Weyl semimetals and altermagnets represent two distinct classes of quantum materials exhibiting nontrivial topological and magnetic order, respectively. Here we report the realization of a Weyl nodal-loop altermagnet in Cr$_7$Se$_8$, combining neutron diffraction and first-principles calculations. The hexagonal system hosts a coplanar $120^\circ$ compensated magnetic order on a triangular lattice, which breaks inversion-time-reversal and translation-time-reversal symmetries simultaneously while preserving a crystalline mirror plane. The resulting electronic structure features linearly dispersing nodal loops close to the Fermi level ($E_F$) confined to the mirror-invariant $k_z=0$ plane. Along high-symmetry directions, the crossings near $E_F$ form Dirac-like fourfold degeneracies in the absence of spin-orbit coupling; at generic momenta, these crossings split into twofold and form continuous Weyl-like nodal loops protected by mirror symmetry. The momentum-dependent spin polarization exhibits an $f$-wave-like pattern characteristic of odd-parity altermagnets.

In this article, we study the existence of normalized solutions to the following mixed nonlinear Choquard equation with exponential growth \begin{align*} \left\{ \begin{aligned} \mathcal{L}u+λu \; &=\; Λ(I_α\ast F(u))F'(u), \quad \text{in }\mathbb{R}^{2}, \int_{\mathbb{R}^{2}}|u|^{2}\,dx \; &=\; a^{2}, \end{aligned} \right. \end{align*} where $\mathcal{L}= -Δ+(-Δ)^s$, $0<s<1$, $a>0$, $I_α$ is the Riesz potential of order $α\in (0,2)$, $Λ>0$ is a parameter and $λ\in \mathbb{R}$ appears as a Lagrange multiplier. Here, the nonlinearity $F$ has exponential growth in $\mathbb{R}^{2}$. Using variational methods, we prove the existence of normalized solution in the Pohožaev manifold. Moreover, we discuss the regularity result and the construction of the Pohožaev identity, essential for the existence. \keywords{Normalized solutions; Nonlinear Schrödinger equations; Choquard nonlinearity; Critical exponential growth; Trudinger-Moser inequality}

We found sets of exact analytic quasi-stationary states of a massive scalar field in a dyonic Kerr-Sen black hole~(DKSBH) background in the maximally extended spacetime region. A central novelty is the use of horizon-regular ingoing Eddington-Finkelstein coordinates, which enables a direct and unambiguous imposition of the ingoing boundary condition at the horizon. The exact radial solutions are in the form of confluent Heun functions. Imposing regularity at spatial infinity enforces a series truncation condition, yielding an exact quantization of the quasi-stationary frequencies. The spectrum exhibits a rich multi-branch structure, which we show splits into two distinct classes: modes that are insensitive to the black hole spin and charges and modes that explicitly depend on them. We uncover a clear asymmetry between co-rotating and counter-rotating configurations, driven by the spin-angular momentum coupling, as well as a systematic shift of the spectrum induced by electric and magnetic charges. The physical branches exhibit a universal behavior: modes with positive real frequency possess positive imaginary parts and therefore grow exponentially in time, whereas modes with negative real frequency are damped and decay. This suggests that positive-energy excitations in the region behind the outer horizon including the inner region of the inner horizon which contains the closed-timelike-curve, exponentially destabilize the background spacetime, supporting Hawking's chronology protection conjecture. In addition, the purely imaginary modes contain no oscillatory component and hence do not propagate through the spacetime, preventing traveling excitations along closed timelike curves and remaining consistent with the conjecture.

Direct-to-cell (DTC) satellite communication is regarded as one of the most recent technologies that provides global connectivity. However, with the growing number of wireless users and devices, the design of DTC communications must satisfy the requirements of high-scale capabilities and efficient spectrum utilization. To this end, integrating satellite communications with advanced multiple-access techniques, such as non-orthogonal multiple access (NOMA), has attracted considerable interest in developing NOMA-DTC communications. In this article, we first introduce the fundamentals of NOMA-DTC communications, including architectural fundamentals, system design aspects, and potential applications. Given the various cooperative modes and the still-evolving satellite network (SatNet) architectures, such as cooperative SatNets and multi-tier SatNets, we explore protocols that suit future SatNets and enhance system performance. Furthermore, a case study is conducted to investigate the benefits of NOMA schemes for DTC communications and to compare them with OMA schemes. Finally, to inspire further research, several opportunities for NOMA-DTC communications are presented.

For a given subset $S\subseteq V(G)$ of a graph $G$, the pair $(G,S)$ is \emph{knitted} if for every partition of $S$ into non-empty subsets $S_1, S_2, \ldots, S_t$, there exist pairwise disjoint connected subgraphs $C_1, C_2, \ldots, C_t$ in $G$ such that $S_i\subseteq V(C_i)$ for all $1 \le i \le t$. A graph $G$ is \emph{$\ell$-knitted} if $(G,S)$ is knitted for every subset $S\subseteq V(G)$ of size $\ell$. In this paper, we prove that every $8\ell$-connected graph is $\ell$-knitted. We subsequently apply this result to Hadwiger's Conjecture, which states that every $k$-chromatic graph contains a $K_k$-minor. Specifically, we demonstrate that the vertex connectivity of any minimal counterexample to Hadwiger's Conjecture is at least $\lceil k/8 \rceil$, improving upon the previous lower bound of $\lceil 2k/27 \rceil$ established by Kawarabayashi (2007). Our proof corrects a gap in the argument of Kawarabayashi-Yu~(2013) and establishes the claim stated without proof in Liu--Rolek--Yu~(2019).

We provide a systematic framework for computing the logarithmically divergent part of one-loop partition functions on product spaces $\mathrm{AdS}_{d_A} \times S^{d_S}$ of arbitrary dimension. By expanding the higher-dimensional kinetic operators in spherical harmonics, we reduce the ($d_A+d_S$)-dimensional spectral problem to an infinite tower of $d_A$-dimensional determinants, which are then represented via spectral $ζ$-function methods. We isolate the logarithmic divergences arising from the interplay between the individual AdS determinants and the infinite Kaluza-Klein sum, carefully accounting for the contributions of zero modes on the sphere that produce additional AdS determinants. We test this framework on different fields and apply it to the complete multiplet of 11-dimensional supergravity on $\mathrm{AdS}_4 \times S^7$. We recover in a 4d language the result of arXiv:1210.6057, namely that the only non-vanishing logarithmic divergence originates entirely from the 2-form AdS mode in the ghost sector, reproducing the well-known $\frac{1}{4}\log N$ correction to the ABJM free energy predicted by supersymmetric localization.

We formulate a program to prove the categorical local Langlands conjecture (CLLC) of Fargues-Scholze, for all quasisplit $p$-adic groups where the Fargues-Scholze $L$-parameters agree with the semisimplification of a known "automorphic" local Langlands parametrization. A key working hypothesis - which we expect to prove elsewhere jointly with Hamann - is the compatibility of the enhanced Whittaker coefficient functor $c_ψ$ with Eisenstein series. For $\mathrm{GL}_n$, we show that this hypothesis alone implies the full CLLC. For more general groups $G$, we prove an induction principle which reduces CLLC for $G$ to CLLC for all proper Levi subgroups together with a very small amount of information about $G$. This principle applies unconditionally to many classical groups with current technology. Along the way, we establish many foundational results. In particular: - We prove a very strong finiteness theorem for spectral constant term functors. - We prove a spectral analogue of Bernstein's finite global dimension theorem for $p$-adic Hecke algebras. - We introduce and develop the theory of admissible ind-coherent sheaves and admissible duality on derived stacks. - We prove a duality theorem for the spectral action. Using all of these results, we unconditionally define a new and explicit functor $t_ψ$ from the spectral side to the automorphic side, which is defined on enough ind-coherent sheaves to control the entire conjecture.

We compute the mean interval between successive returns of the apparent geocentric solar longitude $λ$ to a fixed value $L \in \{0^\circ, 45^\circ, 90^\circ, \ldots, 315^\circ\}$, averaged over a multi-millennium window; this gives eight ``mean years'' against which calendar leap rules can be tuned: four cardinal-point years (equinoxes and solstices); four cross-quarter years. The construction is built on Meeus's low-precision solar theory (Astronomical Algorithms, 2nd ed., 1998), itself a low-order truncation of Newcomb's Tables of the Sun re-expanded around J2000.0. Where Meeus presents polynomial coefficients without justification, we draw on Smart's Textbook on Spherical Astronomy (6th ed., revised by Green, 1977) for the underlying derivations. Numerical accuracy is validated against the cardinal-point intervals tabulated in Meeus, More Mathematical Morsels, 2002. We close with a derivation of the secular drift equation, showing that, regardless of how well a leap rule is tuned, the slow shrinkage of the tropical year produces a quadratic cumulative error that reaches one day in $\sim$5{,}700 years for any fixed intercalation rule.

We assess the impact of future measurements of dihadron production in semi-inclusive deep-inelastic scattering from the CLAS12 and proposed SoLID experiments at Jefferson Lab, as well as from the ePIC experiment at the future Electron-Ion Collider (EIC), on the transversity parton distribution functions (PDFs) and the corresponding tensor charges of the nucleon. To this end, we generate pseudo-data for these experiments for a proton target (CLAS12 and ePIC) and a $^3$He target (SoLID and ePIC), and we include these pseudo-data in the JAMDiFF global analysis of existing experimental dihadron data. We find that future data from Jefferson Lab will significantly reduce uncertainties in the transversity PDFs in the region of intermediate-to-large quark momentum fractions $x$, while the EIC will provide strong constraints across the entire range of $x$, allowing for the first experimental test of the predicted small-$x$ behavior of the transversity PDFs. In discussing the reduction of uncertainties in the tensor charges, we also compare the results from the data analyses with those from lattice QCD, highlighting scenarios in which compatibility or tension between the two would arise.

Data scarcity limits inference in many scientific and policy domains. Survey data are essential for decision-making, but sparse samples often fail to capture fine spatial granularities. We evaluate normalizing flows, a generative model that learns complex data distributions and can be conditioned on exogenous contextual features, in controlled data scarcity scenarios. Across eight household survey datasets spanning six low-income or middle-income countries in the humanitarian domain, we show that context-conditioned generative models can refine sub-national survey distributions under severe data scarcity, and that performance increases systematically with the richness of the conditioning information. These findings support a general principle for survey data augmentation: generative models can improve sub-national estimates when the sparse sample retains sufficient support and contextual covariates encode relevant local heterogeneity. By learning full conditional distributions rather than point estimates, the approach provides fine-grained evidence for humanitarian decision-making and resource allocation.

We present both static and quasi-static direct numerical simulations of Rayleigh-Bénard convection in a quasi-one-dimensional domain, revealing for the first time a clear subcritical transition to turbulence in a buoyancy-driven flow. Within a narrow range of Rayleigh number (Ra), three coexisting flow states are identified: steady convection, oscillatory chaos, and intermittent turbulence. The transitions between these states are accompanied by abrupt jumps in both the Nusselt number (Nu) and Reynolds number (Re), the key global transport quantities in buoyancy-driven flows. Additionally, they exhibit pronounced hysteresis, forming three distinct hysteresis loops in the Nu-Ra plane: normal, reverse, and anomalous loops. More importantly, we show that the steady convection state is linearly stable against infinitesimal perturbations but can transition to intermittent turbulence when subjected to finite-amplitude disturbances, which is a defining hallmark of subcriticality. Thus, contrary to the prevailing view that the transition from convection to turbulence is supercritical, our results demonstrate that buoyancy-driven turbulence can emerge via a subcritical route, paving the way for a unified framework that describes instability mechanisms in both buoyancy-driven and shear-driven flows.

Photonic time crystals (PTCs) are temporally periodic media whose Floquet spectra can exhibit momentum gaps, parametric amplification, and effective non-Hermitian descriptions, making them an idealized setting for vacuum amplification and nonequilibrium light-matter dynamics. Their classical electrodynamics is now well developed; the quantum side is less so, and this focused review is an attempt to organize what exists. We trace that account from temporal boundaries to homogeneous Floquet media and light-matter dynamics. A single temporal boundary induces Bogoliubov mode mixing and photon-pair creation; in homogeneous bulk media, momentum conservation isolates counter-propagating $(k,-k)$ sectors and yields a two-mode $SU(1,1)$ squeezing structure. Temporal periodicity promotes this to a Floquet problem with band and momentum-gap regimes, compactly described in a fixed Nambu basis. We then relate PTCs to the dynamical Casimir effect and parametric amplification, which share the same pair-creation mechanism but organize it through discrete resonances rather than a momentum-resolved bulk spectrum. We close with light-matter settings: spontaneous-emission decay and modulation-assisted excitation, atom-PTC dynamics, LDOS-based observables and their limits, and finite, dispersive, and experimentally accessible platforms.

This work introduces a self-optimizing virtual processor (VP) for numerical array programs that shifts parallelization from a manual developer task to a cooperative, agent-like runtime mechanism. Instead of relying on centralized task-graph scheduling, static compiler optimization, or explicitly annotated parallel constructs, the VP uses a decentralized network of cooperative execution segments, derived from the stream of numerical instructions and their data dependencies at runtime. Each segment makes only local decisions about when, where, and how to prepare and execute its computation, including task placement, kernel preparation, and data movement. No central scheduler or mapper instance determines the execution globally; instead, scheduling itself is parallelized and distributed over time - asynchronously and strictly dependency driven. The overall execution strategy emerges from concurrently executing local segments, continuously responding to data availability, cost estimates, system state, hardware capabilities, and problem size. While preserving the sequential semantics of the program our VP automatically exploits parallelism across large program regions rather than being limited to individual loop bodies, modules, or explicitly marked parallel sections; developers are not required to design or encode a parallelization strategy. The current VP primarily targets low-latency strong scaling on local heterogeneous hardware, covering workloads from small, latency-sensitive array operations to large data-parallel computations. The current implementation targets the predefined array instruction set of the ILNumerics ONAL domain-specific language, accessible https://github.com/ILNumerics/ILNumerics.ONAL , while the underlying concept is applicable to general array-based numerical programming models such as MATLAB and NumPy.

Variational excited-state density functional theory (DFT) enables the calculation of excited states at a cost comparable to ground-state calculations, but single-configuration approaches often suffer from spin contamination. We implement restricted open-shell Kohn-Sham (ROKS) DFT, which recovers spin-pure singlet excitation energies via the variational minimization of a weighted combination of mixed-spin and triplet configurations, within the plane-wave projector augmented-wave framework of VASP. The energy functional is optimized using a preconditioned conjugate-gradient or a direct inversion in the iterative subspace algorithm, and analytical atomic forces are derived. The implementation is validated for eight organic molecules by comparison to the Q-Chem quantum chemistry code, yielding mean deviations of approximately $30\,\mathrm{meV}$. As a solid-state application, we investigate the three lowest lying excitations of MgO with a neutral oxygen vacancy. For a dielectric-dependent hybrid functional, vertical excitation energies from ROKS and time-dependent density functional theory (TDDFT) differ on average by about $0.21\,\mathrm{eV}$. The Franck-Condon shifts deviate on average by $0.14\,\mathrm{eV}$ between the two methods and mass-weighted displacements between the excited states and the ground state by $0.12\,\mathrm{amu}^{1/2}$ Ang. Additional calculations at the PBE level reveal that these properties depend less strongly on the DFT functional for ROKS than for TDDFT. These results demonstrate that ROKS provides excitation energies and excited-state forces with an accuracy similar to TDDFT while retaining the favorable scaling of ground-state DFT, making it a promising approach for affordable excited-state simulations in extended systems.

We revisit electric and magnetic surface charges and edge modes in four-dimensional Maxwell theory and QED on a spacetime with a finite spatial boundary. Using the S-wall, which implements electromagnetic duality, we clarify the dual structure of surface charges. We show that, for the standard Neumann and Dirichlet boundary conditions, large gauge transformations and the corresponding shifts of edge modes are gauge redundancies rather than physical boundary symmetries. We also consider singular large gauge transformations and interpret them as insertions of Wilson or 't Hooft loops on the boundary. For modified boundary conditions, we show that large gauge transformations can become genuine physical boundary symmetries generated by topological surface operators, and that the corresponding edge modes can become physical. We further construct new boundary conditions that are electromagnetic duals of the modified boundary conditions.

A novel inequality involving the residual entropy and genuine multi-entropy was proposed in \cite{Balasubramanian:2025hxg} for tripartite holographic pure states, using which it was argued, that purely GHZ-like tripartite entanglement is not allowed in holography. In this work, we generalize this holographic signal inequality to mixed states. In a minimal extension, we compute the reflected genuine multi-entropy following \cite{Yuan:2024yfg} and find a class of holographic geometries that violate this minimally extended inequality due to vanishing Markov gap. We can symmetrize this prescription, where instead of computing the residual entropy on the given mixed state $ρ_{ABC}$, we compute it on its canonical purification. The inequality is restored on the canonically purified state, as expected. Finally, we conjecture a new inequality for tripartite holographic states and give supporting evidence.

We study the $ϕ$ meson at finite temperature and finite momentum using QCD sum rules. The presence of medium breaks the Lorentz invariance, and induces distinct in-medium modifications of the transverse and longitudinal modes at finite momentum. We find that, with increasing momentum, the masses of both modes increase and a clear transverse--longitudinal splitting develops. The splitting is found to grow with temperature and to be mainly generated by the dimension-four spin-dependent thermal condensates.

The dynamics of Schrödinger equation with time dependent potentials of general time dependence is considered. It is shown that for localized in space potentials, there is propagation of regularity which is uniformly bounded in higher Sobolev norms. Unlike the cases where the solution scatter, and then propagation is proved via a standard bootstrap argument, the solutions considered here have a part that does not scatter, as expected in general. For this we introduce propagation estimates that work directly in (e.g.) $H^2(\mathcal{R}^3).$

Entanglement is a key resource for fundamental tests of physics and emerging quantum technologies. In quantum optics, two perspectives on entanglement coexist. In the continuous-variable framework, entanglement is understood as holding between optical modes. In contrast, discrete-variable quantum optics focuses on quantum correlations in degrees of freedom such as polarization that label fixed numbers of photons. In this paper, we show that entanglement can transcend this separation. Spontaneous parametric down-conversion inherently generates correlations in optical phase space, photon number, and labelling degrees of freedom simultaneously. In polarization, this structure is traditionally described by macroscopic Bell states. Existing witnesses, however, fail to detect the genuine hybrid entanglement of these states, which goes beyond the continuous-discrete-variable categorization. Here, we lay the groundwork for a general framework unifying continuous- and discrete-variable notions of entanglement. In particular, we derive an operational witness providing a sufficient criterion for genuine hybrid number-polarization entanglement and outline its experimental implementation. Finally, we discuss exemplary states which, together with our results on macroscopic Bell states, motivate a broader classification of genuine hybrid quantum correlations.

The recent cosmological picture contains a significant tension indicating that our standard $Λ$CDM picture may be incomplete. Early Dark Energy models can alleviate the Hubble tension, by assuming an early acceleration that could explain the divergence between the early and late-time cosmological data. We investigate the implications of Early Dark Energy models on the Big Bang Nucleosynthesis processes by considering several cosmological models, including a model assuming a simple cosmological constant, alongside with varying equations of state dark energy models. We construct a simulator through a nested sampling algorithm, with the help of which we estimate the upper bounds for model parameters, and determine the maximum allowable dark energy density contribution during the radiation-dominated era. Our results are obtained through the \href{https://github.com/croi900/eden}{eden} program. We show that for a linear or polytropic equation of state, the dark energy density is constrained to less than $10^{-13}$ MeV$^4$ and $10^{-5}$ MeV$^4$, respectively, at the 95\% confidence level. Furthermore, we identify a temperature-dependent equation of state of dark energy as the most physically compelling framework, which remains consistent with primordial abundances for coupling parameters $\lesssim 10^{-2}$. This model successfully allows for high-temperature deviations from the standard $Λ$CDM expansion history, while rapidly diluting to obtain standard general relativistic results in the weak freeze-out era.

Orbital order describes a quantum state where occupied orbitals line up in a periodic pattern. While orbital physics plays a fundamental and universal role in strongly correlated electron systems, the existence and particularly the band structure fingerprint of orbital order remain a long-standing mystery. Here, we report the discovery of rare earth 5d-orbital order developed by the surface states of intermetallic compound Tb2CoAl4Ge2. Angle-resolved photoemission spectroscopy reveals characteristic nematic features like Fermi surface deformation and band split. These experimental observations can be described by a ferro-orbital order term in the mean-field Hamiltonian. The structural and magnetic origin of such order is excluded by systematic high-resolution neutron powder diffraction and scanning tunnelling microscopy measurements. Our results provide strong evidence for a pure surface orbital order scenario avoiding complications from structural distortion as in colossal magnetoresistance manganites, magnetic order as in iron-based superconductors, and charge transfer p-orbital order in cuprates.

Cosmology where the effective dark energy crosses $w=-1$ can be realized in Horndeski gravity with shift symmetric terms plus a linear potential. We highlight the special role of the nearly conserved scalar charge. The theory is highly predictive for the early phantom behavior and we identify three ways to cross $w=-1$. None of them recreate conditions indicated by current data very well. The major lesson is that such modified gravity with a potential lacking a cosmological constant and only crossing $w=-1$ once (hence the less elaborate models) has difficulty fitting current data. We provide an online interactive application solving the system of evolution equations, for the reader to explore various scenarios at will.

Ensemble forecasts are commonly used to support decision-making and policy planning across various fields because they often offer improved accuracy and stability compared to individual models. As each model has its own unique characteristics, understanding and measuring the value of each constituent model can support the construction of effective ensembles. The R package modelimportance provides tools to quantify how each component model contributes to the accuracy of ensemble performance for both point and probabilistic forecasts. The package supports multiple ensemble methods and multiple model importance metrics. Additionally, the software offers customizable options for handling missing values. These features enable the package to serve as a versatile tool for researchers and practitioners. It helps not only in constructing an effective ensemble model across a wide range of forecasting tasks, but also in understanding the role of each model within the ensemble and gaining insights into individual models themselves. This package follows the 'hubverse' framework, which is a collection of open-source software, tools and data standards developed to promote collaborative modeling hub efforts and simplify their setup and operation. Doing so enables seamless integration and flexibility with other forecasting tools and systems, allowing many analyses to be performed on existing hubs.

We develop dimension-reduction-free tests for the slope function in functional linear regression when the functional regressor may be endogenous or measured with error. The tests are based on a functional moment condition induced by an auxiliary functional variable and do not require estimation of the slope function. This feature is particularly useful in infinite-dimensional settings, where the identification and regularization conditions needed for consistent estimation are often strong and difficult to verify. The proposed procedures remain asymptotically valid under weak or even failed relevance of the auxiliary variable, and they are consistent against fixed alternatives that are detectable through the moment operator. We establish the asymptotic null distribution, consistency against detectable alternatives, and local power under drifting alternatives. We also derive the locally optimal test within a class of weighted test statistics. Feasible critical values for implementation of the tests are obtained from data. Simulations show reliable size control and competitive power, including under weak relevance. We illustrate the method using a functional regression analysis of residential electricity demand and temperature distributions in South Korea.

Starting from the exact Projected Central Limit Theorem on hyperspheres, we rederive the Beta distribution for subsystem occupation probabilities and Lubkin's purity formula from elementary hyperspherical moments, quantifying the finite-size ``platykurtic'' suppression of tails relative to the Gaussian approximation used in standard eigenstate-thermalization and typicality treatments. Our main new result concerns the bipartite quantum mutual information $\langle I(A{:}B)\rangle$ for Haar-random pure states. We show that its full asymptotic expansion in $1/N$ admits a Bernoulli-factorized form in which every order $k \ge 1$ carries the symmetric factor $(d_A^{2k}-1)(d_B^{2k}-1)$ and all higher odd-order corrections vanish identically. Through an exact algebraic reorganization of Page's formula (conjectured in Ref.~\cite{Page1993} and subsequently proven~\cite{Foong1994, SanchezRuiz1995, Sen1996}), we establish that the leading finite-size correction separates into a dominant $\mathfrak{su}(d_A) \otimes \mathfrak{su}(d_B)$ bipartite quantum coherence contribution $(d_A^2 - 1)(d_B^2 - 1)/(2N)$ and a subtracted classical-probability (Cartan $\otimes$ Cartan) contribution $(d_A - 1)(d_B - 1)/(2N)$, and we trace this separation to the difference between diagonal and eigenvalue entropies via Schur's majorisation theorem, with the dimensional counts $(d-1)$ and $(d^2-1)$ acquiring meaning through the Cartan structure of the generalised Bloch decomposition. These results admit a single non-perturbative closed form: the exact typical mutual information factors as $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, with $\mathcal{G}$ given by an explicit Bose--Einstein integral whose asymptotic expansion in $1/N$ reproduces the Bernoulli series.

The average bipartite quantum mutual information $\langle I(A{:}B)\rangle$ of Haar-random pure states can be expressed exactly through Page's formula in terms of digamma functions. We show that this quantity admits a single non-perturbative closed form: $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, where $\mathcal{G}$ is given by an explicit convergent integral over a Bose--Einstein kernel. The overall factor $(d_A^2-1)(d_B^2-1)=\dim[\mathfrak{su}(d_A)]\cdot\dim[\mathfrak{su}(d_B)]$ is exact, not merely asymptotic. The asymptotic expansion of $\mathcal{G}$ in $1/N$ yields a Bernoulli-factorised series whose coefficients involve $ζ(1{-}2k)$; this series diverges, and our integral is its exact Borel sum. The integral representation also makes $\langle I\rangle < (d_A^2{-}1)(d_B^2{-}1)/(2N)$ manifest via a scale-inversion symmetry of the kernel. Our derivation traces the mutual information's structure to an exact decomposition of Page's entropy into a diagonal (Dirichlet) contribution and a Schur-majorisation eigenvalue correction, whose assembly into the mutual information cleanly separates classical from quantum correlations.

We investigate when plane-wave memory admits standard outgoing free data beyond the idealized sandwich-wave approximation. For a Brinkmann plane wave with profile $A(U)$, the commonly used condition $A(U)|_{U\to\infty}=0$ is not sufficient to guarantee ordinary asymptotically free motion. From the integral form of the transverse geodesic equation, we derive weighted decay criteria which divide the asymptotic dynamics into strongly asymptotically free, weakly asymptotically free, and non-asymptotically free motions. These motions are realized explicitly by the new analytical solutions of three typical examples: a Scarf profile, an inverse-cubic profile, and an inverse-square profile. A surprising feature is that the drift correction in the weakly asymptotically free motion affects only trajectories with nonzero outgoing velocity and therefore does not obstruct displacement memory. We further express the classification in terms of the accumulated tidal matrix, showing that it is an intrinsic curvature effect rather than a coordinate artifact.

Resonance phenomena are central to many quantum systems, where resonant states are typically characterized by pole singularities of the S-matrix. In this work, we employ the complex scaling method (CSM) in conjunction with exact WKB analysis to elucidate the geometric structure of scattering problems that encompass both bound and resonant states. By analyzing the continuum spectrum via the exact WKB framework, we derive the S-matrix for the inverted Rosen--Morse potential and reveal its underlying complex-geometric features. Furthermore, we reinterpret the Aguilar--Balslev--Combes theorem, the foundation of CSM, from a geometric perspective, and discuss the physical significance of the Siegert boundary condition within a rigorously defined modified Hilbert space. Our analysis bridges scattering cross-sections and spectral theory, offering new geometric insights into quantum resonance and scattering phenomena.

The 15-minute city concept, which advocates cities where essential services are accessible within a 15-minute walk or bike ride, has gained significant attention in recent years. However, despite being celebrated for promoting sustainability, large-scale empirical evaluations of the effectiveness of the 15-minute concept in reducing emissions remain limited. To address this gap, we investigate whether cities with better walking accessibility to services, such as 15-minute cities, are associated with lower transportation emissions. Analysing 662 cities worldwide, we find that cities with better walking accessibility to services emit less CO2 per capita for transport. An increase of 10 percentage points in the share of residents living in 15-minute accessible areas is associated with an approximate 5% reduction in transport-related CO2 emissions per capita. Moreover, among cities with similar levels of accessibility, those covering larger areas and exhibiting lower population densities tend to emit more. Our findings highlight the effectiveness of decentralised urban planning, especially the proximity-based 15-minute city, in promoting sustainable mobility. At the same time, our results also emphasise the need to integrate local accessibility with urban compactness - both in terms of population density and of urbanised area - to support sustainable mobility.

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, improve on the weighted large sieve inequality beyond what was heuristically accessible by the earlier approach. We also discuss the optimality of this approach.