories as well as an independent probe of the expansion of the Universe. In this paper, we show for the first time that the BTFR is also simultaneously directly sensitive to the cosmological parameters $\Omega_m$ and $\sigma_8$, the astrophysical feedback from supernovae (SNe) and active galactic nuclei (AGN), and the mass of warm dark matter particles $M_{\rm wdm}$, and can therefore be used as novel, direct probe of cosmology and fundamental physics. We perform simulation-based inference on the large DREAMS cosmological magneto-hydrodynamic simulations suite and train deep neural networks in the form of normalizing flows to estimate the posterior distributions of $\Omega_m$, $\sigma_8$, $M_{\rm wdm}$ and the three astrophysical free parameters, given a BTFR measurement. Our framework is able to recover unbiased values for $\Omega_m$ and $\sigma_8$, with subpercent deviations accuracy and a $\sim 2.6\%$ and $\sim 3.9\%$ median precision, respectively, to capture the warm dark matter particle mass $M_{\rm wdm}$ within a $\sim 30-35\%$ precision, as well as to constrain the SN feedback parameters (but not the one regulating AGN feedback). We conclude that, beyond its usage as a distance indicator and to constrain the baryon cycle and the feedback mechanisms shaping galaxy formation and evolution, the BTFR constitutes a direct independent probe of cosmology and fundamental physics and opens new promising avenues, to be explored with the future Square Kilometer Array.

This paper proposes an integrated framework for machine learning-guided quota optimization applied to multi-round sorority recruitment, a small two-sided market where approximately 100 potential new members (PNMs) are matched to three chapters through a structured process governed by the Release Figure Methodology (RFM). Our framework combines a Random Forest classifier trained on historical registration data to generate PNM-chapter compatibility scores, integer linear programs for Round~2 and Round~3 invitation quota optimization balancing fairness, coverage, and efficiency objectives, and a Deferred Acceptance algorithm for final matching. Applied to five years of de-identified recruitment data from a small Midwestern university, and working with only 282 matched training pairs, the compatibility model achieves a cross-validated ROC-AUC of 0.5822, reflecting the inherent difficulty of predicting social compatibility from pre-recruitment registration data in a data-limited setting. Because fairness and coverage constraints dominate quota allocation under noisy scores, the framework is designed to degrade gracefully when ML signal is weak. Optimized quotas closely align with actual coordinator decisions for active chapters, and the Deferred Acceptance algorithm replicates actual 2025 recruitment outcomes with a 96.4% individual-level agreement rate and a 100% match rate across 56 PNMs. An interactive web application implementing the framework is made available to recruitment coordinators. These results support the viability of data-driven approaches to small-market matching with broader applicability to other constrained two-sided markets.

We say a word in a free group is ``partially positive'' if there is at least one generator which appears in the word, but does not appear with negative exponent. We show that one-relator groups with a partially positive relator are residually solvable. This extends a theorem of Baumslag showing one-relator groups with positive relators are residually solvable.

The invariant two-point function of a scalar field in de Sitter spacetime can be expressed in terms of Legendre functions via Lorentzian harmonic analysis. Using this structure together with the Mehler--Fock transform, we obtain an integral representation of the completed Riemann $\xi$-function in which the Legendre kernel appears naturally. Motivated by this correspondence, we introduce a retarded propagator whose spectral weight is given by the $\xi$-function and analyze it within the framework of Krein space quantization, where sign-indefinite spectral measures are admissible. This construction yields a geometric and spectral interpretation of the $\xi$-function restricted to the critical line and relates the asymptotic spacing of its zeros to a mass--time scaling in de Sitter geometry. The results provide a novel interpretive framework linking de Sitter quantum field theory, harmonic analysis, and analytic number theory.

Electron transfer through chiral structures can exhibit spin asymmetry, known as the chiral-induced spin selectivity effect, whose microscopic origin remains an open question. While path-interference within the chiral moiety has been proposed as a key mechanism, its experimental validation requires precise and versatile tunability of system parameters. Here we implement a programmable quantum simulation of spin-dependent electron transfer in a donor--chiral-bridge--acceptor model using a trapped ion. The bridge is encoded in internal states of the ion with tunable nearest- and next-nearest-neighbor couplings, while donor and acceptor states are coupled via a spectator bosonic motional mode. We observe spin-dependent interference within the bridge, and further reveal spin-dependence in donor-to-acceptor transfer dynamics, controlled by amplitude and phase of the coupling parameter. Our results identify interference among spin-dependent pathways as a microscopic origin of spin-dependent transfer, and open a route toward quantum simulations of complex chiral lattices with multi-level and bosonic degrees of freedom.

Compact graphs are graphs for which the fractional automorphism polytope has no genuinely fractional vertices. This paper proposes a quantum analogue of this idea by evaluating the fundamental magic unitary of the quantum automorphism group on states, which we show to produce a closed convex set of doubly stochastic matrices sitting between the classical automorphism polytope and the full fractional automorphism polytope. Our main result is that the natural quantum analogue of compactness is classical, that is, a quantum compact graph is classically compact. We also relate this set to the quantum orbital algebra and obtain a hierarchy of classical and quantum compactness pseudo notions. The framework recovers familiar consequences of compactness through commutants and suggests quantum analogues of generous transitivity and distance-transitivity. We also isolate examples and open problems indicating where quantum symmetries may strictly refine the classical compactness theory.

Resolution spaces play a central role in constructing solutions for nonlinear partial differential equations. One of the main goals in the area of nonlinear dispersive PDEs has been to construct effective resolution spaces which capture the known bilinear restrictions estimates for free solutions. In this paper we propose a new structure for the Schrödinger equation which effectively replicates the classical bilinear $L^2_{t,x}$ estimate. In addition, the new structure has the property that its "dual" is an effective candidate for a space for the forcing in the linear inhomogeneous Schrödinger equation, a feature that has been elusive so far in the literature. As an application, we show how these structures can recover the known global well-posedness results for derivative NLS with null structure, with Schrödinger Maps being one such model.

We investigate several astrophysical motivated properties of a four-dimensional black hole solution in the framework of $f(R)$ gravity. The model is characterized by a single additional parameter, $\alpha$, which encodes nontrivial deviations from General Relativity. Using this black hole spacetime as the background geometry, we analyze: (i) the quasinormal modes of massless scalar perturbations (employing several complementary methods), and (ii) the greybody factors associated to the propagation of massless test scalar fields. Regarding quasinormal modes, we study the response of black holes to massless scalar perturbations using three independent approaches: the well-established sixth-order WKB semi-analytic method, analytic expressions, and the analytical expression in the eikonal limit. We examine the behavior of both the real and imaginary parts of the quasinormal frequencies as functions of the parameter $\alpha$, the overtone number $n$, and the multipole number $\ell$. In addition to that, we compute the greybody factors within the WKB approximation for various combinations of the relevant parameters. In particular, we focus on the propagation of test massless scalar fields and investigate how the greybody factors depend on the multipole number $\ell$, and the free parameter $\alpha$. The impact of the aforementioned parameters on the absorption cross-section and the reflection and transmission coefficients is studied in detail.

We continue the study of locally finite-dimensional traces introduced in our earlier work. We give new characterizations of LFD traces in terms of finite-rank projections in irreducible representations and in the socle of the bidual. We show that, for separable nowhere scattered \(C^*\)-algebras, the set of all LFD traces is convex. We also prove that quasidiagonal traces form a face of the trace simplex and record applications to strongly self-absorbing \(C^*\)-algebras. We construct an exact tracially AF algebra not KK-equivalent to any nuclear \(C^*\)-algebra.

Earth observation (EO) is essential to understanding the Earth system, enabling the transformation of planetary properties into measurable variables that can be analysed, compared, and modelled. In recent decades, EO capabilities have grown rapidly, accompanied by an even faster expansion in the number and variety of available EO instruments. Today, EO includes instruments deployed on satellites, airborne platforms, and terrestrial or in-situ systems. However, despite this proliferation of instruments, users often lack a single, reliable source describing their existence and key characteristics. Although existing data catalogues have substantially improved dataset discovery, they primarily describe data products rather than providing persistent, curated metadata about the instruments that produced them. Here we present Awesome Earth Observation Instruments, an open, standardized, and community-oriented registry providing machine-readable metadata for EO instruments. The catalogue is hosted on GitHub and allows contributors to submit instrument metadata following a common schema. The schema combines a lightweight core with modular extensions covering spectral, geometric, and data access-related metadata, enabling both standardization and flexibility across diverse EO systems. All submissions undergo automated schema validation and human review. Because the schema is open, versioned, and extensible, the catalogue can continuously evolve as new instruments and metadata requirements emerge. This facilitates the discovery, interpretation, and analysis of EO data in light of instrument characteristics. To support programmatic access and interoperability, we further envisage an API for integration within common EO analysis environments. The catalogue is openly available at this https URL.

Given a finite group $G$ acting orientably on a surface $S$ of genus $\sigma \geq 2$, the group $G$ acts faithfully on the homology group $H_{1}(S;\mathbb{Z})$, preserving the symplectic intersection form. The action on $S$ and the homology is determined by a \emph{generating vector}, a tuple of elements of $G$, generating $G$ and satisfying certain properties. In this note we show how to compute the homology representation, using the generating vector, when $S/G$ has genus 0 and the genus is suitably low. A $2\sigma \times 2\sigma$ representing matrix can be determined for any element in the group, usually for a small set of generators. The matrices are computed with respect to an auto-generated basis for the cellular homology of $S$, using a regular $CW$ structure on $S$, derived from the $G$ action. We demonstrate the application of these results by computing \emph{invariant theta characteristics} of the Riemann surfaces $S$ with the algorithm implemented using Sage.

The capacity of dynamical systems to resist and recover from perturbations, broadly referred to as resilience, is commonly expressed by two complementary quantities: ecological and engineering resilience. As many complex systems exhibit critical transitions, or tipping points, understanding how these resiliences jointly change nearby them is central to characterising and anticipating such shifts. Here, we develop a theoretical framework that clarifies this for bifurcation-induced tipping, i.e., critical transitions triggered by the crossing of a local bifurcation. Using normal form theory, we derive explicit scaling laws for commonly used resilience metrics as functions of the distance to the bifurcation point in parameter space, and show that these extend to general models up to a scaling factor. They are particularly relevant for detecting tipping, where the relative behaviour of metrics matters more than their absolute values. The rates at which metrics decrease as the bifurcation is approached depend on both the type of bifurcation and the metric considered. Furthermore, our results show that, sufficiently close to a local bifurcation, resiliences are intrinsically linked. Our predictions, which replace previously proposed scalings based on heuristic arguments, are validated for three representative models covering all commonly encountered local bifurcations in one-dimensional systems.

We study the dichotomy between compact satellite and ring formation in proto-planetary disks. Specifically, we examine the behavior of a model system of $N$ identical particles locked into circular, gravitationally-bound orbits around a central body. We treat interactions as dominated by inter-particle collisions, and extract an effective two-particle interaction function based on both theory and simulations. We then demonstrate that the expected dynamics are equivalent to a variant of the Kuramoto model, which undergoes a phase transition as parameters vary. This offers a novel potential explanation for the transition between formation of rings versus moons.

We present a Bayesian calibration layer for slopsquat detectors -- those that flag hallucinated package imports in code produced by large language models (LLMs). Where existing pipelines emit binary decisions (flag / do-not-flag), our layer emits a Beta-posterior probability per detection, derived from a 3-category epistemic taxonomy that explicitly classifies each prior as empirically calibrated, constructively argued, or engineering-judgement-traced. Beyond the primary 200/404 registry channel, the calibrated layer exploits PyPI metadata signals -- package age, release count, author descriptor, summary -- to surface registered-but-suspicious packages that a binary registry detector misses, which is the realistic post-LLM-emission attacker regime. The resulting risk-aware primitive is directly consumable by downstream CI gates and supports principled threshold decisions across detection rules. We evaluate the calibration on a merged corpus of 1,734 Python snippets -- a stratified 189-prompt BigCodeBench slice plus a 100-prompt niche-library stress-test set, generated across a six-model panel spanning four cloud models (Claude-Sonnet-4.6, Mistral-Large, DeepSeek-v4-pro, DeepSeek-R1) and two local open-weight code models (Mistral Codestral, Meta CodeLlama). Against a re-implemented binary baseline inspired by Mahmud et al. -- which shares its registry oracle with our ground truth and therefore serves as a degenerate upper bound rather than a genuine competitor -- the calibrated layer reproduces the strict-registry detections and introduces well-calibrated additional flags on the metadata channel. We assess detector asymmetry with a McNemar paired test and calibration with both a flagged-subset Expected Calibration Error and a strictly proper full-corpus Brier score.

Metastable states and their minimum energy pathways (MEPs) are central to understanding transformations and phase stability in complex materials, yet mapping transition pathways between competing states remains computationally demanding and experimentally challenging. Here, we introduce a hybrid solid-state nudged elastic band (SSNEB) framework that integrates two pretrained machine learning models, EquiformerV2 (eqV2) and the equivariant Smooth Energy Network (eSEN), with DFT for energy, force, and stress evaluations. Applied to three solid-state systems, CsPbI$_3$, GaN, and TiO$_2$, our framework achieves up to a 7-fold speedup while converging to the same pathways predicted by first-principles calculations. Moreover, the hybrid SSNEB framework enables systematic benchmarking of existing ML models, providing both efficiency and reliability for predicting MEPs across various materials.

The Fermi paradox is sharpened, not weakened, by plausible extrapolations of artificial intelligence, autonomous robotics, in-situ resource utilization, orbital manufacturing, space-based computing, and uncrewed interstellar probes. Once a civilization can design, launch, and maintain autonomous industrial systems beyond its home planet, interstellar expansion no longer requires biological starships or a human-like empire. It can proceed through low-mass probes, robotic seed factories, archival payloads, biological repositories, local computation, and slow replication across nearby stellar systems. This paper proposes the quiet expansion filter: old, stable civilizations that reached autonomous AI-cosmoindustry probably did not arise in the part of the Galaxy capable of reaching the Solar System, because after that threshold interstellar expansion becomes too useful, inexpensive, and rational for all civilizations to refuse; however, successful expansion would be machine-mediated, distributed, low-noise, and partly biological rather than Kardashev-like or imperial. Order-of-magnitude estimates indicate that a single post-threshold civilization could saturate its reachable stellar neighborhood within ~10^7 yr -- less than 0.1% of Galactic age -- at modest energy cost per probe. The novelty of the proposal lies not in any new mechanism but in extending the AI-filter literature toward post-threshold observability predictions. The hypothesis predicts that successful advanced expansion, if present, is more likely to appear as weak artifacts, local probes, small-scale resource processing, exoplanetary anomaly clusters, or techno-biological preservation systems than as galaxy-scale energy harvesting.

We prove that a sequence of constant scalar curvature (CSC) metrics which is collapsing with bounded curvature to a manifold $(X,g_{\infty})$ can be perturbed to a sequence of $\mathcal{N}$-invariant collapsing CSC metrics, under a natural assumption involving the eigenvalues of the drift Laplacian on $(X,g_{\infty})$. This answers a special case of a question of Cheeger-Fukaya-Gromov. We also give some natural conditions on the limiting metric-measure space under which the eigenvalue assumption is automatically satisfied.

We investigate the interactions of the recently confirmed exotic hadron $G(3900)$ with the hadronic medium composed of light mesons produced in heavy-ion collisions. Using an effective Lagrangian approach, we compute both vacuum and thermally averaged cross sections for processes such as $G(3900) + \pi \to D^{(\ast)} + \bar{D}^{(\ast)}$ and their inverse reactions. Following recent proposals, the $G(3900)$ is interpreted as a $P$-wave $D\bar{D}^*/D^*\bar{D}$ molecular state. The resulting thermally averaged cross sections are employed in a rate equation to determine the time evolution of the $G(3900)$ multiplicity, with initial conditions provided by statistical and coalescence hadronization models. We compute the number of $G(3900)$'s produced in central Pb-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV and compare it with the number of produced $Z_c(3900)$'s. We obtain a considerably smaller final yield of $G(3900)$ compared to $Z_c(3900)$.

Quantum Chromodynamics (QCD) provides a remarkable realization of how quantum effects reshape the symmetries of a classical field theory. The axial anomaly links chirality to gauge-field topology and underlies the resolution of the $U(1)_A$ problem, while the trace anomaly generates the intrinsic QCD scale through dimensional transmutation and accounts for most of the mass of hadrons and hence visible matter. Together, these quantum effects reveal the central role of gluonic dynamics and vacuum structure in strong-interaction physics. In this review, we discuss the theoretical foundations and physical consequences of anomalous symmetry breaking in QCD. We examine anomalous Ward identities, the topological structure of gauge fields, instantons, topological susceptibility, and the realization of chiral and scale symmetries in the QCD vacuum. We review their role in the generation of hadron masses, the $\eta'$ mass, and the resolution of the $U(1)_A$ problem, and discuss their manifestations in polarized deep inelastic scattering, nucleon spin structure, and modern studies of hadron structure. Particular emphasis is placed on recent developments connecting vacuum topology, the flavor-singlet axial charge, topological screening, and the proton spin problem. Our aim is to provide a unified perspective on how quantum anomalies connect vacuum structure, hadron properties, and partonic observables, bridging nonperturbative dynamics and perturbative QCD.

We introduce a variant of the Cox ring using $\mathbb{Q}$-Cartier divisors and use this to remedy various deficiencies of nonsimplicial toric varieties. Our main applications are: Cox's ideal-variety correspondence, an explicit classification of subschemes and sheaves in terms of multigraded modules, an associated toric Deligne-Mumford stack, and a stacky extension of the GKZ/Mori theory of toric varieties.

The weak van der Waals (vdW) force in layered chalcogenide materials has enabled the growth of ternary chalcogenide layers using unconventional approaches. Here, we report the molecular beam epitaxy (MBE) growth of Mn2In2Se5, a spin glass material with high level of magnetic frustration, through the heterointegration of MnSe on In2Se3. Directly depositing {\alpha}-MnSe on the vdW In2Se3 layers results in Mn intercalation, transforming the In2Se3 layer into Mn2In2Se5. Large growth windows, including substrate temperatures from 250-450 °C and Se:Mn flux ratio of 1.1-3.1, have been identified for the intercalation process. With an optimized MnSe deposition time, smooth, single-crystalline, and (0001)-oriented Mn2In2Se5 layers with a root-mean-square (RMS) roughness of 1.5 nm can be synthesized. Further extending the MnSe deposition time results in the growth of uniform rock-salt structured {\alpha}-MnSe (111) layers with a thickness of up to 8 nm and a narrow full-width-at-half-maximum (FWHM) of 0.35° in MnSe (222) XRD rocking curves. This report presents a unique approach for the growths of uniform and single-crystalline Mn2In2Se5 vdW layers using MBE, and potentially opens a pathway for synthesis of ternary vdW chalcogenides by intercalation of new atomic species in binary vdW chalcogenides.

We prove a degree-four Sum-of-Squares lower bound for the standard vector-coordinate formulations of mutually unbiased bases. For every dimension $d$ and every proposed number $m$ of bases, we construct a degree-four pseudoexpectation satisfying the orthonormality constraints and the cross-unbiasedness constraints in the quartic equality formulation. The construction is expectation over $m$ independent Haar-random orthonormal bases. We also prove that the same pseudoexpectation satisfies the degree-four localizing constraints for the natural $2\times 2$ Hermitian semidefinite formulation of the cross-coherence inequalities. Consequently, degree-four vector-coordinate SoS cannot refute the existence of $m$ mutually unbiased bases, even when $m>d+1$. In particular, under the two vector-coordinate encodings explicitly described in Randomstrasse101 Open Problem 23, degree-four SoS cannot prove that seven mutually unbiased bases do not exist in $\mathbb C^6$. We contrast this with a centered projector-coordinate Gram formulation, where degree-four SoS already recovers the elementary upper bound $m\le d+1$, giving a simple separation between vector-coordinate and projector-coordinate degree-four relaxations.