Fast Radio Bursts (FRBs) carry fossil information about non-Gaussianity generated during inflation. This primordial signal is most accessible on the largest scales, where the scale-dependent bias correction $\propto f_\mathrm{NL}\,H_0^2/k^2$ dominates, but where systematic effects are also strongest. A central challenge is the degeneracy between the intergalactic-medium electron bias $b_e$ and the primordial non-Gaussianity (PNG) signal, which can degrade $σ(f_\mathrm{NL})$ by orders of magnitude when $b_e$ is marginalised. We show this degeneracy can be broken internally by exploiting the cross-power spectrum $C_\ell^{DΔt}$ between the FRB dispersion measure (DM) field and Shapiro timing delays along multiple interferometric sightlines. The DM field traces the biased electron density, while the Shapiro timing signal probes the Newtonian gravitational potential independently of astrophysical bias. Their cross-correlation is directly proportional to $b_e$, independently of the matter power spectrum, providing a self-calibration of the electron bias. We derive $C_\ell^{DΔt}$ analytically in the Limber approximation and find a correlation coefficient $|ρ(\ell)|\approx 0.51$--$0.79$ across $\ell = 2$--$100$. A joint Fisher matrix analysis over $\{f_\mathrm{NL},\,b_e^0,\,z_\mathrm{fb}\}$ shows that including the cross-spectrum reduces $σ(b_e^0)$ by a factor of $2.1$--$5.1$ relative to a DM-only analysis. After full marginalisation, the joint analysis recovers $σ(f_\mathrm{NL})$ within a factor of $1.0$--$1.9$ of the fixed-bias benchmark, compared with $1.7$--$3.3$ degradation without the cross-spectrum. For a shallow survey with a 500\,AU baseline and $10^4$ FRBs, the joint constraint achieves $σ(f_\mathrm{NL})\approx 790$, within 4\% of the fixed-bias result and a factor $3.3$ better than the marginalised DM-only case.

Motivated by recently discovered connections between matroid depth measures and block-structured integer programming [ICALP 2020, 2022], we undertake a systematic study of recursive depth parameters for matrices and matroids, aiming to unify recently introduced and scattered concepts. We propose a general framework that naturally yields eight different depth measures for matroids, prove their fundamental properties and relationships, and relate them to two established notions in the field: matroid branch-depth and a newly introduced natural depth counterpart of matroid tree-width. In particular, we show that six of our eight measures are mutually functionally inequivalent, and among these, one is functionally equivalent to matroid branch-depth and another to matroid tree-depth. Importantly, we also prove that these depth measures coincide on matroids and on matrices over any field, which is (somehow surprisingly) not a trivial task. Finally, we provide a comparison between the matroid parameters and classical depth measures of graphs.

Database theory is exciting because it studies highly general and practically useful abstractions. Conjunctive query (CQ) evaluation is a prime example: it simultaneously generalizes graph pattern matching, constraint satisfaction, and statistical inference, among others. This generality is both the strength and the central challenge of the field. The query optimization and evaluation problem is fundamentally a "meta-algorithm" problem: given a query $Q$ and statistics $\cal S$ about the input database, how should one best answer $Q$? Because the problem is so general, it is often impossible for such a meta-algorithm to match the runtimes of specialized algorithms designed for a fixed query -- or so it seemed. The past fifteen years have witnessed an exciting development in database theory: a general framework, called PANDA, that emerged from advances in database theory, constraint satisfaction problems (CSP), and graph algorithms, for evaluating conjunctive queries given input data statistics. The key idea is to derive information-theoretically tight upper bounds on the cardinalities of intermediate relations produced during query evaluation. These bounds determine the costs of query plans, and crucially, the query plans themselves are derived directly from the mathematical proof of the upper bound. This tight coupling of proof and algorithm is what makes PANDA both principled and powerful. Remarkably, this generic algorithm matches -- and in some cases subsumes -- the runtimes of specialized algorithms for the same problems, including algorithms that exploit fast matrix multiplication. This paper is a tutorial on the PANDA framework. We illustrate the key ideas through concrete examples, conveying the main intuitions behind the theory.

Blockchain ecosystems face a significant issue with liquidity fragmentation, as applications and assets are distributed across many public chains with each only accessible by subset of users. Cross-chain communication was designed to address this by allowing chains to interoperate, but existing solutions limit communication to directly connected chains or route traffic through hubs that create bottlenecks and centralization risks. In this paper, we introduce xRoute, a cross-chain routing and message-delivery framework inspired by traditional networks. Our design brings routing, name resolution, and policy-based delivery to the blockchain setting. It allows applications to specify routing policies, enables destination chains to verify that selected routes satisfy security requirements, and uses a decentralized relayer network to compute routes and deliver messages without introducing a trusted hub. Experiments on the chains supporting the Inter-Blockchain Communication (IBC) protocol show that our approach improves connectivity, decentralization, and scalability compared to hub-based designs, particularly under heavy load.

Let $E_1,\dots,E_n \subset \mathbb{R}^d$ be compact sets of positive diameter with Feng--Wu thickness at least $c>0$. Feng and Wu proved that $E_1+\cdots+E_n$ has non-empty interior when $n>2^{11}c^{-3}+1$. We show that \[n>\frac{\sqrt d}{(\sqrt{1+c}-1)^2}=\frac{\sqrt d\,(\sqrt{1+c}+1)^2}{c^2}\] already suffices. In particular, since $0<c\le 1$, the bound $n>6\sqrt d\,c^{-2}$ is enough. For fixed dimension $d$, this improves the exponent in $c^{-1}$ from $3$ to $2$, while introducing only an explicit factor of $\sqrt d$. The proof replaces the one-summand-at-a-time enlargement of Feng--Wu by a simultaneous convexification step based on a radius form of the Shapley--Folkman theorem.

Recently, Yamaguchi and Kempf [Phys. Rev. Lett. 136:010801, arXiv:2501.02757] proved that encrypted qubits can be cloned. In this work, we generalize the encrypted cloning protocol and prove that it also applies to higher-order quantum systems. Given that a straightforward generalization of the protocol using the exponential of the shift and phase operators fails to satisfy the unitary requirement for a quantum gate, we propose a different approach. We introduce a new operator to be used in the encryption process and show that it is unitary. We adapt the decryption operator from the reference paper to fit in the framework of multi-level quantum systems. We analyze the circuit implementation of the proposed operators and show that the overhead imposed by larger dimensions scales linearly with qudit dimension.

Considering $Z_4$ symmetry in Type I seesaw scenario, one could obtain mass-squared differences of light neutrinos, mixings and $CP$ violating phase within $3 σ$ confidence level based on neutrino oscillation data. This is possible with only three independent complex parameters for allowed Yukawa couplings and one real mass parameter for heavy right handed neutrino fields around electroweak scale. After considering only three more real parameters as coming from small soft-symmetry breaking terms, the lightest right handed neutrino could be considered as dark matter candidate via freeze-in mechanism and the other two heavier right handed neutrinos through their decays, could generate the baryonic asymmetry of the universe naturally via resonant leptogenesis.

At least one large-body (diameter > 1.1 km) hypervelocity cratering event occurred during ~ 0.8-0.90 Ma (Zhamanshin, Kazakhstan) in the Middle Pleistocene Transition period. Analysis designed to reduce uncertainty in the dimensions of the Zhamanshin structure employing high resolution topography demonstrated that it likely generated a ~ 26.5 km diameter multi-ring crater. This is at least two times larger than the current best estimates. Using a range of accepted impactor sizes, velocities, compositions, and angles of impact, such impacts typically yield kinetic energies of impact over 240,000 Megatons (TNT). Explosive energetic events of this magnitude (e.g., Yellowstone Caldera) at other times (K-Pg) have created global environmental effects. The factor of two discrepancy in the dimensions of Zhamanshin increases the kinetic energy yield by factors of 7-10, with significantly larger environmental consequences. This justifies examination of rapid climate transitions linked to biological consequences, including those related to environmental perturbations, at ~0.9 Ma.

Intrinsic topological superconductors with protected boundary modes obeying non-Abelian statistics constitute a vanishingly small class of quantum materials. A defining spectroscopic signature of such phases is the presence of in-gap topological surface states (TSS). However, despite extensive theoretical proposals, their unambiguous experimental identification has remained elusive. Here we use vector magnetic-field scanning tunnelling microscopy to obtain direct spectroscopic evidence of TSS in the spin-triplet superconductor UTe2. Atomic-scale spectroscopy reveals striking site-dependent superconductivity: Te sites host a large in-gap density of states that nearly fills the superconducting gap, whereas neighboring atomic sites remain gapped. Upon application of a magnetic field, the in-gap states on the Te sites are selectively suppressed, yielding a spatially homogeneous superconducting state with a markedly deeper gap relative to zero field. This site-selective gap evolution is in quantitative agreement with theoretical predictions for TSS in UTe2 that possess dominant Te-orbital character. Spectral-function calculations incorporating the Zeeman coupling reproduce the observed magnetic-field response. Our results provide a spectroscopic fingerprint of the long-sought TSS in superconductors and establish UTe2 as a compelling system for exploring intrinsic topological superconductivity.

We give counterexamples to a problem of M. Kac in the Scottish Book, which asks whether a certain nonlinear operation on two characteristic functions characterizes Laplace distributions, in analogy with the Cramér--Lévy theorem for Gaussian distributions. We then give an affirmative answer to a refined version of the problem. Finally, we develop a general framework for such characterization problems, construct generalized counterexamples, and pose some open questions.

Motivated by the study of unlikely intersection in the moduli space of rational maps, we initiate our investigation on algebraic dynamics for families of regular polynomial skew products in this article. Our goals are threefold. (1) We classify special loci -- which contain a Zariski dense set of postcritically finite points -- in the moduli space of quadratic regular polynomial skew products. More precisely, special loci include families of homogeneous polynomial endomorphisms, families of split endomorphisms, and polynomial endomorphisms of the form $(x^2,y^2+bx)$ up to conjugacy. As a consequence, we verify a special case of a conjecture proposed by Zhong. (2) Let $F_t$ be a family of regular polynomial skew products defined over a number field $K$ and let $P_t, Q_t\in K[t]\times K[t]$ be two initial marked points. We introduce a good height $h_{P_t}(t)$ which is built from the theory of adelic line bundles for quasi projective varieties. We show that the set of parameters $t_0\in \overline{K}$ for which $P_{t_0}$ and $Q_{t_0}$ are simultaneously $F_{t_0}$-preperiodic is infinite if and only if $h_{P_t}=h_{Q_t}$. (3) As an application of $h_{P_t}$, we show that, under some degree conditions of $P_t$, if there is an infinite set of parameters $t_0$ for which the marked point $P_{t_0}$ is preperiodic under $F_{t_0}$, then the Zariski closure of the forward orbit of $P_t$ lives in a proper subvariety of $\mathbb{P}^2$. As a by-product, we conditionally verify a special case of a conjecture of DeMarco--Mavraki which is a relative version of the Dynamical Manin--Mumford Conjecture.

Ferrimagnetic spinel materials of formula AB2X4, where A and B are transition metals and X is oxygen or sulphur, hold promise for the realization of multiferroic characteristics. In this work, we report synthesis of spinel CoMn2O4 and explore its magnetic, dielectric, and ferroelectric aspects and their correlations. Polycrystalline CoMn2O4 was synthesized by using the conventional solid-state method. The X-ray diffraction (XRD) and Raman spectroscopy confirmed the phase purity of the synthesized compound. The crystal structure was identified with tetragonal symmetry (I41/amd space group). DC magnetization measurements indicate two magnetic transitions: one at temperature T1 ~ 186 K, followed by another Yafet-Kittel (YK) ferrimagnetic transition at T2 ~ 86 K. A frequency independent anomaly in the temperature dependent dielectric permittivity is observed near the low magnetic ordering temperature (T2). This reflects the possibility of the correlation between lattice dynamics and spin ordering in spinel CoMn2O4. A substantial exchange bias was also observed below T2 ~ 86 K. The change in dielectric permittivity in the presence of applied magnetic field follows the square of the magnetization dependence, which is consistent with Ginzburg-Landau theory. However, the detailed pyroelectric current measurements reveal the absence of intrinsic ferroelectric order.

Given a characteristic initial value problem with smooth data representing a dynamical event horizon settling down to that of Kerr in the subextremal, strictly rotating range with suitable upper and lower bounds, we prove that a weak null singularity forms, across which the spacetime metric is continuously extendible but not Lipschitz extendible. The bulk of the proof is a stability argument showing that a dynamical Teukolsky field can be approximated by a linear Teukolsky field, whose linear instability was proved in previous works.

Quantum coherence provides a controllable thermodynamic resource that can raise or lower the effective temperature of a cavity mode, enabling efficiency tuning in quantum heat engines. Here, we derive analytic expressions for the effective engine temperature, demonstrating the enhanced temperature tunability achievable via $N$-level ground-state coherence. We further unify ground- and excited-state coherence within a single analytic framework, revealing their interplay as a mechanism for thermodynamic control. Such quantum resources serve as tunable parameters that enable switching between heating, cooling, and cancellation regimes, driving the effective temperature from near-zero to divergence. Ultimately, our framework connects and generalizes previous models of quantum heat engines, and we identify rubidium atoms as a promising candidate for experimentally realizing these coherence-assisted effects.

Translating statistical methods into reliable software is a persistent bottleneck in quantitative research. Existing AI code-generation tools produce code quickly but cannot guarantee faithful implementation -- a critical requirement for statistical software. We introduce StatsClaw, a multi-agent architecture for Claude Code that enforces information barriers between code generation and validation. A planning agent produces independent specifications for implementation, simulation, and testing, dispatching them to separate agents that cannot see each other's instructions: the builder implements without knowing the ground-truth parameters, the simulator generates data without knowing the algorithm, and the tester validates using deterministic criteria. We describe the approach, demonstrate it end-to-end on a probit estimation package, and evaluate it across three applications to the authors' own R and Python packages. The results show that structured AI-assisted workflows can absorb the engineering overhead of the software lifecycle while preserving researcher control over every substantive methodological decision.

We extend a recently developed Big Bang Nucleosynthesis (BBN) code, {\tt PRyMordial}, to constrain a broad class of Grand Unified Theories to which BBN is sensitive, since these lead to varying fundamental couplings. A previously developed self-consistent perturbative analysis of the effects of these variations has been implemented in {\tt PRyMordial}, leading to robust constraints of the value of the fine-structure constant, $α$, at the BBN epoch using current observations of Helium-4 and Deuterium abundances. We explored two different viable scenarios, relying on alternative assumptions on the gravitational sector: the variation of the gravitational coupling can be implemented by varying either particle masses, or Newton's gravitational constant. For the variation of masses, we obtained at $68\%$ confidence level a constraint on the relative variation of $α$, between the BBN epoch and the present-day laboratory value, of $Δα/α=2\pm51$ ppm (parts per million), while for the variation of Newton's constant the analogous constraint is $Δα/α=2\pm22$ ppm. We also show that, given these constraints, these models do not provide a solution to the cosmological Lithium problem.

Joint analyses of high-resolution CMB temperature maps with galaxy surveys provide a unique way to reconstruct the radial velocity field of the underlying matter distribution via the kinematic Sunyaev-Zeldovich (kSZ) effect. Using data from the Atacama Cosmology Telescope (ACT) DR6 and the Dark Energy Spectroscopic Instrument (DESI) DR2, we present radial velocity reconstructions for luminous red galaxies (LRGs), emission-line galaxies (ELGs), and quasars (QSOs). Leveraging the spectroscopic data, we are able to reliably model the foreground contamination and report a negligible impact on our main observables. We detect the velocity-galaxy cross-correlation at $17.0σ$ for LRGs, and for the first time, at $8.3σ$ for ELGs and $6.8σ$ for QSOs. We further report the first detection of the velocity-velocity correlation using LRGs at $3.1σ$, as well as the highest cumulative detection of the kSZ effect to date at $20.8 σ$. Similarly to previous results, we find a lower amplitude of the kSZ signal compared to our fiducial halo model prediction and electron profile assuming a Battaglia profile. Combining these new observables, we obtain constraints on local-type primordial non-Gaussianity (PNG): $f_{\rm NL}^{\rm loc} = 15.9_{-34.4}^{+34.6}$ at 68\% confidence, which represents the tightest constraint to date derived from the velocity field. The measurements presented here already exhibit lower noise on a per-mode basis than the galaxy auto-correlation on the largest scales, $k<0.004~\rm{Mpc^{-1}}$, highlighting the key role these observables will play in the context of future CMB experiments such as the Simons Observatory.

We simulate the dynamics of pelagic \emph{Sargassum} rafts as systems of finite-size floating particles, governed by a Maxey--Riley law with nonlinear elastic interactions. Using surface ocean currents and wind data from reanalysis systems for clump transport, we computed trajectories within a domain covering the tropical and subtropical north Atlantic. The subsequent motion is reduced using Ulam's discretization method into a time-inhomogeneous Markov chain that simulates a background \emph{Sargassum} concentration. Bayesian inversion, combined with nonautonomous transition path theory, was used to infer the origin of the first significant recorded bloom in the tropical North Atlantic, which unfolded in April 2011. Both methodologies independently identified the bloom's origin as near the West African coast, up to two years before it was detectable via satellite imagery on the basin's western side. This finding supports anecdotal evidence of \emph{Sargassum} strandings on the Ghanaian coast in 2009. Moreover, it correlates with unusual environmental conditions -- such as increased nutrient loads from significant upwelling linked to a pronounced Dakar Niña and Saharan dust deposition -- that promote bloom proliferation. Additionally, it aligns with the observation that the species of \emph{Sargassum} in the 2011 bloom differ from those in the Sargasso Sea, which might otherwise be considered a natural origin.

We introduce the Legendre bundle, a geometric structure encoding the essential duality of dually flat (Hessian) manifolds, and demonstrate that both exponential families in information geometry and a natural class of quantum field theories -- which we term Hessian QFTs -- arise as distinct realisations of this single framework. The Legendre bundle is shown to carry a canonical para-Kähler structure.

We present a thorough magnetization and magneto-transport study of the diluted Fe-intercalated CrTe2 family member, Fe1/5CrTe2, a van der Waals ferromagnet. Fe1/5CrTe2 shows an elevated Curie transition temperature of 182 K in comparison to the Fe1/3CrTe2 composition, indicating the sensitive role of Fe concentration in modulating magnetic exchange interactions within the CrTe2 framework. The saturated magnetization exhibits a quadratic dependence with temperature, indicating the presence of long-wavelength spin fluctuations. Analysis of the temperature dependent resistivity reveals a dominant T3/2 contribution over the typical T2 behavior, signaling substantial coupling between conduction electrons and localized spins. The magnetoresistance shows a linear and non-saturating negative field dependency throughout a wide temperature range below TC, which is compatible with the increasing suppression of spin-disorder dispersion related to ferromagnetic spin fluctuations. A thorough analysis of the anomalous Hall effect (AHE) shows that extrinsic skew-scattering contribution, which is associated to Fe-related disorder, dominates the anomalous Hall response. The systematic separation of intrinsic and extrinsic components reveals that, over a wide temperature range, the intrinsic anomalous Hall conductivity scales linearly with the saturation magnetization, despite the substantial extrinsic dominant background. The linear behavior of intrinsic anomalous Hall conductivity with magnetization is in line with a long wavelength spin-fluctuation framework, where thermal spin disorder lowers net magnetization without significantly altering the underlying electronic structure. These findings reveal Fe1/5CrTe2 as a newly investigated van der Waals ferromagnet where spin fluctuations and disorder coexist with a well-defined intrinsic Berry-curvature contribution to the Hall response.

We show an example of a function and a collision kernel for which the entropy production increases in time when we flow it by the space-homogeneous Boltzmann equation. The collision kernel is not any of the physically motivated kernels that are commonly used in the literature. In this particular setting, our result disproves a conjecture of McKean from 1966.

We investigate the impact of the quantum effects of protons on the electronic structure of high-pressure H$_3$S, a benchmark hydrogen-rich superconductor with a critical temperature ($T_c$) exceeding 200 K. Using Nuclear-Electronic Orbital Density Functional Theory (NEO-DFT), we treat hydrogen nuclei quantum mechanically on the same footing as electrons within a first-principles framework. Our calculations reveal that nuclear quantum effects (NQEs) induce subtle modifications to the electronic band structure and density of states (DOS) near the Fermi energy, including features associated with van Hove singularities. However, the resulting changes in the DOS would increase $T_c$ by only a few percent. On the other hand, calculations of the phonon dispersion with the NEO-DFT method show large changes in the hydrogen-dominated phonons that arise from a stiffening of the S-H bonds due to NQEs. These findings imply that the experimentally observed reduction in $T_c$ upon deuteration arises predominantly from changes in the phonon properties, while NQEs-induced modifications to the electronic structure itself are minimal.

We propose a globally-admissible phenomenological spectral density of the bath for the non-Markovian Brownian motion of an optomechanical resonator, motivated by the near-resonance experimental observation of a non-Ohmic spectrum in [Nat. Commun. 6, 7606 (2015)]. To avoid divergences arising from a naive global extrapolation, we construct this phenomenological bath spectral density that reproduces the observed local-power-law behavior near the mechanical resonance while remaining well defined globally, ensuring the finiteness of the bath-induced renormalizations and quadrature fluctuations of the resonator. The corresponding model of the structured environment produces a nonlocal mechanical susceptibility whose analytic pole structure encodes the observed linewidth. The resulting dissipation kernel exhibits a power-law-modulated exponential decay with transient negativity, signaling strong memory effects. In the weak-coupling regime, the optical readout based on homodyne detection enables near-resonance spectroscopy and, with a calibrated drive on the resonator, permits, in principle, the reconstruction of the full mechanical susceptibility, thereby providing access to both the dissipative and dispersive bath contributions. Our results provide a consistent route from locally-inferred spectral properties to globally-admissible open-system descriptions and establish a framework for probing structured environments in cavity optomechanics.

Integer Quadratic Programming (IQP), $\min\{x^T Q x + c^T x : Ax \le b,\, x\in\Z^n\}$, is a fundamental problem in combinatorial optimization. While the convex and concave special cases admit polynomial-time algorithms for fixed~$n$, the general indefinite case is considerably harder: it was only recently shown to lie in NP, and the FPT algorithm, due to Lokshtanov, establishes fixed-parameter tractability parameterized by $n$ and the largest coefficient~$L$ without giving an explicit running time. We give the first single-exponential algorithm for IQP, solving it in time $ \bigl(n\,L^n_A\,Δ(A)\,L_Q\bigr)^{O(n)}\cdot\mathrm{poly}(φ), $ which is $(nL)^{O(n^2)}\cdot\mathrm{poly}(φ)$ in general using the same parameterization. We achieve improvements for structured cases like total unimodularity and further state explicit complexity results for a number of FPT algorithms and optimization problems. The single-exponential bound is achieved via curvature batching: we classify kernel directions by the sign of their quadratic curvature and observe that when no negative-curvature direction exists, all gradient constraints can be imposed simultaneously in a single batch. This replaces the chain of determinant squarings inherent in sequential branching with a single polynomial inflation, after which the remaining problem is an ILP. As a secondary contribution, we give an explicit bound for concave integer minimization over a polytope $\{Ax \le b\} \cap \Z^n$ whose parametric complexity depends only on the constraint matrix~$A$ and is independent of the right-hand side~$b$.

Public attitudes toward artificial intelligence (AI) and driving safety are typically studied in isolation using variable-centered methods that assume population homogeneity, yet risk perception theory predicts that these evaluations covary within individuals as expressions of underlying worldviews. This study identifies latent profiles of AI risk perception among U.S. adults and tests whether these profiles are differentially associated with community driving safety concerns. Latent class analysis was applied to nine AI risk-perception indicators from a nationally representative survey (Pew Research Center American Trends Panel Wave 152, n = 5,255); Bolck-Croon-Hagenaars corrected distal outcome analysis tested class differences on nine driving-safety outcomes, and survey-weighted multinomial logistic regression identified demographic and ideological predictors of class membership. Four classes emerged: Moderate Skeptics (17.5%), Concerned Pragmatists (42.8%), AI Ambivalent (10.6%), and Extreme Alarm (29.1%), with all nine driving-safety outcomes significantly differentiated across classes. Higher AI concern mapped monotonically onto greater perceived driving-hazard severity; the exception, comparative evaluation of AI versus human driving, was driven by trust rather than concern level. The cross-domain covariation provides person-level evidence for the worldview-based risk structuring posited by Cultural Theory of Risk and yields a four-class segmentation framework for AV communication that links AI risk orientation to transportation safety attitudes.

In this addendum we extend Theorem 4.6 on the negative binomial distribution in `Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics' (Journal of Mathematical Biology 92:40, 2026; arXiv:2503.21403). We prove that the fractional linear lower bound to the negative binomial generating function derived there is indeed valid for every $x\in[0,1]$, and not only for $x\in[0,P^\infty_{\rm NB}]$, where $P^\infty_{\rm NB}$ is the extinction probability of the associated Galton-Watson process.

We present a comprehensive Bayesian study of the $B_c$ meson spectrum using non-relativistic Cornell and logarithmically modified Cornell potentials, introducing the logarithmic term as the minimal deformation that preserves short-range Coulombic and long-range linear confinement while adding controlled flexibility at intermediate distances to probe the sensitivity of higher excited states to the confining form. Model parameters are sampled via Markov chain Monte Carlo (MCMC), enabling rigorous propagation of correlated uncertainties to all predictions. Spin-dependent interactions are treated perturbatively, with unequal heavy-quark masses accounted for consistently. Both potentials reproduce the known states within uncertainties, with small errors for low-lying states that grow for higher radial and orbital excitations. Analyzing radial and orbital Regge trajectories using linear and nonlinear parametrizations, we observe pronounced nonlinearity for low $S$-waves trending toward linearity at higher excitations. The modified potential yields modest, systematic shifts in higher excited states, reflecting the logarithmic correction's impact. We provide updated theoretical predictions for excited $B_c$ states with uncertainties, serving as benchmarks for ongoing and future experiments.

The vertices of a $k$-token graph of a graph $G$ correspond to $k$ indistinguishable tokens placed on $k$ different vertices of $G$. Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex of $G$, we define a generalization of token graphs, which we call generalized token graphs or simply supertoken graphs, which have different applications. Depending on the above conditions, different families of graphs (such as the Cartesian $k$-th power of $G$ by itself) are obtained, and we present some of their properties, including order, size, and connectivity.

In the contest design problem, there are $n$ strategic contestants, each of whom decides an effort level. A contest designer with a fixed budget must then design a mechanism that allocates a prize $p_i$ to the $i$-th rank based on the outcome, to incentivize contestants to exert higher costly efforts and induce high-quality outcomes. In this paper, we significantly deepen our understanding of optimal mechanisms under general settings by considering nonconvex objectives in contestants' qualities. Notably, our results accommodate the following objectives: (i) any convex combination of user welfare (motivated by recommender systems) and the average quality of contestants, and (ii) arbitrary posynomials over quality, both of which may neither be convex nor concave. In particular, these subsume classic measures such as social welfare, order statistics, and (inverse) S-shaped functions, which have received little or no attention in the contest literature to the best of our knowledge. Surprisingly, across all these regimes, we show that the optimal mechanism is highly structured: it allocates potentially higher prize to the first-ranked contestant, zero to the last-ranked one, and equal prizes to the all intermediate contestants, i.e., $p_1 \ge p_2 = \ldots = p_{n-1} \ge p_n = 0$. Thanks to the structural characterization, we obtain a fully polynomial-time approximation scheme given a value oracle. Our technical results rely on Schur-convexity of Bernstein basis polynomial-weighted functions, total positivity and variation diminishing property. En route to our results, we obtain a surprising reduction from a structured high-dimensional nonconvex optimization to a single-dimensional optimization by connecting the shape of the gradient sequences of the objective function to the number of transition points in optimum, which might be of independent interest.

This paper derives a lower bound on the spacing between adjacent zeros of the confluent hypergeometric function $Φ(a,b,z)$ when $a$ is variable and $(b,z) \in \mathbb{R}^+$ are known and fixed. Monotonicity of the bound is established, and the results are used to assess the accuracy of asymptotic approximations for the first passage probability of a Wiener process.