AI for Mathematics (AI4Math) has emerged as a distinct field that leverages machine learning to navigate mathematical landscapes historically intractable for early symbolic systems. While mid-20th-century symbolic approaches successfully automated formal logic, they faced severe scalability limitations due to the combinatorial explosion of the search space. The recent integration of data-driven approaches has revitalized this pursuit. In this review, we provide a systematic overview of AI4Math, highlighting its primary focus on developing AI models to support mathematical research. Crucially, we emphasize that this is not merely the application of AI to mathematical activities; it also encompasses the development of stronger AI systems where the rigorous nature of mathematics serves as a premier testbed for advancing general reasoning capabilities. We categorize existing research into two complementary directions: problem-specific modeling, involving the design of specialized architectures for distinct mathematical tasks, and general-purpose modeling, focusing on foundation models capable of broader reasoning, retrieval, and exploratory workflows. We conclude by discussing key challenges and prospects, advocating for AI systems that go beyond facilitating formal correctness to enabling the discovery of meaningful results and unified theories, recognizing that the true value of a proof lies in the insights and tools it offers to the broader mathematical landscape.

We study the entanglement complexity of a system consisting of two simple-closed curves (self-avoiding polygons) that span a lattice tube, referred to as a 2SAP. 2SAPs are of interest as the first known model of confined ring polymers where the linking probability goes to 1 exponentially with the size of the system. Atapour et al proved this in 2010 by showing that all but exponentially few sufficiently large 2SAPs contain a pattern that guarantees the 2SAP is non-split, provided that the requisite pattern fits in the tube. This result was recently extended to all tubes sizes that admit non-trivial links. Here we develop and apply knot theory results to answer more general questions about the entanglement complexity of 2SAPs. We first extend the 1992 concept of a good measure of knot complexity to a good measure, $F$, of spanning-link complexity for $k$-component links. Using tangle products, we show, for example, that the more complex the prime knot decomposition of any component of a given link type, the greater its $F$-measure. We then prove that all but exponentially few size $m$ 2SAPs have $F$ complexity that grows at least linearly in $m$ as $m\to \infty$. We establish that good measures of knot complexity yield good measures of spanning-link complexity. We also establish conditions whereby more general link invariants can yield good measures. In particular, we establish that measures based on several classical invariants are good measures by our definition, eg bridge number or the splitting number. Finally, we consider how the tube dimensions affect which links are embeddable as 2SAPs as well as geometric restrictions on the entanglement complexity of the embeddings. For example, we establish that there are two-component links that occur as 2SAPs in a given tube size only when one of the components is forced into a non-minimal bridge number conformation.

We develop a compact transmon emitter/detector (TED) superconducting circuit and demonstrate its dual functionality as a single-photon source and detector. In our setup, photons emitted by a source TED are transmitted via a meter-long coaxial cable, routed through a circulator, and captured by a measurement TED. Both TED modules operate with nominally identical parameters, highlighting the flexibility of this novel architecture. Furthermore, we introduce an efficient microwave photon detection scheme tailored to the TED. Using this setup, we detect 60% of the emitted Fock state photons and infer a 95% detection efficiency at the input of the measurement TED, which we calibrate against coherent state measurements. The reset and photon emission/detection processes each require approximately $2\,μs$, yielding a minimum protocol duration of $4\,μs$ as constrained by our chosen TED parameters. Ultimately, the TED demonstrates a new use case for the recently developed double transmon coupler (DTC): a compact, drop-in, tunable, and transition-selective link between a coherent data transmon and a waveguide. Circuits like the TED will play a vital role in quantum information processing by facilitating unconditional fast reset, enabling microwave photon metrology, and serving as nascent quantum communication interfaces (QCIs). QCIs mediate entanglement distribution between quantum processing units (QPUs) within a cryostat or interface with microwave-to-optical transducers for long-range quantum networking.

This paper analyzes and explicitly solves a class of long-term average impulse control problems and a related class of singular control problems. The underlying process is a general one-dimensional diffusion with appropriate boundary behavior. The model is motivated by applications such as the optimal long-term management of renewable resources and financial portfolio management. A large class of admissible policies is identified over which the agent seeks to maximize her long-term average reward, consisting of a running reward and income from either discrete impulses or singular actions. In addition to the long-term average objective, we will briefly consider the long-term expected total reward functional and its relation to overtaking optimality. Sensitivity analysis with regard to the parameters of the impulse control model are performed. Key connections between the impulse and singular control problems are displayed.

Asymptotic normality is frequently observed in large combinatorial structures, rigorously established for many quantities such as cycles or inversions in random permutations, the number of prime factors of random integers, and various parameters of random graphs. In this paper, we investigate whether this normal limit behavior extends to the topological invariants of geometric spaces. We show that the Betti numbers of the moduli space of rational curves with $n$ marked points $\overline{\mathcal{M}}_{0,n}$ and the Fulton-MacPherson configuration space $\mathbb{P}^1[n]$ are asymptotically normally distributed. Based on numerical evidence and established log-concavity, we conjecture that the Betti numbers of the quotients of these spaces by the symmetric group $\mathbb{S}_n$ are also asymptotically normally distributed. In contrast, we provide examples of geometric spaces that do not follow this Gaussian law.

Non-orthogonal multiple access (NOMA) is a promising technique for optical wireless communication (OWC), enabling multiple users to share the optical spectrum simultaneously through the power domain. However, imperfect channel state information (CSI) and residual decoding errors deteriorate NOMA performance, especially in realistic dense-user indoor scenarios. In this work, we model an OWC system that integrates light detection and localization (LiDAL) and random linear network coding (RLNC) within a NOMA framework. LiDAL exploits spatio-temporal information to improve user CSI, while RLNC enhances data resilience in the successive decoding process, resulting in a LiDAL-assisted RLNC-NOMA OWC system. Power allocation (PA) is crucial in this system due to complex interactions between multiple users and the coding and detection processes, but optimizing continuous PA dynamically can be computationally prohibitive. To address this, we adopt a deep reinforcement learning (DRL) framework to efficiently learn near-optimal PA strategies. In particular, a DRL-based normalized advantage function (NAF) algorithm is proposed to maximize the average sum rate, and its performance is compared to deep deterministic policy gradient (DDPG), gain ratio PA (GRPA), and exhaustive search. The results indicate that NAF closely matches exhaustive search, is 39% faster than DDPG, and improves the average sum rate by 4.6% over GRPA, while accounting for user location estimation errors.

Plastic deformation and fracture of FCC metals involve the formation of stable or unstable stacking faults (SFs) on (111) plane. Examples include dislocation cross-slip and dislocation nucleation at interfaces and near crack tips. The stress component normal to (111) plane can strongly affect the SF energy when the stress magnitude reaches several to tens of GPa. We conduct a series of DFT calculations of SF energies in six FCC metals: Al, Ni, Cu, Ag, Au, and Pt. The results show that normal compression significantly increases the stable and unstable SF energies in all six metals, while normal tension decreases them. The SF formation is accompanied by inelastic expansion in the normal direction. The DFT calculations are compared with predictions of several representative classical and machine-learning interatomic potentials. Many potentials fail to capture the correct stress effect on the SF energy, often predicting trends opposite to the DFT calculations. Possible ways to improve the ability of potentials to represent the stress effect on SF energy are discussed.

At its lowest frequencies, LIGO is limited by noise in its many degrees of freedom of suspended optics, which, in turn, introduce noise in the interferometer through their feedback control systems. Nonlinear interactions are a dominant source of low-frequency noise, mixing noise from multiple degrees of freedom. The lowest-order form is bilinear noise, in which the noise from two feedback-controlled subsystems multiplies to mask gravitational waves. Bilinear couplings require control trade-offs that simultaneously balance high- and low-frequency noise. Currently, there is no known lower limit to bilinear control noise. Here, we develop benchmark cost functions for bilinear noise and associated figures of merit. Linear-quadratic-Gaussian control then establishes aggressive feedback that saturates the lower bounds on the cost functions. We then develop a mixed LQG and $H_\infty$ approach to directly compute stable, robust, and optimal feedback, using the LIGO's alignment control system as an example. Direct computations are fast while ensuring a global optimum. By calculating optimal robust control, it is possible to construct the lower bound on controls noise along the Pareto front of practical controllers for LIGO. This method can be used to drastically improve controls noise in existing observatories as well as to set subsystem control noise requirements for next-generation detectors with parameterized design.

We theoretically investigate a non-Hermitian optical dimer whose parameters are renormalized by dispersive and dissipative backaction from the coupling of the passive cavity with a ring-trapped Bose-Einstein condensate. The passive cavity is driven by a two-tone control laser, where each tone is in a coherent superposition of Laguerre-Gaussian beams carrying orbital angular momenta $\pm \ell \hbar$. This imprints an optical lattice on the ring trap, leading to Bragg-diffracted sidemode excitations. Using an exact Schur-complement reduction of the full light-matter dynamics, we derive a frequency-dependent self-energy and identify a static regime in which the atomic response produces a complex shift of the passive optical mode. This renormalized dimer supports a tunable exceptional point, enabling spectroscopic signatures in the optical transmission due to a probe field, which can in turn be utilized for estimating the winding number of the persistent current. Exploiting the associated half-integer topological charge, we propose a digital exceptional-point-based sensing scheme based on eigenmode permutation, providing a noise-resilient method to sense superfluid rotation without relying on fragile eigenvalue splittings. Importantly, the sensing proposals are intrinsically nondestructive, preserving the coherence of the atomic superfluid.

Fast phase logic (FPL) is a novel digital superconductor electronic (SCE) logic family that employs multiple junction types, including switching 0-Josephson junctions (0-JJs), non-switching 0-JJ stacks, and $π$-JJs. FPL enables flexible, automatable cell layouts, faster pulse propagation, reduced bias current via phase-shifting $π$-JJs, and minimized inductive loops, thereby reducing susceptibility to trapped flux and crosstalk. A fabrication process to support FPL is proposed. NbTiN superconductors offer small grain sizes, smooth surfaces, and thermal stability up to 400~$^\circ$C, while high-$J_c$, self-shunted JJs enable compact devices. AlN dielectrics provide good crystal matching to NbTiN, improving superconducting properties. Projections indicate that FPL, combined with the proposed process, can achieve a two-order-of-magnitude increase in integration density over conventional RSFQ logic and a five-fold reduction in supply current. The increased density reduces latency and improves computational throughput, while NbTiN-based devices provide higher output voltage and impedance, improving compatibility with CMOS circuits. Further fabrication advancements, such as higher-$J_c$ NbTiN-based JJs, higher processing temperatures, and stacked JJ structures, could enhance FPL implementation and scalability toward very large-scale integration (VLSI). FPL has the potential to significantly advance SCE technology, with near-term applications in accelerator cores for signal processing and artificial intelligence, and long-term potential in supercomputing. Its advantages are evaluated through an architectural study of a fast Fourier transform (FFT) circuit, with comparisons to CMOS and SFQ technologies.

We propose a novel approach for studying small-time asymptotics of the fractional heat content of $C^2$ non-characteristic domains in Carnot groups. Denoting the sub-Laplacian operator by $\mathcal{L}$, the fractional heat content of a bounded domain $Ω$ is defined as $Q^{(α)}_Ω(t)=\int_Ωu_α(x,t) dx$, where $u_α$ is the solution to the heat equation corresponding to the fractional sub-Laplacian $\mathcal{L}_α:=\mathcal{L}^{α/2}$ with Dirichlet boundary condition on $Ω$. We prove that for $1\le α\le 2$, there exists explicit rate function $μ_α: (0,\infty)\to (0,\infty)$ such that \begin{align*} \lim_{t\to 0}\frac{|Ω|-Q^{(α)}_Ω(t)}{μ_α(t)}=|\partial Ω|_H, \end{align*} where $|Ω|$, $|\partial Ω|_H$ are the volume and horizontal perimeter of $Ω$ respectively. Moreover, the rate function $μ_α$ coincides with the same for the Euclidean case.

We optimize theoretically and experimentally the performances of the recently demonstrated modulation transfer protocol [D.-A. Trinh, K. V. Adwaith, M. Branco, A. Rouxel, S. Welinski, P. Berger, F. Goldfarb, and F. Bretenaker, Applied Physics Letters 125, 154001 (2024)] aiming at extending the bandwidth of quantum RF receivers based on hot Rydberg atoms. This optimization relies on tuning the parameters of the phase modulation of the coupling beam, which is converted by the nonlinear response of the atoms into an amplitude modulation of the probe beam. We develop a theoretical model to optimize both the modulation frequency and the modulation amplitude of the coupling beam, thereby maximizing the atomic response. Once optimized, the sensitivity to detuned RF fields of this modulation transfer protocol is measured and compared with that of the conventional protocol. This comparison shows that the new protocol permits a strong increase in the detection bandwidth. Indeed, it outperforms the usual one as soon as the RF signal to be measured is detuned by more than a few MHz. We illustrate the capability of this modulation transfer protocol to enhance the detection bandwidth by showing experimentally how it permits to bridge the gap between two Rydberg transitions separated by 166 MHz. In all cases, the experimental results are in good agreement with the simulations.

We study the homogeneous Kuramoto model on a graph and the geometry of its underlying optimization landscape $\min_{\boldsymbol θ\in \mathbb R^n}-\sum_{1\leq i,j\leq n} A_{ij}\cos(θ_i-θ_j).$ This problem admits a dual interpretation. On the one hand, it can be viewed as an unconstrained optimization problem, seeking configurations of points on the unit circle that minimize the energy function. On the other hand, the same function serves as a Lyapunov potential governing the dynamics of the homogeneous Kuramoto model. A central question is to identify which graphs induce a benign energy landscape, in the sense that every second-order stationary point is a global minimizer, corresponding to the fully synchronized state. In this case, the graph is said to be globally synchronizing. Most existing results establish global synchronization by exploiting the fact that the complete graph is globally synchronizing, and by showing that graphs sufficiently close to it inherit this property. In contrast, we uncover a fundamentally different mechanism: on highly-structured graph classes, namely quasi-threshold graphs, global synchronization unfolds through a sequential process of local synchronization that propagates along their underlying skeletons. Our approach relies on a detailed analysis of the phasor geometry at second-order stationary points of the nonconvex energy landscape.

In this paper, using some arithmetic properties of Jacobi sums, we investigate some products involving Jacobi sums and reveal the connections between these products and certain cyclotomic matrices. In particular, as an application of our main results, we confirm a conjecture posed by Z.-W. Sun in 2019, and obtain a stronger result.

The effective dynamics of the Janis-Newman-Winicour spacetime inspired by loop quantum gravity is studied. Two different schemes are considered to regularize the Hamiltonian constraint for the quantum dynamics. In the $μ_0$ scheme in which the quantum parameters are treated as constants, the equations of motion generated by the effective Hamiltonian are solved analytically. The resulting quantum-corrected effective spacetime obviously extends the effective spacetime previously obtained in the literature. In the new effective spacetime, the naked singularity and the central singularity presented in the classical JNW spacetime are resolved by a series of quantum bounces. In the scheme of choosing the quantum parameters as Dirac observables, the effective dynamics is also solved in the light of the solution in $μ_0$ scheme. It turns out that the resulting effective spacetime has singularities due to the appearance of the zero points of the time reparametrization functions. Hence, the effective theory in this scheme does not remain valid throughout the full spacetime.

We propose a scalar-mediated Self-Interacting Dark Matter (SIDM) model to address small-scale structure anomalies such as the core-cusp and diversity problems. The model is composed by a leptophilic scalar mediator and a pseudo-Dirac dark matter candidate with a mass splitting of 100ev.We imposed a dark discrete $\mathbb{Z}_2$ symmetry forbids tree-level elastic scattering. Therefore creates kinematic threshold that suppresses scattering in ultra-faint satellite galaxies while enabling large self-interaction cross-sections in dwarf galaxies via resonant enhancement. To satisfy Big Bang Nucleosynthesis (BBN) requirements, we introduce a dimension-5 magnetic dipole operator that enable the decay of the excited state ($χ_2 \rightarrow χ_1 γ$). This operator also provides a unique, low-threshold signal for direct detection experiments, characterized by a distinct $1/E_R$ recoil spectrum. We identify a benchmark parameter space around ($m_χ\approx 40$ GeV, $m_ϕ\approx 20$ MeV) where non-perturbative coupled-channel dynamics successfully reconcile astrophysical observations with cosmological bounds, including CMB constraints on annihilation.

We study the correlation functions between the dynamical variables and between their conjugate momenta at sites of a harmonic lattice in the $d$-dimensional Euclidean space. We show that at the thermodynamic limit, they can be expressed in terms of Lauricella's C-type hypergeometric series. Furthermore, using these expressions, we explicitly demonstrate that the correlators near the center of the lattice satisfying Diriclet boundary conditions coincide with those for the lattice with the periodic boundary conditions. By utilizing these expressions, we expect to make it easier to create programs that compute fast and highly precise for the quantum information quantities of subsystems within lattices.

Incompressible flow solvers based on strong-form meshfree methods represent arbitrary geometries without the need for a global mesh system. However, their local evaluations make it difficult to satisfy incompressibility at the discrete level. Moreover, the collocated arrangement of velocity and pressure variables tends to induce a zero-energy mode, leading to decoupling between the two variables. In projection-based approaches, a spatial discretization scheme based on a conventional node-based moving least-squares method for the pressure causes inconsistency between the discrete operators on both sides of the Poisson equation. Thus, a solenoidal velocity field cannot be ensured numerically. In this study, a numerical method for the incompressible Navier-Stokes equations is developed by introducing a local primal-dual grid into the mesh-constrained discrete point method, enabling consistent discrete operators. The constructed virtual dual cell is defined solely from the local connectivity among nodes, and thus the method retains its meshfree nature. To achieve a consistent coupling between velocity and pressure variables under the primal-dual arrangement, time evolution converting is applied to evolve the velocity on cell interfaces. For numerical validation, a linear acoustic equation is solved to confirm the effectiveness of the staggered-variable arrangement based on the local primal-dual grid. Then, incompressible Navier-Stokes equations are solved, and the proposed method is demonstrated to ensure a local divergence-free velocity field up to an arbitrarily small discrete error, achieve the expected spatial convergence order, and accurately reproduce flow features over a wide range of Reynolds numbers.

The anomalous magnetic moment of a lepton encodes the fraction of the lepton's interaction strength with an external magnetic field, which is generated by quantum corrections. Lepton anomalous magnetic moments are sensitive probes of fundamental interactions and play a pivotal role in the quest for "new physics" that may be able to explain the shortcomings of the Standard Model. This chapter introduces the basic concepts and describes the calculation of the individual contributions arising from electromagnetism, the strong and the weak interactions.

Efficient charge-carrier injection from air-stable electrodes into organic semiconductors (OSCs) is essential for fabricating solution-processed organic optoelectronic devices under ambient conditions. Today, this is typically achieved by incorporating doped OSC interlayers, introducing self-assembled dipole monolayers, or adding mobile ions to the active material (AM). Here, we demonstrate an alternative approach that eliminates the need for additional injection layers or ionic additives. We achieve this by blending the dipolar compound TMPE-OH into the electroluminescent polymer Super Yellow (SY) and depositing this sole AM between two air-stable electrodes, forming a single-layer, dipole-doped OLED (D-OLED). By tracking its transient voltage-luminance response, performing impedance spectroscopy, and comparing these characteristics with two other single-layer device concepts, i.e. a neat-SY OLED without a dipolar compound and a light-emitting electrochemical cell (LEC) containing mobile ions, we can establish that the auxiliary dipoles in the D-OLED reorient under the applied driving voltage, enabling immediate luminance turn-on and lowering the injection barriers at both electrodes. Finally, we demonstrate that the D-OLED achieves current efficacies comparable to those of SY OLEDs incorporating dedicated injection layers or LECs. Our study establishes dipolar doping as a practical strategy for efficient bipolar charge injection from air-stable electrodes in solution-processed organic semiconductor devices.

In recent years, the quantum Mpemba effect (QME), which occurs when an out-of-equilibrium system reaches equilibrium faster than another that is closer to equilibrium, has attracted significant attention from the scientific community as an intriguing and counterintuitive phenomenon. It generalizes its classical counterpart by extending the concept beyond temperature equilibration. This paper approaches the QME in Markovian open quantum systems from different perspectives. First, we propose a physical mechanism based on decoherence-free subspaces. Second, we show that an exponential enhancement of the decay rate toward equilibrium, scaling with system size, can be obtained, leading to an extreme version of the phenomenon in Markovian open quantum systems. Third, we study the strong Mpemba effect through the unravelings of Davies maps, revealing subtleties in the choice of figures of merit used to identify the QME. Finally, we propose a microscopic model to gain deeper insight into bath dynamics in this context.

In this work we extend our study of the effect of certain crystallographic defects on the spin-1/2 Kitaev honeycomb spin liquid (arXiv:2511.19409), focusing on its gapless phase and contrasting with the gapped phase. We identify a Stone-Wales (SW) local defect consisting of a 90$^\circ$ bond rotation that preserves Kitaev bond labels for edge-sharing octahedra and thereby enables exact solvability. These SW-type defects involve odd-sided plaquettes with $\pm π/2$ fluxes, but can be created locally. An isolated defect hosts a time-reversal pair of ground-state flux configurations with large net chirality. Certain excitations are also chiral. The chirality manifests in Majorana local Chern marker and in scalar spin chirality, producing electronic orbital magnetization. T-matrix analysis and numerics at finite defect density $n_d$ show that defect chiralities generate a topological gap of $11 n_d$ protecting a Chern number $C=\pm 1$. Emergent ferromagnetic long range Ising interactions $r^{-γ}$ with $2<γ< 3$ between defect chiralities lead to a finite temperature $T_c$ phase transition into the chiral spin liquid. The $T_c$ is proportional to $n_d$ and diverges when $γ\rightarrow 2$. We also consider additional solvable impurity potentials and find that $γ$ can be reduced to below $2.3$ and correspondingly enhance $T_c$. Our results offer applications to 2D Dirac cone systems with a finite density of fluctuating Ising magnetic impurities and to identifying spin liquids with lattice defects.

We investigate gluonic hidden-charm tetraquark states composed of two valence quarks, two valence antiquarks and an explicit valence gluon. In the color configuration $[\bar{3}_c]_{c q}\otimes[8_c]_{G}\otimes[3_c]_{\bar{c}\bar{q}}$, a complete set of eight interpolating currents is constructed for states with quantum numbers $^{PC}=0^{++}$, $0^{-+},$ $0^{--}$, and $0^{+-}$. The corresponding mass spectra are systematically analysed within the QCD sum rule framework, including nonperturbative condensate contributions up to dimension eight. Our numerical analysis indicates the possible existence of six gluonic hidden-charm tetraquark states exhibiting stable behaviour in the adopted Borel windows. By replacing the charm quark with the bottom quark, masses for the corresponding hidden-bottom partners are also estimated. Possible production mechanisms and dominant decay channels are discussed, providing phenomenological guidance for experimental searches. These predicted states may be accessible at current and forthcoming facilities, including Belle II, PANDA, SuperB and LHCb, and thus offer an opportunity to probe explicit gluonic degrees of freedom in multiquark systems and deepen our understanding of nonperturbative QCD.

The paper presents results of investigation of a natural Ib-IaA diamond containing Y-defects from Yubileinaya kimberlite pipe, Yakutia. Analysis of spatial distribution of nitrogen-related A and C centers and intensity of Infra-red absorption at Raman frequency (1332 cm-1) reveals anticorrelation between these defects. Transmission electron microscopy of a zone with abundant Y-defects shows presence of dislocations in various configurations and numerous clusters of point defects generated by non-conservative dislocation motion. Extended defects with shape resembling thin (1-3 nm) rhombic plates with the largest dimension up to 20 nm are observed. Analysis of contrast of these defects shows that they represent nanosized voids (vacancy clusters). It is suggested that the defects were formed by annihilation of dislocation dipoles with subsequent growth by consumption of vacancies produced by non-conservative motion of dislocations. Upon excitation by 787 nm laser, numerous narrow photoluminescecne lines are observed between 800-900 nm; their intensity and position show pronounced spatial heterogeneity on scale of few microns. Qualitatively similar behaviour of photoluminescence was earlier noted for hydrogenated nanodiamonds. It is suggested that unusual behaviour of the luminescence lines may be explained by recombination processes at internal walls of the discovered nanovoids.

We propose an information -- theoretic framework for quantifying Kochen-Specker contextuality. Two complementary measures are introduced: the mutual information energy, a state-independent quantity inspired by Onicescu's information energy that captures the geometric overlap between joint eigenspaces within a context; and an operational measure based on commutator expectation values that reflects contextual behavior at the level of measurement outcomes. We establish a hierarchy of bounds connecting these measures to the Robertson uncertainty relation, including spectral, purity-corrected, and operator norm estimates. The framework is applied to the Klyachko-Can-Binicioğlu-Shumovsky (KCBS) scenario for spin-1 systems, where all quantities admit closed-form expressions. The Majorana-stellar representation furnishes a common geometric platform on which both the operational measure and the uncertainty products can be analyzed. For spin-1, this representation yields a three-dimensional Euclidean-like visualization of the Hilbert space in which, states lying on a plane exhibit maximum uncertainty for the observable along the perpendicular direction; simultaneous optimization across all KCBS contexts singles out a unique state on the symmetry axis. Notably, states achieving the optimal sum of uncertainty products exhibit vanishing operational contextuality, while states with substantial operational contextuality satisfy a nontrivial Robertson bound -- the two extremes are achieved by distinct quantum states.

We propose a new consistency condition for the compatibility of a gravitational effective field theory in AdS with a dual holographic description in terms of a family of large-$N$ CFTs. Using large-$N$ factorization of correlation functions combined with a properly defined notion of single- and multi-particle operators, we argue that the cubic scalar bulk couplings for fields dual to operators with extremal arrangements of the conformal dimensions, i.e. $Δ_i=Δ_j+Δ_k$, should vanish. We apply this criterion to the 4d $\mathcal{N}=1$ effective supergravity theory describing the simplest DGKT AdS$_4$ vacua in type IIA string theory and show that it is non-trivially satisfied. In addition, we calculate explicitly all non-vanishing three-point correlation functions of low-lying scalar operators in the putative 3d CFTs dual to these AdS$_4$ string theory backgrounds.

In this second part of the series, we investigate the uniqueness of positive fixed points of the Poincare map associated with the 3-dimensional Lotka-Volterra competition model with seasonal succession. Building on our first part of the series on the classification of 33 dynamical equivalence classes (regardless of the uniqueness of positive fixed points), we demonstrate in this paper that classes 26 and 27 may indeed exhibit multiple positive fixed points. This reveals a fundamental distinction from both its 2-dimensional analogue and the classical 3-dimensional competitive Lotka-Volterra model. More concretely, by focusing on the model with identical growth and death rates, we establish an equivalent characterization for the (non)uniqueness of positive fixed points. Based on this characterization, we further show that classes 19-25 and 28-33 admit a unique positive fixed point and exhibit trivial dynamics: all trajectories converge to some fixed point (corresponding to harmonic solutions). In contrast, classes 26 and 27 possess richer dynamical scenarios: there can contain a continuum of invariant closed curves, on which orbits may be periodic (corresponding to subharmonic solutions), dense (corresponding to quasi-periodic solutions), or may even consist entirely of positive fixed points (which exhibits the nonuniqueness of positive fixed points).

The description of the internal spaces of fermion and boson fields with "basis vectors", which are the superposition of odd and even products of the operators $γ^a$, offers in $d=2(2n+1)$-dimensions, such as $d=(13+1)$, a unified picture of all so far observed fermions (quarks, leptons, antiquarks and antileptons that appear in families) and bosons (gravitons, photons, weak bosons, gluons and scalars), under the condition that all fields have non-zero angular momenta only in the $d =(3+1)$, $SO(3,1)$, of ordinary space-time. Bosons, which also carry the spatial index $α$ (which is for tensors and vectors $μ=(0,1,2,3)$ and for scalars $σ\ge 5$) appear in two orthogonal groups. In any $d=2(2n+1)$-dimensional space the number of internal states of fermions in all families and their Hermitian conjugate partners is equal to the number of internal states of boson states. The article presents general properties of massless fermion and boson fields and their mutual interactions in this theory, which determine the Lagrangian density of both fields and their interactions. It particularly illustrates "basis vectors" and their properties in $d=(13+1)$ and $d=(5+1)$. The article presents new results and discusses open problems in this theory.

In all previous calculations of the non-local sub-leading contribution to the inclusive penguin decay $\bar B \to X_s γ$ due to the interference of the electroweak operators ${O}_1^c$ - ${O}_{7γ}$ the local Voloshin term was subtracted. In view of the ongoing analysis at order $α_s$, we present a calculation of the complete non-local contribution which takes into account the high correlation between the uncertainties of the local Voloshin and the non-local term of the previous analyses. The new calculation has a high impact on the range of the non-local contribution.

Light scattering by spherical-shaped particles of sizes comparable to the wavelength is foundational in many areas of science, from chemistry to atmospheric science, photonics and nanotechnology. With the new capabilities offered by machine learning, there is a great interest in end-to-end differentiable frameworks for scattering calculations. Here we introduce PyMieDiff, a fully differentiable, GPU-compatible implementation of Mie scattering for layered, spherical particles in PyTorch. The library provides native, autograd-compatible spherical Bessel and Hankel functions, vectorized evaluation of Mie coefficients, and APIs for computing efficiencies, angular scattering, and near-fields. All inputs - geometry, material dispersion, wavelengths, and observation angles and positions - are represented as tensors, enabling seamless integration with gradient-based optimisation or physics-informed neural networks. The toolkit can also be combined with "TorchGDM" for end-to-end differentiable multi-particle scattering simulations. PyMieDiff is available under an open source licence at https://github.com/UoS-Integrated-Nanophotonics-group/MieDiff.