Motivated by color coherence and decoherence effects in the QCD medium, we propose a theoretical framework that combines vacuum-like emissions and medium-induced radiation to study jet quenching and its dependence on jet cone sizes and substructure. In our approach, a jet produced at a hard scale $Q$ first undergoes vacuum-like evolution, as described by the well-established generating-function method in the double logarithmic approximation. These vacuum-like emissions generate subjets at an infrared momentum scale $Q_0$. Each subjet then experiences medium-induced energy loss as described by the BDMPS-Z formalism. By modeling the QCD bulk medium using OSU (2+1)-dimensional viscous hydrodynamics and treating $Q_0$ together with the jet-quenching parameters at the initial proper time of the hydrodynamic evolution as free parameters, our approach provides a very good description of the inclusive jet modification factor $R_{AA}$ for large-radius jets and its dependence on jet substructure in 0-10% PbPb collisions at $\sqrt{s_{NN}} = 5.02~\rm{TeV}$, as measured by the ATLAS experiment.

In 1992, Bollobás and Meir showed that for every $k \geq 1$ there exists a constant $c_k$ such that, for any $n$ points in the $k$-dimensional unit cube $[0, 1]^k$, one can find a tour $x_1, \dots, x_n$ through these $n$ points with $\sum_{i = 1}^n |x_i - x_{i + 1}|^k \leq c_k$, where $x_{n + 1} = x_1$ and $|x - y|$ is the Euclidean distance between $x$ and $y$. Remarkably, this bound does not depend on $n$, the number of points. They conjectured that the optimal constant is $c_k = 2 \cdot k^{k / 2}$ and showed that it cannot be taken lower than that. This conjecture was recently revised for $k = 3$ by Balogh, Clemen and Dumitrescu, who showed that $c_3 \geq 2^{7/2} > 2 \cdot 3^{3/2}$. It remains open for all $k > 2$, with the best known upper bound $c_k \leq 2.65^k \cdot k^{k / 2} \cdot (1 + o_k(1))$. We significantly narrow the gap between lower and upper bounds on $c_k$, reducing it from exponential to linear. Specifically, we prove that $c_k \leq 2\mathrm{e}(k + 1) \cdot k^{k / 2}$ and $c_k = k^{k / 2} \cdot (2 + o_k(1))$, the latter establishing the conjecture asymptotically. We also obtain analogous results for related problems on Hamiltonian paths, spanning trees and perfect matchings in the unit cube. Our main tool is a new generalization of the ball packing argument used in earlier works.

The Wasserstein-Fisher-Rao (WFR) distance on $S^{2}$ has recently been shown to coincide with a classical elastic distance between $S^{2}$-immersions in the theory of Riemannian shape analysis. While this correspondence holds in dimension $2$, the analogous statement fails in general on $S^{1}$ and, in the case of convex curves, it cannot be derived from existing two-dimensional arguments. In this paper, we establish that for convex absolutely continuous immersions of $S^{1}$ in the plane, the shape distance induced by the square root velocity transformation (SRVT) is indeed equivalent to the WFR distance acting on their associated length measures. The proof exploits a monotonicity principle for optimal transport on the universal cover of the circle, which in turn guarantees the existence of an optimal reparametrization achieving the SRVT infimum and enables a one-dimensional unbalanced optimal transport reformulation. Motivated by this equivalence, we further investigate the role of sparsity in shape optimization problems formulated in terms of length measures and regularized by the WFR distance. We study linear optimization over the corresponding balls, for which we prove a finiteness result when the reference measure is discrete, and propose a convex, positively one-homogeneous regularizer suitable for conditional gradient algorithms.

In this first-of-its-kind survey conducted on heavy and deformed nuclei in the actinide region of the nuclear chart, we have charted nuclear ground state properties predicted by Fayans energy density functionals (EDFs), focusing in particularly on octupole deformability. Compared to earlier studies with Skyrme-based EDFs, we found similar region of octupole deformed nuclei. Moreover, Fayans EDFs were found to provide accurate predictions for various ground state properties, when compared to experimental data. Comparison to Skyrme-based EDF shows similar trends in various nuclear properties.

We present an extended version of the \texttt{ePump} framework that enables the simultaneous profiling of proton parton distribution functions (PDFs) and the strong coupling $α_s$ using new experimental data. By promoting $α_s$ to a fit parameter within the Hessian updating formalism, the method performs coherent updates of $\{\text{PDFs},α_s\}$ while preserving parameter correlations and the full covariance structure. Validation studies based on CTEQ-TEA analyses with collider data demonstrate that the upgraded \texttt{ePump} accurately reproduces the shifts in PDFs, the preferred $α_s(m_Z)$, and the associated uncertainty reductions obtained in full global fits, including those inferred from Lagrange--Multiplier scans; small deviations arise only for data sets whose $χ^2$ profiles exhibit nonlinear behavior. Applications to representative collider measurements illustrate the impact on the gluon distribution and on precision observables such as the Higgs boson production cross section via gluon fusion. This enhanced framework provides a fast and reliable tool for assessing the effects of new data on the global QCD parameter space, offering near-global-fit accuracy at a fraction of the computational cost.

Conjunctive Hierarchical Secret Sharing (CHSS) is a type of secret sharing that divides participants into multiple distinct hierarchical levels, with each level having a specific threshold. An authorized subset must simultaneously meet the threshold of all levels. Existing Chinese Remainder Theorem (CRT)-based CHSS schemes either have security vulnerabilities or have an information rate lower than $\frac{1}{2}$. In this work, we utilize the CRT for polynomial ring and one-way functions to construct an asymptotically perfect CHSS scheme. It has computational security, and permits flexible share sizes. Notably, when all shares are of equal size, our scheme is an asymptotically ideal CHSS scheme with an information rate one.

Lepton number violating (LNV) $A(e^-,e^+)X$ double charge exchange (LDCE) reactions on nuclei at accelerator facilities with multi-GeV beams are proposed as a probe for physics beyond the Standard Model (BSM). A second order formalism is presented including LNV dynamics by the left-right symmetric model (LRSM). For practical studies a phenomenological model is used to estimate LDCE cross sections numerically. Sizable cross sections are predicted for multi-GeV beam energies. LDCE reactions proceed preferentially by the energy-momentum dependent left-right mixing terms. While Majorana mass terms are negligible for light neutrinos, they may become sizable for heavy neutral leptons at higher beam energies. In the ~10 GeV region inclusive total LDCE cross sections of about $100\times|Γ_{BSM}|^2$~fb in units of the BSM vertices are predicted, increasing strongly with energy and target mass. LDCE experiments seem to be feasible with existing equipment under full laboratory control and free of the constraints imposed on decay or capture experiments of nuclear and hadronic systems.

We investigate the behavior of scalar quantum fields in cosmological backgrounds under modified dispersion relations, specifically focusing on how ultraviolet asymptotics influence field quantization. We establish the conditions for both the validity of the adiabatic approximation and the unitary equivalence between quantizations defined via different time variables. Our analysis reveals that while superluminal modified dispersion relations consistently yield unitarily equivalent quantizations, asymptotically subluminal behaviors can lead to inequivalent physical descriptions. By applying adiabatic regularization to the two-point correlation function, we demonstrate that the ultraviolet scaling of the frequency uniquely dictates the required subtraction order. These results are illustrated through applications to standard, superluminal Corley--Jacobson, and Unruh dispersion relations.

Out-of-plane and in-plane electric polarization, which rarely coexist in a two-dimensional (2D) ferroelectric material, offer different advantages in ferroelectricity-based devices. Here, we report the coexistence of in-plane and out-of-plane electric dipoles, along with various phase transitions, in 2D van der Waals layered TlGaS2 single crystal. Quantum paraelectricity was observed along both in-plane and out-of-plane directions of the TlGaS2 crystal. Detailed investigation of the quantum paraelectric soft-mode behavior reveals a close correlation between the electric dipoles and the off-center displacement of Tl1+ ions with 6s2 lone pairs in TlGaS2. Anomalies near temperatures of about 120 K and 60-75 K in dielectric and/or infrared spectra indicate the existence of local or weak long-range structural transitions in TlGaS2. Our results provide important experimental evidence for elucidating the phase transitions and coexistence of in-plane and out-of-plane electric dipoles in 2D layered TlGaS2.

A finite group $G$ is said to be Cayley integral if every undirected Cayley graph $\operatorname{Cay}(G,S)$ on $G$ is integral. In this paper, we introduce three natural extensions of this concept; namely as: Cayley colour integral, $\mathfrak{F}$-Cayley colour integral and normal Cayley integral groups. We characterize the first two families in its entirety. The last family of groups is shown to be coinciding with inverse semi-rational groups introduced by Chillag and Dolfi, thereby providing an alternative characterization for the same. We also establish an inclusion hierarchy among these families.

Stochastic epidemic models can estimate infection and removal rates, and derived quantities such as the basic reproductive number ($R_0$), when both infection and removal times are observed. In practice, however, removal times are often available while infection times are not, and existing methods that rely only on removal times can become unstable or biased. We study inference for stochastic SIR/SEIR models in a partial--observation setting. We develop imputation--based estimators that use a small calibration sample of fully observed infectious periods, derive closed--form expressions for the pairwise exposure terms they require, and use a studentized parametric bootstrap for bias correction and uncertainty quantification. In simulations, removal time--only methods performed poorly in moderate to large $R_0$ scenarios, while observing even tens of complete infectious periods substantially improved the estimation of the infection rate. A reanalysis of the 1861 Hagelloch measles outbreak under simulated missingness recovered stable qualitative differences in transmission between school classes. Based on our results, we advocate for the targeted collection of a modest number of complete infectious periods as a means of improving surveillance in the early stages of an epidemic.

After Taiwan's legalization of same-sex marriage in 2019, LGBTQ+ communities continue to face hostility on social media. Using the lens of hermeneutical injustice and autonomy, we examine how technological conditions affect LGBTQ+ individuals' identity exploration, narrative seeking, and community resilience. We conducted a multi-stage study with Taiwanese LGBTQ+ individuals, including in-depth interviews, participatory design workshops, and evaluation sessions. Participants described fragile yet creative strategies such as seeking validation in online interactions, reframing hostile content through theory, and relying on allies. Building on these insights, we designed and evaluated a retrieval-augmented, LLM-powered chatbot with four modes of interaction: reflection, validation, discussion, and allyship. Findings show that the system fosters hermeneutical autonomy by helping participants reframe hostile narratives, validate lived experiences, and scaffold identity exploration, while reducing the hermeneutical labor of navigating social media hostility. We conclude by outlining design implications for AI systems that advance hermeneutical autonomy through fluid self-representation, contextualized dialogue, and inclusive community participation.

We calculate the correction exponents in the chiral Heisenberg model in the $1/N$ expansion. These exponents are related to the slopes of $β$ functions at the phase transition point. We present the results at order $1/N^2$ and check that they agree with the results of the $ε$ expansion near $d = 4$. We find that one of the correction exponents diverges as $d \to 3$. We argue that the appearance of the pole is a rather general phenomenon and is associated with operator mixing involving the system of four-fermion operators. After analyzing the operator mixing structure, we propose a resummation procedure which modifies the exponents already at leading order. We also perform calculations directly in the three-dimensional model and find complete agreement with the resummed exponents.

A systematic investigation of the odd-mass neutron-rich bromine isotopes $^{87-91}$Br has been carried out within the configuration-constrained cranked shell model (CSM) framework. The calculated kinematic moments of inertia and angular-momentum alignments reproduce the experimental trends with excellent agreement for the proposed quasiparticle configurations, thereby supporting the assigned band structures. The total Routhian surface calculations reveal pronounced $γ$-softness in $^{87}$Br, shape coexistence in $^{89-93}$Br, and a gradual prolate-to-oblate transition approaching $N=56$. These results demonstrate that the CSM provides a reliable description of rotational behavior and shape evolution in this mass region, achieving a better level of accuracy, while offering a transparent interpretation of quasiparticle configurations and shape-driving effects.

The renowned theorem of Dirac states that if $G$ is a graph with minimum degree at least $n/2$ then $G$ has a Hamilton cycle. A natural generalisation asks what properties of an edge-colouring of $G$ guarantee the existence of a properly edge-coloured Hamilton cycle in $G$. This concept can be further generalised as follows: an \emph{incompatibility system} for $G$ is a set~$\mathcal{F}$ of `forbidden' pairs of adjacent edges, that is, $\mathcal{F}\subseteq \{\{uv,vw\}\in \binom{E(G)}2\}$. A cycle in $G$ is then \emph{compatible} if no two of its edges form a pair in $\mathcal{F}$. The system $\mathcal{F}$ is called \emph{$μn$-bounded} if for all $v\in V(G)$ and $uv\in E(G)$, there are at most $μn$ pairs $\{uv,vw\}\in \mathcal{F}$. How small must $μ$ be to guarantee the existence of a compatible Hamilton cycle in $G$? Krivelevich, Lee and Sudakov showed that $μ=10^{-16}$ suffices (for $n$ large), while an example of Bollobás and Erdős shows that $μ\leq 1/4$ is necessary. We significantly reduce this gap for large graphs of minimum degree at least $(1/2+\varepsilon)n$, by showing that $μ=1/8$ suffices but $μ\leq 1/6$ is necessary for such graphs. In fact, we give more precise bounds which are functions of $δ(G)/n$.

Quantum-correlated networks distribute quantum resources such as squeezed and entangled states. These states are central to modern quantum technology, including photonic quantum computing, quantum communications, non-destructive biological sensing and gravitational-wave detection. Even for squeezed states of light - the most robust quantum-correlated resource - loss-induced decoherence remains the dominant obstacle to strong quantum advantage in in large-scale interferometric and networked quantum systems. Common design assumption in these applications is treating mismatches between spatial modes as a small, incoherent loss. Here we show that this picture can fail: coherent spatial-mode mixing with higher-order spatial modes can produce an apparent loss exceeding 100% relative to the initial squeezing, a regime we term hyperloss. We experimentally demonstrate hyperloss in a minimal two-node quantum network: with only 8% mode mismatch, a 5.8dB squeezed state is converted into an effectively thermal state with no quadrature squeezing, eliminating the quantum advantage. Because the effect is coherent, it is controllable: lost correlations can be recovered by tuning differential spatial-mode phases (e.g., Gouy-/propagation-phase). We demonstrate this recovery experimentally, not only eliminating the hyperloss, but even significantly suppressing the mode mismatch loss, with 15% geometric mismatch acting like only ~2.8% effective loss. Hyperloss is a design-limiting mechanism for all quantum networks with squeezed light, from from photonic quantum processors to large-scale interferometers and distributed quantum-sensing networks. Our results provide a practical route to avoid hyperloss and turn mode mismatch into an explicit, phase-aware design parameter for future quantum technologies.

Elasticity has long been regarded as a property exclusive to material media. Here we uncover its hidden existence in the spin degree of freedom. We introduce spin elasticity-an intrinsic mechanism that governs recoverable deformation of spin morphology. This discovery reveals a previously unrecognized universality: elasticity operates in both matter and spin spaces, underpinning structural integrity across physical realms. By establishing the missing spin counterpart, this work completes the elastic picture and points toward a broader paradigm where elasticity transcends its conventional boundaries.

Motivation: Quickly obtaining high-quality MRI from accelerated acquisitions is important to mitigate motion artifacts, maintain patient comfort, and improve clinical efficiency. Goals: To obtain high-quality dynamic MRI using efficient, personalized models. Approach: We propose a novel explicit representation learning approach using Gaussian splatting. Multiple Gaussian primitives are trained to represent the underlying tissue. We extend the Gaussian splatting framework to model anatomical motion, enabling learning an efficient, explicit representation of dynamic MRI. Results: Gaussian splats can be trained in 60s with 0.5ms/dynamic inference time. High-quality cardiac MRI is obtained at R=16. We show that the properties of the Gaussians directly encode physiological properties.

We investigate the kinetic energy cascade in zero-temperature quantum turbulence. Using simple theoretical arguments and unprecedented numerical simulations, we unveil an universal mechanism transferring energy directly from large to very small scales, thus bypassing the Kolmogorov-like local energy cascade and resulting in nonclassical energy spectra. This mechanism rests both on the vast separation of scales typical of superfluid helium-4 flows and on the alignment between quantum vortices and large-scale velocity gradients, in direct analogy with vortex stretching in classical flows.

These notes are from three lectures given at the 54th Saas-Fee Advanced Course of the Swiss Society of Astrophysics and Astronomy in January 2025. This chapter reviews the dramatic evolution in our understanding of supermassive black holes in the first billion years, from ground-based discoveries to recent space-based infrared observations with JWST. Section 1 introduces AGN and quasars to contextualise observations at the highest redshifts. Section 2 reviews the pre-JWST understanding of early quasars, including personal accounts of how key discoveries were made. Section 3 examines how JWST is transforming the field, from black hole mass measurements and host galaxy characterisation to large-scale environmental studies, and identifies emerging directions.

High power end pumped solid state lasers often operate in regimes where pump induced heating creates a strong refractive index gradient (thermal lensing) that governs resonator stability and mode quality. When the pump is absorbed according to the Beer Lambert law, the thermal load, and hence the GRIN strength, vary along the crystal length, so the standard ABCD matrix of a constant-gradient GRIN element is no longer directly applicable. Here, we derive a closed-form ABCD transmission matrix for a thermally loaded laser crystal pumped by atop-hat beam while explicitly accounting for axial absorption. Starting from the steady-state heat equation, we obtain the temperature field and the associated thermo-optic index profile. We then solve the paraxial eikonal ray equation analytically and express the transfer-matrix elements in terms of Bessel and Neumann functions. The resulting matrix is validated against the conventional slab product method and shown to recover the uniform-medium and constant gradient GRIN limits. Finally, we illustrate its utility by model ing Bessel Gaussian beam propagation through the axially varying thermally induced GRIN medium.

In moiré graphene systems, electronic interactions lift spin and valley degeneracies, leading to symmetry-broken ground states. In helical trilayer graphene (HTG), we uncover a distinct interaction-driven mechanism in which the roles of sublattice-polarized valence and conduction bands are cyclically reversed. Using scanning nano-SQUID magnetometry, we detect a series of sharp magnetic signatures consistent with seesaw-like transitions, where occupied and unoccupied valence and conduction bands interchange repeatedly with doping, accompanied by a novel form of magnetic hysteresis. These transitions occur entirely within metallic regimes and leave only weak fingerprints in transport measurements. Self-consistent Hartree-Fock calculations reveal that interactions reorganize all eight low-energy flat bands, driving abrupt changes in orbital magnetization. Our results establish HTG as the first system where electronic interactions provide doping-controlled access to all three internal degrees of freedom - spin, valley, and sublattice - introducing a new class of correlated phase transitions.

Given $a,b,c\in\mathbb N$, let $D_{a,b,c}$ be the tournament on $a+b+c$ vertices obtained by replacing the vertices of the directed triangle $C_3$ with transitive tournaments $TT_a$, $TT_b$, and $TT_c$, respectively. Keevash and Sudakov (2009) showed that every sufficiently large oriented graph $G$ on $n$ vertices with $δ^{0}(G)\geqslant (1/2-o(1))n$ contains a $C_3$-tiling, equivalently a $D_{1,1,1}$-tiling, covering all but at most three vertices. We generalize this result to arbitrary blow-ups $D_{a,b,c}$. Specifically, for any fixed $a,b,c$, every sufficiently large oriented graph $G$ on $n$ vertices with $δ^{0}(G)\geqslant (1/2-o(1))n$ contains a $D_{a,b,c}$-tiling covering all but at most $2(a+b+c)-3$ vertices. Moreover, this bound is essentially sharp. We also establish a stronger stability result: if $(a+b+c)\mid n$, then either $G$ contains a $D_{a,b,c}$-factor, or $G$ is close to an extremal graph. Our interest in $D_{a,b,c}$ is also motivated by oriented Turán theory: a seminal theorem of Bollobás and Häggkvist (1990) shows that a tournament $T$ is Turánable (i.e., contained in every sufficiently large regular tournament) if and only if $T\subseteq D_{s,s,s}$ for some $s$. Complementing our tiling results, we also investigate related semi-degree thresholds for powers of directed cycles and paths. In particular, we present two $n$-vertex constructions that give lower bounds, showing that the minimum semi-degree thresholds for $C^2_l$ with $l\not\equiv 0\pmod 6$ and for $P^2_l$ with $l\geqslant 7$ are at least $4n/9$ and $3n/8$, respectively.

Altermagnetism, a recently identified magnetic phase that combines vanishing net magnetization with momentum-dependent spin splitting, challenges the conventional dichotomy between ferromagnets and antiferromagnets. While several candidate materials have been proposed, direct experimental evidence linking crystal symmetry, electronic structure and d-wave spin polarization remains scarce. Here we report the visualization of a metallic d-wave altermagnet in KV2Se2O. Through spin-selective scanning tunneling microscopy powered by a topological insulator tip, we uncover symmetry-protected momentum-dependent spin splitting that follows a characteristic d-wave form factor. Our results establish KV2Se2O as a tunable platform to study the interplay between spin-valley locking, Fermi-surface instability and unconventional magnetism, and open a pathway toward symmetry-engineered spintronics without net magnetization.

Evaluating treatment effect heterogeneity across patient subgroups is a fundamental aspect of clinical trial analysis. Yet, these analyses have inherent limitations due to small sample sizes and the substantial number of subgroups investigated. Statisticians in regulatory agencies and pharmaceutical companies have begun considering shrinkage methods grounded in Bayesian statistical theory. These methods incorporate priors on treatment effect heterogeneity, which operationally shrink raw subgroup treatment effect estimates towards the overall treatment effect. Various shrinkage estimators and priors have been proposed, yet it remains unclear which methods perform best. This work provides a unified presentation, software implementation (in the R package bonsaiforest2), and simulation comparison of one-way and global shrinkage methods for continuous, binary, count, and time-to-event endpoints. One-way models fit a separate shrinkage model for each subgrouping variable, whereas global models fit a model including all subgroup indicators at once. Both can derive standardized subgroup-specific treatment effects. Across all simulation scenarios, shrinkage methods outperformed the standard subgroup estimator without shrinkage in terms of mean squared error. They were also more efficient in identifying a non-efficacious subgroup. Global shrinkage models tended to have smaller mean squared error and less dependence on hyperprior parameters than one-way models, but also exhibited slightly larger bias and worse frequentist coverage of associated credible intervals. For both models, hyperprior choices anchored in trial assumptions about the anticipated size of the overall treatment effect performed well. We conclude that some degree of shrinkage is preferable to none and advocate for the routine inclusion of shrunken estimates in clinical forest plots to facilitate more robust decision-making.

A search for resonances with masses from 50 to 300 GeV decaying into a quark-antiquark pair is presented. The search uses proton-proton collision data at $\sqrt{s}$ = 13 TeV collected by the CMS experiment at the CERN LHC in 2016$-$2018, corresponding to an integrated luminosity of 13 fb$^{-1}$. Two coupling scenarios are considered, with the resonances coupled either equally to all flavors of quarks or preferentially to bottom quarks. The search targets resonances produced in association with hard initial-state radiation, resulting in a large-radius jet with a two-pronged substructure. The ParticleNet algorithm is used to distinguish resonance decays to bottom quark pairs from lighter quark pairs and to suppress background processes. The invariant jet mass spectrum is scrutinized for peaking excesses over a falling background. No evidence for such resonances is observed. Limits are set on the couplings of new scalar and vector resonances to quarks, representing the most stringent limits to date in the mass range of 50$-$250 GeV.

Chiral-induced spin selectivity (CISS) is a striking phenomenon in which spin-unpolarized electrons become spin-polarized after traversing a chiral medium. Theoretical studies have shown that spin-orbit coupling, geometric chirality, and dephasing act cooperatively for this effect to emerge. Inspired by this, we demonstrate a solid-state realization of CISS in an engineered InAs/GaSb quantum well where geometric chirality and dephasing can be introduced controllably. Introducing a chiral structure produces a clear spin polarization whose sign reverses when the chirality is flipped, and whose magnitude grows systematically with the number of dephasing electrodes, while achiral configurations exhibit no spin selectivity. The polarization remains robust even under strong Anderson disorder, showing that the engineered chiral structures provides an intrinsically stable route to spin-selective transport. These results establish a solid-state platform in the topological quantum well system for controllably generating the CISS effect.

We investigate finite-width optical-beam propagation in a medium with uniform torsion described by the geometric theory of a continuous distribution of screw dislocations. Starting from the Riemann--Cartan framework that yields torsion-induced circular birefringence for local plane waves, we construct a minimal paraxial beam model in which the same contortion-driven helicity splitting remains explicit. We show that uniform torsion breaks the degeneracy between the two circular-polarization sectors and induces a geometric rotation of the polarization that scales with both the propagation distance and the radial position in the beam. As a consequence, a finite-width beam develops spatially varying polarization textures across its transverse profile, naturally described by the Stokes parameters. We introduce beam-level observables based on the integrated Stokes vector, the transverse inhomogeneity of the polarization texture, and the number of resolved radial polarization domains, thereby connecting the torsion parameter to experimentally accessible beam diagnostics. The paper combines two complementary levels of description: an analytic short-distance regime, used to isolate the geometric mechanism, and full paraxial propagation including diffraction, used to test the robustness of the predicted textures. Within the cylindrically symmetric minimal model, the most robust structured-light signature of uniform torsion is beam-resolved polarization structuring, whereas strong orbital-angular-momentum conversion is not expected without additional azimuthal structure. We also identify the geometric ingredient required for genuine torsion-assisted spin--orbit conversion beyond the minimal radial model: an effective azimuthal geometric connection.

We consider magnetic Schrödinger equations with sublinear magnetic potentials and subquadratic electric potentials on $\mathbb{R}^{d}$, as well as generalizations thereof. We obtain new results on the global well-posedness of the Cauchy problem with initial data in magnetic modulation spaces $M^{p}_{A}(\mathbb{R}^{d})$. Our results are achieved by approximating the solution in phase space using the magnetic Hamiltonian flow. This method includes the potentials as part of the generalized Schrödinger operator instead of treating them as perturbations, and thereby allows us to deal with unbounded potentials. For $A \equiv 0$, the space $M^{p}_{A}(\mathbb{R}^{d})$ reduces to the usual modulation space $M^{p}(\mathbb{R}^{d})$, for which relevant known results for the usual Schrödinger equation can be recovered.

We study an inverse spectral problem for singular AKNS operators based on spectral data associated with two distinct values of the effective angular momentum parameter $κ\,$. Our main focus is the local inverse problem near the zero potential. For the pairs $(κ_1,κ_2)=(0,1)$, $(1,2)$ and $(0,3)\,$, we establish local uniqueness. For $(0,2)\,$, we prove that the Fréchet differential of the spectral map at the origin is injective, while the question whether its range is closed remains open.