Fast radio bursts (FRBs) can provide a measure of the Hubble constant $H_0$ that is independent of the constraints set by the cosmic microwave background (CMB) and the type Ia supernovae (SNIa), thereby arbitrating the Hubble tension. In the literature, the methodology proposed by Macquart et al. has been widely used, in which the contributions to the dispersion measure (DM) from the intergalactic medium (IGM, $\rm DM_{IGM}$) and the host galaxy ($\rm DM_{host}$) are described by probability distribution functions. Within the Macquart et al. methodology, it has been found that the parameter $F$, which quantifies the strength of the baryon feedback in galaxies, must be bound by an artificially narrow prior to result in a Hubble constant $H_0$ that is consistent with the ones derived from the CMB and SNIa studies. A recent study using ${\cal O}(100)$ localized FRBs found that this also causes the fraction of baryon mass in the IGM, $f_{\rm IGM}$, to approach its upper bound of 1. In the present work, using 125 localized FRBs, we find an unusually low $H_0$ when using a model with a loose prior on $F$. This model is in fact strongly preferred to the model with the narrow prior when considering the Bayesian evidence and the Akaike and Bayesian information criteria. Instead of modifying $σ_Δ=Fz^{-0.5}$ in the distribution of $\rm DM_{IGM}$, we explore an alternative method of resolving the tension by generalizing the distribution of $\rm DM_{host}$ with varying location and scale parameters $\ell$ and $e^μ$, respectively. We find that $H_0$ can be well consistent with the ones of Planck 2018 and SH0ES for all the models considered in this work, while these generalized models are all strongly preferred to the model with a narrow prior on $F$. Our findings indicate that more realistic distributions of $\rm DM_{host}$ could be the key to using FRBs as an independent measure of $H_0$.

Light and heat drive interfacial chemistry at solid-liquid interfaces, underpinning processes central to sustainable energy conversion, including photoelectrochemical and hydrovoltaic systems. Yet, non-invasive probing of light-induced interfacial dynamics remains challenging due to the weak and spatially complex nature of optical signals. Here, we introduce a nanophotonic platform that enhances second harmonic generation (SHG) from nanostructured interfaces by over two orders of magnitude, enabling real-time, all-optical access to interfacial processes. We develop a rigorous overlap-integral formalism that provides a general quantitative framework for SHG in nanostructured geometries. By accounting for spatially inhomogeneous electromagnetic fields, this approach links the nonlinear response to geometry-dependent near-field and reveals new degrees of freedom, namely independent control of attenuation and phase, which are absent in planar systems. This enables deterministic tuning of surface and electric-field-induced contributions through nanophotonic design. Using in situ SHG at silicon-oxide-electrolyte interfaces, we resolve subtle spectral shifts of ~1.3 nm with electrolyte concentration, indicating coupling between electrical double layer potential and semiconductor polarizability. Under controlled optical excitation, we observe reversible, intensity-dependent modulation of interfacial susceptibility, with a decrease at low intensities consistent with photocharging and an increase at higher intensities due to photothermal effects. These results establish nanophotonic-enhanced SHG as a quantitative and tunable probe of interfacial phenomena, providing a unified framework linking optical response, electrostatics, and geometry, and opening new avenues for controlling interfacial charge and potential with light for applications in energy conversion, catalysis, and nanophotonic devices.

In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo--Pak, proved that the partition function $p(n)$ is eventually log-concave. Inspired by this and other results, Chern--Fu--Tang first conjectured log-concavity of $k$-coloured partitions. Three of the authors and Tripp later proved this conjecture by introducing recursive sequences and a strict inequality for fractional partition functions, giving explicit errors. In this paper, we show that the log-concavity is, in fact, strict for $k\geq 2$. We shed further light on this phenomenon by utilizing Hardy--Littlewood--Pólya's notion of majorizing. We prove that for partitions $\bm{a},\bm{b}$ of $n\in\N$, if $\bm b$ majorizes $\bm a$, then $p_k(\bm{b})>p_k(\bm{a})$. Numerical calculations indicate that our result is sharp.

The article is devoted to the systematic derivation of new representations of the Hu-Meyer formulas. The formula expressing a multiple Wiener stochastic integral through the sum of multiple Stratonovich stochastic integrals and the formula expressing a multiple Stratonovich stochastic integral through the sum of multiple Wiener stochastic integrals are derived for the case of a multidimensional Wiener process. At that several different definitions of the multiple Stratonovich stochastic integral and several variants of sufficient conditions for the validity of the Hu-Meyer formulas are used. In particular, the proof method proposed by the author in 2006 is applied to obtain Hu-Meyers formulas based on generalized multiple Fourier series for the case of a multidimensional Wiener process. Of great importance for the numerical solution of Ito stochastic differential equations is the verification of sufficient conditions for the applicability of the Hu-Meyer formula (based on generalized multiple Fourier series) for the case of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process. In the author's previous works, the indicated conditions were verified for iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case of an arbitrary basis in the Hilbert space) and for iterated Stratonovich stochastic integrals of multiplicities 7 and 8 (the case of two special bases in Hilbert space (the trigonometric Fourier basis and the basis of Legendre polynomials)). Therefore, the results of the article will be usefull for constructing high-order strong numerical methods for non-commutative systems of Ito stochastic differential equations.

We present a general approach to prove the existence, both locally and globally in amplitude, of fully localised multi-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. While one-dimensional localised patterns induced by spatial heterogeneities have been well-studied, proving the existence of fully localised patterns emerging from a Turing instability in higher dimensions remains a key open problem in pattern formation. In order to demonstrate the approach, we consider the two-dimensional Swift--Hohenberg equation, whose linear bifurcation parameter is perturbed by a radially-symmetric potential function. In this case, the trivial state is unstable in a compact neighbourhood of the origin and linearly stable outside. We prove the existence of local bifurcation branches of fully localised patterns, characterise their stability and bifurcation structure, and then rigorously continue solutions to large amplitude via analytic global bifurcation theory. Notably, the primary bifurcating branch in the Swift--Hohenberg equation alternates between an axisymmetric spot and a non-axisymmetric `dipole' pattern, depending on the width of the spatial heterogeneity.

Parallelism has become a central concern in modern decoding frameworks aiming to meet stringent throughput and latency requirements. Guessing Random Additive Noise Decoding (GRAND) is a recently proposed decoding paradigm that tests candidate Error Patterns (EPs) until a valid codeword is found. Among its variants, Soft GRAND (SGRAND) achieves maximum-likelihood (ML) decoding but relies on real-time generation and likelihood ordering of EPs, making parallel execution nontrivial under the ML optimality constraint. In this work, we introduce a unified binary tree representation of EPs, termed the EP tree, which formalizes the hierarchical structure underlying SGRAND and Ordered Reliability Bits (ORB) GRAND algorithms, enabling structured organization of EPs and algorithmic-level parallel exploration. Building upon this unified framework, we propose a parallel design of SGRAND that preserves ML optimality while significantly reducing decoding complexity through pruning strategies and tree-based computation. Furthermore, we develop an enhanced ORBGRAND algorithm based on the same EP tree representation, improving decoding performance toward ML while retaining parallel efficiency. Numerical experiments show that the proposed parallel SGRAND achieves a $3.96\times$ reduction in decoding latency compared with its serial counterpart, while the enhanced ORBGRAND achieves a $4.21\times$ speedup, demonstrating the effectiveness of the unified tree-based framework and its strong potential for future algorithmic and hardware optimizations.

Localized $5d^2$ electrons in a cubic crystal field possess multipoles such as electric quadrupoles and magnetic octupoles. We studied the cubic double perovskite Ba$_2$CaOsO$_6$ containing the Os$^{6+}$ ($5d^2$) ions, which exhibits a phase transition to a `hidden order' below $T^* \sim$ 50 K, by X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD) at the Os $L_{2,3}$ edge. The cubic ligand-field splitting between the $t_{2g}$ and $e_g$ levels of Os $5d$ was deduced by XAS to be $\sim$4 eV. Ligand-field (LF) multiplet calculation under fictitious strong magnetic fields indicated that the exchange interaction between nearest-neighbor octupoles should be as strong as $\sim$1.5 meV if a ferro-octupolar order is stabilized in the `hidden-ordered' state, consistent with the exchange interaction of $\sim$1 meV previously predicted theoretically using model and density functional theory calculations. The temperature dependence of the XMCD spectra was consistent with a $\sim$18 meV residual cubic splitting of the lowest $J_{\rm eff} =$ 2 multiplet state into the non-Kramers $E_g$ doublet ground state and the $T_{2g}$ triplet excited state.

We study the computational complexity of the Local Hamiltonian problem under the promise that its ground state is succinctly represented. We show that the Succinct State 2-Local Hamiltonian problem, for qubit Hamiltonians, is (promise) MA-complete. The approach combines a systematic characterisation of succinct quantum states, defined through arithmetic over specific number fields, with a refined reduction that lowers the locality of Feynman-Kitaev circuit-Hamiltonians from 6 to 2, without increasing particle dimension. This reveals a complexity phase transition, parameterised by locality, and extends the scope of previously known MA-complete problem instances. Our results further clarify how succinctness behaves under circuit-based constructions, and progresses toward a better understanding of the boundary between efficiently describable and efficiently verifiable quantum systems.

The Open Network (TON) blockchain employs an asynchronous execution model that introduces unique security challenges for smart contracts. A primary concern is race conditions arising from unpredictable message processing order. While previous work established vulnerability patterns through static analysis of audit reports, dynamic detection of temporal dependencies through systematic testing remains an open problem. This study proposes a dynamic evaluation methodology based on controlled message orchestration to systematically expose vulnerabilities in asynchronous smart contracts. By synthesizing precise message queue manipulation with differential state analysis and probabilistic permutation testing, we establish a framework (namely, BugMagnifier) for identifying execution flaws that static methods miss. Experimental evaluation demonstrates BugMagnifier's effectiveness through extensive parametric studies on purpose-built vulnerable contracts and five real-world vulnerability cases reproduced from recent security audits. Results reveal message ratio-dependent detection complexity that aligns with theoretical predictions. This quantitative model enables predictive vulnerability assessment while shifting discovery from manual expert analysis to automated evidence generation. By providing reproducible test scenarios for temporal vulnerabilities, BugMagnifier addresses a critical gap in the TON security tooling, offering practical support for safer smart contract development in asynchronous blockchain environments.

We introduce a novel distribution-based estimator for the Hurst parameter of log-volatility, leveraging the Kolmogorov-Smirnov statistic to assess the scaling behavior of entire distributions rather than individual moments. To address the temporal dependence of financial volatility, we propose a random permutation procedure that effectively removes serial correlation while preserving marginal distributions, enabling the rigorous application of the KS framework to dependent data. We establish the asymptotic variance of the estimator, useful for inference and confidence interval construction. From a computational standpoint, we show that derivative-free optimization methods, particularly Brent's method and the Nelder-Mead simplex, achieve substantial efficiency gains relative to grid search while maintaining estimation accuracy. Empirical analysis of the CBOE VIX index and the 5-minute realized volatility of the S&P 500 reveals a statistically significant hierarchy of roughness, with implied volatility smoother than realized volatility. Both measures, however, exhibit Hurst exponents well below one-half, reinforcing the rough volatility paradigm and highlighting the open challenge of disentangling local roughness from long-memory effects in fractional modeling.

Optical frequency combs are utilized in a wide range of optical applications, including atomic clocks, interferometers, and various sensing technologies. They are often generated via four-wave mixing in chip-integrated microring resonators, a method that requires low optical input power due to the high-quality factor of the resonator, making it highly efficient. While the classical properties of optical frequency combs are well established, this work investigates the quantum-mechanical characteristics of the individual comb modes. We derive closed-form analytical expressions describing the squeezing, second-order correlation and joint spectral intensity between the generated signal and idler modes. This comprehensive theoretical framework enables an intuitive understanding and optimization of the quantum features across the comb, revealing conditions for substantial squeezing and entanglement relevant for quantum information processing. Our findings highlight the profound impact of design and dispersion on these quantum properties and offer a foundational tool for chip-integrated quantum applications, including quantum sensing, computing and communication.

The cosmic dipole measured in surveys of cosmologically distant sources is generally found to be in disagreement with the kinematic expectation of the Cosmic Microwave Background (CMB). This discrepancy represents severe tension with the Cosmological Principle and challenges the standard model of cosmology. Here, we present a Bayesian analysis of the tension between datasets used to measure the cosmic dipole. We examine the NRAO VLA Sky Survey (NVSS), the Rapid ASKAP Continuum Survey (RACS) and the Wide-field Infrared Survey Explorer catalogue (CatWISE), and jointly analyse them with the Planck observations of the CMB. Under the kinematic interpretation, we find that Planck is in severe tension with CatWISE above 5$σ$, strong tension with RACS, and moderate tension with NVSS. Moreover, the strong concordance between CatWISE and NVSS suggests that their dipoles arise from a common astrophysical signal. Conversely, the high discordance between RACS and both CatWISE and NVSS indicates a possible systematic difference in the RACS catalogue itself. Whilst the tension between Planck and infrared-selected quasars is already significant, the question of whether or not the dipole in individual radio surveys adds to the challenge against the standard model is yet to be seen. We estimate that $\mathcal{O}(10^6)$ radio sources are required to measure the tension to a significance of 5$σ$. Therefore, in light of the upcoming SKA radio surveys, we are on the cusp of disentangling the anomaly of the cosmic dipole.

Many research questions -- particularly those in environmental health -- do not involve binary exposures. In environmental epidemiology, this includes multivariate exposure mixtures with nondiscrete components. Causal inference estimands and estimators to quantify the relationship between an exposure mixture and an outcome are relatively few. We propose an approach to quantify a relationship between a shift in the exposure mixture and the outcome -- either in the single timepoint or longitudinal setting. The shift in the exposure mixture can be defined flexibly in terms of shifting one or more components, including examining interaction between mixture components, and in terms of shifting the same or different amounts across components. The estimand we discuss has a similar interpretation as a main effect regression coefficient. First, we focus on choosing a shift in the exposure mixture supported by observed data. We demonstrate how to assess extrapolation and modify the shift to minimize reliance on extrapolation. Second, we propose estimating the relationship between the exposure mixture shift and outcome completely nonparametrically, using machine learning in model-fitting. This is in contrast to other current approaches, which employ parametric modeling for at least some relationships, which we would like to avoid because parametric modeling assumptions in complex, nonrandomized settings are tenuous at best. We are motivated by longitudinal data on pesticide exposures among participants in the CHAMACOS Maternal Cognition cohort. We examine the relationship between longitudinal exposure to agricultural pesticides and risk of hypertension. We provide step-by-step code to facilitate the easy replication and adaptation of the approaches we use.

Despite its pervasive implications, many cities worldwide continue to expand in a fragmented, horizontal manner. We analyse urban growth dynamics in 69 Mexican metropolitan areas from 1990 to 2020 using census data, developing a model of urban form change based on population size, density, and spatial configuration. We employ a radial probability density function and the urban expansion factor to create a framework for comparing urban expansion over time and across different regions. Over the past three decades, Mexico's urban population has nearly doubled. However, populations have shifted outward, resulting in a decline of 2.5 million residents in central areas. Our analysis shows that distances from the city centre have increased by 28% on average, driven by population losses in central zones combined with growth in peripheral regions.

Lin, Maldacena, Rozenberg, and Shan (LMRS) presented a new information paradox in black hole physics by noticing that the entanglement and Rényi entropies in a two-sided black hole can become negative when the geometry contains a very large number of matter excitations behind the black hole horizon. While originally this puzzle was presented in the context of BPS two-sided black holes in two-dimensional supergravity, the negativity in fact persists for more general two-sided black holes in the presence of a large number of matter excitations. Since the entanglement and Rényi entropies in ordinary quantum systems cannot be negative, resolving this puzzle is a necessary step towards understanding the quantum mechanical description of black holes. In this paper, we explain how to address the entanglement negativity puzzle, both in the original setting discussed by LMRS and in more general non-supersymmetric settings, by summing over all non-perturbative contributions to the gravitational path integral. We then interpret this result from the point of view of a dual matrix integral, which we use to extend our analysis beyond the regime of validity of the genus re-summation performed in the gravitational path integral. In this regime, positivity is rescued by new saddles of the matrix integral, a one-eigenvalue instanton and a two-eigenvalue instanton. Finally, we formulate a similar puzzle and its resolution using random tensor network techniques.

We incorporate the effect of non-local gravitational self-energy to obtain a neutral, non-singular spacetime geometry. This is achieved by using a non-local gravitational theory inspired by T-duality, where particle mass is not point-like but smeared over a region. This non-local gravitational self-interaction is derived from the Newtonian gravitational potential and energy density, allowing us to define a coordinate-independent quantity. Thus, we incorporate the non-local gravitational field into the spacetime metric. We demonstrate that the total ADM mass is modified by a finite, regularized gravitational mass term, leading to a regular solution of the Ayon-Beato-Garcia type metric but without electric charge. We show the existence of extremal configurations known as \emph{particle-black hole} objects of order of the Planck mass, which are thermodynamically stable, have a vanishing Hawking temperature and could be a viable dark matter candidate.

Text-to-image diffusion models help visualize urban futures but can amplify group-level harms. We propose collective recourse: structured community "visual bug reports" that trigger fixes to models and planning workflows. We (1) formalize collective recourse and a practical pipeline (report, triage, fix, verify, closure); (2) situate four recourse primitives within the diffusion stack: counter-prompts, negative prompts, dataset edits, and reward-model tweaks; (3) define mandate thresholds via a mandate score combining severity, volume saturation, representativeness, and evidence; and (4) evaluate a synthetic program of 240 reports. Prompt-level fixes were fastest (median 2.1-3.4 days) but less durable (21-38% recurrence); dataset edits and reward tweaks were slower (13.5 and 21.9 days) yet more durable (12-18% recurrence) with higher planner uptake (30-36%). A threshold of 0.12 yielded 93% precision and 75% recall; increasing representativeness raised recall to 81% with little precision loss. We discuss integration with participatory governance, risks (e.g., overfitting to vocal groups), and safeguards (dashboards, rotating juries).

Recent high-resolution, high-Reynolds-number simulations have shown that the initial total circulation, quantified by the vorticity packing fraction (VPF), strongly influences the late-time Eulerian statistical equilibria of decaying incom- pressible two-dimensional Navier-Stokes turbulence (Biswas et al., 2022, Physics of Fluids 34, 065101), revealing a transition from point-vortex--dominated to finite-size (patch-vortex) equilibria with increasing vortex packing, and emphasizing the role of of the classical exclusion principle (i.e., incompressibility) and total circulation in determining the final statistical states. The present study examines how the associated Lagrangian tracer transport evolves with VPF across the early (linear-nonlinear turbulence onset), intermediate (turbulence development), and late (coherent dipole evolution) stages, and how it correlates with the corresponding Eulerian states. Turbulence, triggered by the Kelvin-Helmholtz instability and sustained by inverse energy cascades, forms large-scale coherent vortices that govern long-time transport. Tracer dynamics, analyzed via mean-square displacement and position-velocity probability distri- bution functions (PDFs), reveal that increasing VPF accelerates turbulence onset, drives a transition from sub- to super- diffusive transport with decreasing anisotropy in the intermediate stage, and determines late-time behavior dominated by either orbital coherent vortex trapping (sub-diffusive) or linear translational dipole motion (super-diffusive). These dis- tinct long-time transport characteristics, evolving from sub- to super-diffusive behavior with increasing vorticity pack- ing, demonstrate a strong correspondence between the transition from point-vortex- to finite-size-vortex-dominated Eulerian equilibria and the underlying Lagrangian transport in decaying incompressible 2D Navier-Stokes turbulence.

Indirect reciprocity promotes cooperation by allowing individuals to help others based on reputation rather than direct reciprocation. Because it relies on accurate reputation information, its effectiveness can be undermined by information gaps. We examine two forms of incomplete information: incomplete observation, in which donor actions are observed only probabilistically, and reputation fading, in which recipient reputations are sometimes classified as "Unknown". Using analytical frameworks for public assessment, we show that these seemingly similar models yield qualitatively different outcomes. Under incomplete observation, the conditions for cooperation are unchanged, because less frequent updates are exactly offset by higher reputational stakes. In contrast, reputation fading hinders cooperation, requiring higher benefit-to-cost ratios as the identification probability decreases. We then evaluate costly punishment as a third action alongside cooperation and defection. Norms incorporating punishment can sustain cooperation across broader parameter ranges without reducing efficiency in the reputation fading model. This contrasts with previous work, which found punishment ineffective under a different type of information limitation, and highlights the importance of distinguishing between types of information constraints. Finally, we review past studies to identify when punishment is effective and when it is not in indirect reciprocity.

Affine automata provide a finite-state computational model that preserves the linear-algebraic structure of quantum computation while operating entirely over the reals. Recent work has shown that affine automata can far surpass classical probabilistic finite-state verifiers. However, prior constructions relied on arbitrary real-valued transition matrices, leaving open whether the observed power stems from the affine mechanism itself or from computational resources implicitly encoded in irrational or infinite-precision parameters. This paper studies one-way and two-way automata with deterministic and affine states as verifiers in Arthur--Merlin proof systems under the restriction that every affine transition matrix has rational entries, and shows that the resulting rational model still supports the main verification advantages of affine finite-state verification. At the one-way level, we verify benchmark nonregular languages that are provably hard or impossible for classical two-way probabilistic verifiers. At the two-way level, we achieve weak verification of every Turing-recognizable language, strong bounded-error verification for every language in $\mathbf{ATIME}(2^{O(n)})$, and perfect-completeness strong verification for every language in $\mathbf{PSPACE}$. These results establish that the remarkable verification power of affine finite-state automata is structural.

Large language models(LLMs) are increasingly integrated with external systems through the Model Context Protocol(MCP),which standardizes tool invocation and has rapidly become a backbone for LLM-powered applications. While this paradigm enhances functionality,it also introduces a fundamental security shift:LLMs transition from passive information processors to autonomous orchestrators of task-oriented toolchains,expanding the attack surface,elevating adversarial goals from manipulating single outputs to hijacking entire execution flows. In this paper,we identify and characterize a systematic privacy-leakage attack pattern,termed Parasitic Toolchain Attacks,instantiated as MCP Unintended Privacy Disclosure(MCP-UPD). These attacks require no direct victim interaction;instead,adversaries embed malicious instructions into external data sources that LLMs access during legitimate tasks. Unlike traditional prompt injection and tool poisoning attacks,our attack targets the interconnected toolchain itself,assembling multiple legitimate tools into a coordinated workflow whose combined behavior accomplishes malicious objectives. In MCP-UPD,the malicious logic infiltrates the toolchain and unfolds in three phases:Parasitic Ingestion,Privacy Collection,and Privacy Disclosure,culminating in stealthy exfiltration of private data. Our root cause analysis reveals that MCP lacks both context-tool isolation and least-privilege enforcement,enabling adversarial instructions to propagate unchecked into sensitive tool invocations. To assess the severity,we design MCP-SEC and conduct the first large-scale security census of the MCP ecosystem,analyzing 12230 tools across 1360 servers. Our findings show that the MCP ecosystem is rife with real-world exploitable gadgets and diverse attack methods,underscoring systemic risks in MCP platforms and the urgent need for defense mechanisms in LLM-integrated environments.

We derive two fundamental trade-offs for general stochastic limit cycles in the weak-noise limit. The first is the dissipation-coherence trade-off, which was discovered and proved under additional assumptions by Santolin and Falasco [Phys. Rev. Lett. 135, 057101 (2025)]. This trade-off bounds the entropy production required for one oscillatory period using the number of oscillations that occur before steady-state correlations are disrupted. The second is the thermodynamic speed limit, which bounds the entropy production by the Euclidean length of the limit cycle. These trade-offs are obtained by substituting mutually dual observables, derived from the stability of the limit cycle, into the thermodynamic uncertainty relation. This fact allows us to regard the dissipation-coherence trade-off as the dual of the thermodynamic speed limit. We numerically demonstrate these trade-offs using the noisy Rössler model. We also apply the trade-offs to stochastic chemical systems, where the diffusion coefficient matrix may contain zero eigenvalues. Furthermore, we show that the dissipation-coherence trade-off is always achievable by appropriately modifying the diffusion coefficient matrix based on the phase reduction.

The minimum information copula principle initially suggested in \cite{MeeBed97} is a maximum entropy-like approach for finding the least informative copula, if it exists, that satisfies a certain number of expectation constraints specified either from domain knowledge or the available data. We first propose a generalization of this principle allowing the inclusion of additional constraints fixing certain higher-order margins of the copula. We next show that the associated optimization problem has a unique solution under a natural condition. As the latter problem is intractable in general we consider its version with all the probability measures involved in its formulation replaced by checkerboard approximations. This amounts to attempting to solve a so-called discrete $I$-projection linear problem. We then exploit the seminal results of \cite{Csi75} to derive an iterated procedure for solving the latter and provide theoretical guarantees for its convergence. The usefulness of the procedure is finally illustrated via numerical experiments in dimensions up to four with substantially finer discretizations than those encountered in the literature.

We extend Stein's method to include independence with respect to an auxiliary random variable, for any law for which a Stein characterization does exist. This extends the current literature on the problem. Using tools from the Malliavin calculus, an application to the law of the invariant measure of an ergodic diffusion is given to illustrate the theory.

In [26] it had been found that gravitational particle production (to be more specific, gravitational vacuum polarization) results in an effective increase in the directly measured value of the Hubble constant $H_0$ while it does not affect the value of the Hubble constant derived from energy densities $\bar{H}_0$. It had also been pointed out that this may explain why the Hubble constant $H_0$ determined from direct measurements (such as in SN Ia measurements) and the Hubble constant determined from indirect measurements (such as in CMB calculations in the framework of $Λ$CDM) are different. In the present study, first I correct a misidentification in \cite{Erdem-Universe}, namely, $\hat{H}_0=\left(\frac{\bar{H}_0}{H_0}\right)\bar{H}_0$ (rather than $\bar{H}_0$) is the value of the Hubble constant measured in CMB and BAO measurements. Then I extend the analysis to the $σ_8$ tension, and to determination of the Hubble constant through observations of fast radio bursts. It is observed that inclusion of the effect of gravitational vacuum polarization essentially does not neither mitigate nor exacerbate the $σ_8$ tension (while it mitigates or relieves the Hubble tension). This result is significant in the light of the studies in literature that question existence of a true $σ_8$ tension. Moreover, the present framework predicts that the value of the Hubble constant measured in fast radio bursts is $\hat{H}_0$ as in CMB and BAO measurements. This may be checked with observations in future after more precise and conclusive measurements of $\hat{H}_0$, $\bar{H}_0$, $H_0$.

In this paper, we introduce the group version of a Lie-Leibniz triple, which we call a Lie group-rack triple. We define a Lie group-rack triple whose tangent structure is a Lie-Leibniz triple, which is a generalization of an augmented Lie rack whose tangent structure is an augmented Leibniz algebra. We show that any finite-dimensional Lie-Leibniz triple can be integrated to a local Lie group-rack triple by generalizing the integration procedure of an augmented Leibniz algebra into an augmented Lie rack.

Photon-pair sources based on spontaneous four-wave mixing (SFWM) in integrated photonics are often spectrally static. We demonstrate and model a fundamental thermo-optical mechanism that modulates photon-pair generation in amorphous silicon (a-Si) thin films and metasurfaces via SFWM. Femtosecond-pulsed excitation yields g2(0) higher than 400 in unpatterned a-Si, confirming high-purity nonclassical emission. Resonant a-Si metasurfaces produce photon pairs at rates exceeding 3.8 kHz under 0.6 mW pump power through Mie-type modes. Pump absorption induces localized heating that redshifts resonances, altering modal overlap and SFWM efficiency, leading to deviations from the quadratic power scaling expected in the undepleted regime. Coupled electromagnetic and heat-transfer simulations quantitatively reproduce these trends. Polarization-resolved measurements show nearly isotropic nonlinear responses, with 3 times higher third-order susceptibility of a-Si compared to poly-Si. This work positions a-Si as a bright, CMOS-compatible quantum photonics platform and identifies thermo-optical detuning as a key mechanism that should be considered-and potentially harnessed-in integrated photon-pair sources.

We develop an information-theoretic framework for bounding the supremum of stochastic processes, offering a simpler and sharper alternative to classical chaining and slicing arguments for generalization bounds. The key idea is a lifting argument that produces information-theoretic analogues of empirical process bounds, such as Dudley's entropy integral. Lifting introduces permutation symmetry, yielding sharp bounds when the classical Dudley integral is loose. This gives a simple proof of the majorizing measure theorem via the sharpness of Dudley's entropy integral for stationary processes, a result known well before the proof of the majorizing measure theorem. Furthermore, the information-theoretic formulation provides soft versions of classical localized complexity bounds in generalization theory, but is simpler and does not require the slicing argument. We apply this approach to empirical risk minimization over Sobolev ellipsoids, obtaining sharp convergence rates in settings where previous methods are suboptimal.

We study five pencils of projective quartic Delsarte K3 surfaces. Over finite fields, we give explicit formulas for the point counts of each family, written in terms of hypergeometric sums. Over the complex numbers, we match the periods of the corresponding family with hypergeometric differential operators and series. We also obtain a decomposition of the $L$-function of each pencil in terms of hypergeometric $L$-series and Dedekind zeta functions. This gives an explicit description of the hypergeometric motives geometrically realised by each pencil.

Young neutron stars born with magnetic fields $B\gtrsim 10^{16}$ G become hyperactive as the field inside the star evolves through ambipolar diffusion on a timescale $\sim 10^9$ s. We simulate this process numerically and find that it can eject magnetic loops from the star. The internal magnetic field first diffuses to the crust surrounding the liquid core and then erupts from the surface, taking a significant amount of crustal material with it. The eruption involves magnetic reconnection, generating a giant gamma-ray flare. A significant fraction of the eruption energy is carried by the neutron-rich crustal material, which must go through a phase of decompression and nuclear heating. The massive ejecta should produce additional emission components after the giant flare, including radioactively powered gamma-rays, optical emission, and much later a radio afterglow. The predicted eruptions may rarely happen in observed magnetars in our galaxy, which are relatively old and rarely produce giant flares. The model can, however, explain the extremely powerful flare from SGR 1806-20 in December 2004, its ejecta mass, and afterglow. More active, younger magnetars may produce frequent crustal eruptions and form unusual nebulae. Such hyperactive magnetars are candidates for the central engines of cosmological fast radio bursts (FRBs). We argue that each eruption launches an ultrarelativistic magnetosonic pulse leading the ejecta and steepening into a relativistic shock capable of emitting an FRB.