Understanding how group interactions influence opinion dynamics is fundamental to the study of collective behavior. In this work, we propose and study a model of opinion dynamics on $d$-uniform hypergraphs, where individuals interact through group-based (higher-order) structures rather than simple pairwise connections. Each one of the two opinions $A$ and $B$ is characterized by a quality, $Q_A$ and $Q_B$, and agents update their opinions according to a general mechanism that takes into account the weighted fraction of agents supporting either opinion and the pooling error, $α$, a proxy for the information lost during the interaction. Through bifurcation analysis of the mean-field model, we identify two critical thresholds, $α_{\text{crit}}^{(1)}$ and $α_{\text{crit}}^{(2)}$, which delimit stability regimes for the consensus states. These analytical predictions are validated through extensive agent-based simulations on both random and scale-free hypergraphs. Moreover, the analytical framework demonstrates that the bifurcation structure and critical thresholds are independent of the underlying topology of the higher-order network, depending solely on the parameters $d$, i.e., the size of the interaction groups, and the quality ratio. Finally, we bring to the fore a nontrivial effect: the large sizes of the interaction groups, could drive the system toward the adoption of the worst option.

Combining the methods of Brian and Stuart with the classical Dvoretzky theorem, we show that no infinite-dimensional Banach space contains a barrelled subspace of (algebraic) dimension $<\mbox{cov}(\mathcal{N})$, the covering number of the Lebesgue null ideal $\mathcal{N}$. Consequently, every infinite-dimensional normed barrelled space has dimension $\ge\mbox{cov}(\mathcal{N})$ and so it is consistent with \textsf{ZFC} that no normed barrelled space has dimension equal to the bounding number $\mathfrak{b}$.

This paper proposes methods for producing compound selection decisions in a Gaussian sequence model. Given unknown, fixed parameters $μ_ {1:n}$ and known $σ_{1:n}$ with observations $Y_i \sim \textsf{N}(μ_i, σ_i^2)$, the decision maker would like to select a subset of indices $S$ so as to maximize utility $\frac{1}{n}\sum_{i\in S} (μ_i - K_i)$, for known costs $K_i$. Inspired by Stein's unbiased risk estimate (SURE), we introduce an almost unbiased estimator, called ASSURE, for the expected utility of a proposed decision rule. ASSURE allows a user to choose a welfare-maximizing rule from a pre-specified class by optimizing the estimated welfare, thereby producing selection decisions that borrow strength across noisy estimates. We show that ASSURE produces decision rules that are asymptotically no worse than the optimal but infeasible decision rule in the pre-specified class. We apply ASSURE to the selection of Census tracts for economic opportunity, the identification of discriminating firms, and the analysis of $p$-value decision procedures in A/B testing.

Suppose $J = (f_1, \dots, f_n)$ is an $n$-generated ideal in any ring $R$. We prove a general Briançon-Skoda-type containment relating the integral closure $\overline{J^{n+k-1}}$ with ordinary powers $J^k$. We prove that our result implies the full Briançon-Skoda containment $\overline{J^{n+k-1}} \subseteq J^k$ for pseudo-rational singularities (for instance regular rings), and even for the weaker condition of birational derived splinters. Our methods also yield the containment $\overline{J^{n+k}} \subseteq J^k$ for Du Bois singularities and even for a characteristic-free generalization. Our Briançon-Skoda-type theorem also implies well-known closure-based Briançon-Skoda results $\overline{J^{n+k-1}} \subseteq (J^k)^{\mathrm{cl}}$ where, for instance, $\mathrm{cl}$ is tight or plus closure in characteristic $p > 0$, or $\mathrm{ep}$ closure or extension and contraction from $\widehat{R^+}$ in mixed characteristic. Our proof relies on a study of the tensor product of the derived image of the structure sheaf of a partially normalized blowup of $J$ with the Buchsbaum-Eisenbud complex (equivalently the Eagon-Northcott complex) associated to $(f_1,\dots,f_n)^k$. As an application of our results and methods above, we prove the uniform Artin-Rees theorem and the uniform Briançon-Skoda theorem for quasi-excellent, respectively quasi-excellent reduced, rings of finite dimension, answering conjectures of Huneke.

Molecular hydrogen (H$_2$) is one of the key chemical species that controls and shapes a wide spectrum of astrophysical processes from galaxy evolution to planet formation. Although catalyzation on dust grain surfaces is the dominant formation channel of H$_2$ in the interstellar medium, its efficiency across $20-200~\rm K$ has remained not fully understood. Here, using multiscale simulations combining ab-initio-level machine learning force fields, constrained path-integral Monte Carlo, and kinetic Monte Carlo, we perform a systematic, quantum-mechanical study of the full H$_2$ formation sequence, including hydrogen adsorption, diffusion, association and desorption. We explicitly consider the decoupling of gas and dust temperatures, making our results applicable to photon-dominated regions (PDRs) and dense cold clouds. Our results show that on the bare, crystalline surfaces studied here (graphitic and silicate grains), physisorbed hydrogen is negligible, and nuclear quantum effects (NQEs) in chemisorbed hydrogen atoms are essential for efficient formation at low temperatures, overcoming the classical Boltzmann suppression. This work presents a quantitative NQEs-inclusive study on silicate surfaces (exemplified by enstatite) and graphitic grains, revealing surface-specific adsorption behavior. These findings provide a first-principles quantum foundation for interstellar H$_2$ formation, complementing empirical multipliers, and enable new observational constraints on dust composition and molecular cloud evolution. The framework also extends to other astrochemical reactions on dust grains under full NQEs.

We extend the method of modulus of continuity for solutions of parabolic equations--as used, for instance, to prove the Fundamental Gap Conjecture--to solutions of non-local heat equations on R^n and in dimension one with a non-local Neumann boundary condition. Specifically, we show that if a solution of a non-local heat equation has an initial modulus of continuity satisfying simple criteria, then this modulus of continuity is preserved at all subsequent times. In the process of trying to generalise our result in one dimension, we found a counterexample suggesting that a non-local analogue of the Payne-Weinberger inequality would depend on more than the diameter of a bounded (convex) domain.

We propose a localized consensus-based method for sampling from non-Gaussian distributions, a task that frequently arises when solving Bayesian inverse problems. Our method arises from an alternative derivation of consensus-based sampling (CBS). Starting from ensemble-preconditioned Langevin dynamics, we replace the potential by its Moreau envelope -- a smoother approximation -- in order to replace the gradient in the Langevin equation with a proximal operator. We then approximate this operator by a weighted mean. In the limit of infinitely smoothing the potential to a quadratic function, this procedure recovers the standard CBS dynamics. In addition, outside this limit, we retrieve a refined variant of polarized CBS. We call the resulting algorithm localized consensus-based sampling, since particles interact more with nearby particles than with faraway ones. Our method is affine-invariant, exact for Gaussian targets in the mean-field limit, and demonstrates improved robustness over polarized CBS in numerical experiments. Like other consensus-based methods, localized CBS is gradient-free and easily parallelizable.

Observing the circumgalactic medium (CGM) in emission lines from ionized gas enables direct mapping of its spatial and kinematic structure, offering new insight into the gas flows that regulate galaxy evolution. Using the high-resolution Figuring Out Gas & Galaxies In Enzo (FOGGIE) simulations, we generate mock emission-line maps for six Milky Way-mass halos. Different lines (e.g., H$α$, OVI) trace distinct CGM phases and structures, highlighting the importance of observations in multiple species. We quantify the observable CGM mass fraction as a function of instrument spatial resolution and surface brightness sensitivity, finding that sensitivity is the dominant factor limiting detectability across all ions. At fixed sensitivity, higher spatial resolution reveals more structures; at fixed spatial resolution, higher sensitivity recovers a higher percentage of the total mass. We explore CGM kinematics by constructing emissivity-weighted projected velocity maps and comparing line-of-sight velocities between emission lines. OVI shows the largest kinematic deviation from H$α$, while MgII and SiII most closely follow HI velocities. Distinguishing these phases out to 50kpc from the galaxy center requires spectral resolution better than 30km/s for most ion pairs. Additionally, separating inflowing from outflowing gas based on projected kinematics also requires high spectral resolution: at 30km/s, more than 80% of gas above the emission detection threshold can be distinguished kinematically, but this fraction drops to <40% with a resolution of 200km/s. Our results provide predictions for future UV and optical instruments, showing that recovering the multiphase structure and kinematics of circumgalactic emission will require both high sensitivity and fine kinematic resolution.

We present a novel realization of Dirac leptogenesis based on the post-inflationary reheating phase of the early universe. An asymmetry generated within the scalar sector via CP-violating and out-of-equilibrium inflaton decays is transferred to chiral neutrinos through Yukawa interactions and then to baryons via electroweak sphalerons. We describe in detail a minimal realization of this mechanism that naturally accommodates small neutrino Yukawa couplings and results in contributions to the effective number of relativistic species, $N_{\text{eff}}$, testable in upcoming cosmological observations.

Suspended carbon nanotubes hosting electrostatically defined quantum dots allow for exceptionally strong and tunable electromechanical coupling as well as mechanical modes that can reach the quantum ground state of motion simply by cryogenic cooling. This makes them a unique platform for quantum simulation of electron-phonon coupling. Here, we propose an experimentally realisable setup with two such carbon nanotubes in parallel, each hosting four quantum dots. Our system not only exhibits phonon-mediated electron-electron attraction, but also supports a robust, maximally entangled Bell phase at mesoscopic scales shared across the subsystems. These features highlight its potential as a simulator of strongly correlated quantum systems.

A comprehensive investigation of nonradial oscillations in neutron star (NS) admixed with gravitationally bounded dark matter (DM) is carried out within the framework of full general relativity. The relativistic mean field (RMF) formalism is employed to illustrate the hadronic equation of state (EOS), while a physically motivated, gravitationally captured, non-uniform fermionic Higgs-portal DM component is incorporated to model DM-admixed NS. The DM distribution is characterized by two free parameters: $αM_χ$, an effective control parameter that combines the DM concentration and the DM candidate mass, and $β$, a steepness parameter controlling the DM density distribution. The quasi normal mode (QNM) characteristics such as fundamental ($f$) mode frequency and its corresponding gravitational-wave (GW) damping time ($τ$) is calculated for DM-admixed NS by solving the general relativistic perturbed equations involving axial as well as polar modes. The study demonstrates how the inclusion of DM distribution modifies the $f$-mode frequency and enhances the damping rate, reflecting a stronger coupling between matter and spacetime perturbations. Considering DM effects, the correlation analysis among DM model parameters, NS observables and QNM characteristics also carried out. Analytic fits for the $f-C-τ$ and $f-Λ-τ$ relations are constructed and calibrated for DM-admixed NS models. Building upon asteroseismic universal relations (URs), multimessenger constraint from the GW170817 event is employed by mapping the tidal deformability $Λ_{1.4}$ into the $(f_{1.4},τ_{1.4})$ space, thereby providing observational bounds on the oscillation properties of canonical DM-admixed NS model.

A vector field $V$ on any (semi-)Riemannian manifold is said to be mixed Killing if for some nonzero smooth function $f$, it satisfies $L_VL_Vg=fL_Vg$, where $L_V$ is the Lie derivative along $V$. This class of vector fields, as a generalization of Killing vector fields, not only identify the isometries of the manifolds, but broadly also contain the class of homothety transformations. We prove an essential curvature identity along those fields on any (semi-)Riemannian manifold and thus generalize the Bochner's theorem for Killing vector fields in this setting. Later we study it in the framework of almost coKähler structure and we prove that the Reeb vector field $ξ$ on an almost coKähler manifold is mixed Killing if and only if the operator $h=0$. Moving further, we completely classify almost coKähler manifolds with $ξ$ mixed Killing vector field in dimension 3. In particular, if $ξ$ on an $η$-Einstein almost coKähler manifold is mixed Killing, then the manifold is of constant scalar curvature with $h=0$. Also we show that on any $(κ,μ)$-almost coKähler manifold, $ξ$ is mixed Killing if and only if the manifold is coKähler. In the end we present few model examples in this context.

Entanglement preservation in noisy quantum memories represents a central challenge in quantum information science. While experiments have shown that electromagnetically induced transparency (EIT) memories can store entangled photons, a quantitative theoretical analysis of whether nonlocal quantum correlations can survive storage loss induced by ground-state decoherence remains limited. Here we combine the dark-state polariton formalism with a reduced density-operator treatment to derive an EIT-specific effective pure-loss description for the retrieved photonic state in the ground-state-decoherence-limited regime. The analysis reveals that decoherence transforms an initially pure Bell state into a mixed state with a vacuum component and predicts a protocol-dependent storage-efficiency benchmark of 89.7% for violating the chosen unconditional Clauser-Horne (CH) inequality. Above this benchmark, the retrieved photonic state violates the CH inequality without post-selection, whereas below it, this unconditional CH violation is no longer obtained. This framework provides a quantitative theoretical description of entanglement retention, retrieved photonic density operators, and protocol-dependent Bell-test benchmarks in EIT quantum memories.

We present evidence of the dynamical hysteretic nature of dissipation in unsteady turbulent flows. Wind tunnel experiments and direct numerical simulations in oscillating flows show that, at stationary mean Reynolds number, the dissipation constant is larger for decelerating flows. Consequently, a periodic behavior of the flow produces a hysteresis cycle, whose area scales with a parameter combining the Strouhal number and the relative amplitude of the forcing. This phenomenon can be explained and quantified through the influence of the unsteady term in the Karman-Howarth equation, with implications for a wide range of out-of-equilibrium systems.

We study parameterized elliptic systems on symmetric domains with additional system symmetries. We prove the existence of continua of nontrivial solutions bifurcating from the constant branch determined by a critical point of the potential, without assuming nondegeneracy, via the degree for equivariant gradient maps. Our assumptions are formulated in terms of the right-hand side. When the domain is a compact symmetric space, the bifurcating solutions break symmetry at every nonzero level. Under additional assumptions on the right-hand side, the continua are unbounded.

Mergers of white dwarf binaries are a possible progenitor channel for Type Ia supernovae. While white dwarfs are abundant in the universe and relatively well understood, their gravitational wave signals have not yet been directly observed. In order to detect gravitational waves from merging white dwarf binaries, a detector in the mid-band between LVK and LISA appears necessary. In this paper, we compute and discuss the gravitational waves emitted by inspiraling and merging white dwarf binaries, and assess their detectability with proposed space-based atom-interferometer detectors such as MAGIS Space and AEDGE. Gravitational waves from massive white dwarf binaries can be observed for many years before merger, offering a unique early warning of their final explosion. Our projections suggest that MAGIS Space could detect signals from Type Ia supernova progenitors at least once every four years, while AEDGE could observe at least a few hundred such events annually. The prolonged gravitational wave emission captured by atom-interferometers provides precise sky localisation and can allow observation of the final explosion with electromagnetic telescopes. The combined observation with electromagnetic radiation from the white dwarf binary coalescence could open a new pathway for multi-messenger astronomy involving some of the brightest transient events in the universe.

The number of observed gravitational wave (GW) events is growing fast thanks to rapidly improving detector sensitivities. GWs from compact binary coalescences like Black Holes or Neutron Stars behave like standard sirens and can be used as cosmological probes. To this aim, generally, the observation of an electromagnetic counterpart and the measurement of the redshift are needed. However, even when those are not available, it is still possible to exploit these "dark sirens" via statistical methods. In this work, we explore a method that exploits the information contained in the cross-correlation of samples of GW events with matter over-density tracers like galaxy catalogues. Contrary to other currently employed dark-sirens methods, this approach does not suffer from systematic errors related to the incompleteness of the galaxy catalogue. To further enhance the technique, we implement tomography in redshift space for the galaxy catalogue and luminosity distance space for the GWs. We simulate future data collected by the array of currently existing detectors, namely LIGO, Virgo, and Kagra, as well as planned third-generation ones such as the Einstein Telescope and Cosmic Explorers. We cross-correlate these data with those from upcoming photometric galaxy surveys such as Euclid. We perform a sensitivity forecast employing a full-likelihood approach and explore the parameter space with Monte Carlo Markov Chains. We find that with this method, third-generation detectors will be able to determine the Hubble constant $H_0$ with an error of only 0.7%, which is enough to provide decisive information to shed light on the Hubble tension. Furthermore, for the other cosmological parameters, we find that the GWs and galaxy surveys information are highly complementary, and the use of both significantly improves the ability to constrain the underlying cosmology.

We consider the problem of a 3D-3D-2D mutually coupled solute-solvent-structure three-states system. This describes the interaction of a flexible structure with a polymeric fluid of classical Oldroyd-B type without centre-of-mass diffusion. We construct a unique higher-order regularity notion of a strong solution for the system by decoupling the solute from the solvent-structure subsystem, solving the decoupled system individually, and gluing the solutions through a fixed-point argument. As a requirement for the construction, we rely on a maximal regularity result for the Stokes problem on moving domains with non-trivial boundary conditions; a result that is also of independent interest.

This paper introduces a novel approach to approximating continuous functions over high-dimensional hypercubes by integrating matrix CUR decomposition with hyperinterpolation techniques. Traditional Fourier-based hyperinterpolation methods suffer from the curse of dimensionality, as the number of coefficients grows exponentially with the dimension. To address this challenge, we propose two efficient strategies for constructing low-rank matrix CUR decompositions of the coefficient matrix, significantly reducing computational complexity while preserving accuracy. The first method employs structured index selection to form a compressed representation of the tensor, while the second utilizes adaptive sampling to further optimize storage and computation. Theoretical error bounds are derived for both approaches, ensuring rigorous control over approximation quality. Additionally, practical algorithms -- including randomized and adaptive decomposition techniques -- are developed to efficiently compute the CUR decomposition. Numerical experiments demonstrate the effectiveness of our methods in drastically reducing the number of required coefficients without compromising precision. Our results bridge matrix/tensor decomposition and function approximation, offering a scalable solution for high-dimensional problems. This work advances the field of numerical analysis by providing a computationally efficient framework for hyperinterpolation, with potential applications in scientific computing, machine learning, and data-driven modeling.

We explore a physical model of ordered sums of integers as trains of rods. The trains for a fixed, possibly infinite, set of rods naturally correspond to nodes in a tree; relations among finite linear recursions encoded in the subtrees define algebraic operations on sets of rods. We use this algebra to prove classic identities for recursively defined sequences, to show that some Lucas sequences are divisibility sequences, to characterize two-term linear Fibonacci identities, and to find the cyclotomic polynomial factors of Borwein trinomials. We complement abstractions with lots of examples.

We develop a structure-preserving numerical discretization for the electrostatic Euler-Poisson equations with a constant magnetic field. The scheme preserves positivity of the density, positivity of the internal energy, a minimum principle for the specific entropy, and a total energy balance. The scheme uses an operator splitting approach composed of two subsystems: the compressible Euler equations of gas dynamics, and a source system. The source system couples the electrostatic potential, momentum, and Lorentz force, thus incorporating electrostatic plasma and cyclotron motions. Because of the high-frequency phenomena it describes, the source system is discretized with an implicit time-stepping scheme. We use a PDE Schur complement approach for the numerical approximation of the source system, reducing to a single non-symmetric Poisson-like problem that is solved for each time step. Our focus with the present work is on the efficient solution of problems close to the magnetic-drift limit. Such asymptotic limit is characterized by the co-existence of slowly moving, smooth flows with very high-frequency oscillations, spanning timescales with a difference in excess of 10 orders of magnitude, making their numerical solution quite challenging. We illustrate the capability of the scheme by computing a diocotron instability and present growth rates that compare favorably with existing analytical results. The model, though a simplified version of the Euler-Maxwell system, represents a stepping stone toward electromagnetic solvers that are capable of working in the electrostatic and magnetic-drift limits as well as the hydrodynamic regime.

The variation of the oxygen enhancement ratio (OER) across linear energy transfer (LET) currently lacks a comprehensive mechanistic interpretation and a mechanistic model. Our earlier research revealed a significant correlation between the distribution of double-strand breaks (DSBs) within 3D genome and radiation-induced cell death, which offers valuable insights into the oxygen effect. We propose a model where the reaction of oxygen is represented as the probability of inducing DNA strand breaks. Then it is integrated into a track-structure Monte Carlo simulation to investigate the impact of oxygen on the distribution of DSBs within 3D genome. Using the parameters from our previous study, we calculate the OER values related to cell survival. Results show that the incidence ratios of clustered DSBs within a single topologically associating domain (TAD) (case 2) and within frequently interacting TADs (case 3) under aerobic and hypoxic conditions align with the trend in the OER of cell survival across LET. Our OER curves exhibit good correspondence with experimental data. This study provides a potentially mechanistic explanation for changes in OER across LET. High-LET irradiation leads to dense ionization events, resulting in an overabundance of lesions that readily induce case 2 and case 3, which have substantially higher probabilities of cell killing than other damage patterns. This may contribute to the main mechanism governing the variation of OER for high LET. Our study further underscores the importance of the DSB distribution within 3D genome in the context of radiation-induced cell death.

The topological string/spectral theory correspondence establishes a precise, non-perturbative duality between topological strings on local Calabi-Yau threefolds and the spectral theory of quantized mirror curves. While this duality has been rigorously formulated for the closed topological string sector, the open string sector remains less understood. Building on the results of [1-3], we make further progress in this direction by constructing entire, off-shell eigenfunctions for the quantized mirror curve from open topological string partition functions. We focus on local $\mathbb{F}_0$, whose mirror curve corresponds to the Baxter equation of the two-particle, relativistic Toda lattice. We then study the standard and dual four-dimensional limits, where the quantum mirror curve for local $\mathbb{F}_0$ degenerates into the modified Mathieu and McCoy-Tracy-Wu operators, respectively. In these limits, our framework provides a way to construct entire, off-shell eigenfunctions for the difference equations associated with these operators. Furthermore, we find a simple relation between the on-shell eigenfunctions of the modified Mathieu and McCoy-Tracy-Wu operators, leading to a functional relation between the operators themselves.

Estimating the state of an open quantum system monitored over time requires incorporating information from past measurements (filtering) and, for improved accuracy, also from future measurements (smoothing). While classical smoothing is well understood within a Bayesian framework, its quantum generalization has been challenging, leading to distinct and seemingly incompatible approaches. In this work, we demonstrate that quantum state smoothing hinges on a uniquely quantum feature: the fundamental dependence of retrodiction on prior correlations. We introduce auxiliary systems into the prior belief to capture correlations formed during preparation and evolution and develop a comprehensive framework for quantum state smoothing based on extended Bayesian retrodiction. This framework identifies all previous approaches as different choices of the extended prior, and naturally extends it to other choices that have not been considered before. We also give an information-theoretic characterization of the choices of prior, in terms of the average entropy of the smoothed states. Our results establish quantum state smoothing as a fundamentally retrodictive process just like classical smoothing, with proper quantum features clearly identified.

In this paper, we introduce relative Roe functors and show that for every pair of scalable locally compact metric spaces with bounded coarse geometry, the functor of continuous functions and the relative Roe functor, both associated with this pair, are asymptotically adjoint. While this asymptotic adjunction is weaker than the genuine one, it retains sufficient categorical properties to be intuitive and useful in applications. These results can be used to provide an unsuspended description of the Connes-Higson $E$-theory, establish connections between $E_{1}$-theory and extension theory, and express $K$-homology of compact metric spaces in terms the corresponding metric cones.

We study the formation and behaviour of bound states formed outside the horizon of a black hole in the presence of quintessence matter. Calculating the Regge and Wheeler potential for general metric function, we find that the presence of quintessence influences significantly the metric function and the Hawking temperature. We show that large black holes radiate less in the presence of quintessence matter and it seems to live longer, while small black holes radiate more in comparison with the model in the absence of quintessence. Bound states emerge at large enough quintessence parameter $|w|$ or angular momentum.

Ecological models traditionally explain stability and coexistence through pairwise interactions among species. However, interactions can also involve groups of three or more species, higher-order interactions, which recent theory suggests can stabilize communities. Yet, the conditions under which higher-order interactions are sufficient to stabilize coexistence in communities where pairwise and higher-order interactions occur simultaneously remain unknown. This work addresses this gap by analyzing a model of competitive communities that incorporates a proportion of pairwise and higher-order interactions. Using empirical data, numerical simulations, and analytical methods, we show that higher-order interactions alone cannot guarantee coexistence. We find that, while a small fraction of higher-order interactions can stabilize dynamics in communities of identical species, this effect weakens under more realistic conditions, such as variability in birth and mortality rates or explicit interaction structures. Our results challenge the prevailing view of higher-order interactions as a universal stabilizing mechanism, providing quantitative evidence of the joint importance of both pairwise and higher-order interactions, together with network structure and species parameters, for understanding ecological stability.

We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an abstract cluster structure, which captures the essence of cluster mutation at a tropical level and show that cluster algebras, cluster varieties, cluster categories and surface models all have associated abstract cluster structures. For the first two classes, we also show that they can be constructed from abstract cluster structures. By defining a suitable notion of morphism of abstract cluster structures, we introduce a category of these and show that it has several desirable properties, such as initial and terminal objects and finite products and coproducts. We also prove that rooted cluster morphisms of cluster algebras give rise to morphisms of the associated abstract cluster structures, so that our framework includes a version of the extant category of cluster algebras. We can do more, however, because we can relate different types of representation of abstract cluster structures (cluster algebra, varieties, categories) directly via morphisms of their associated abstract cluster structures, even though no direct map from e.g. a cluster category to the associated cluster algebra is possible. In fact, we do much of the above in the setting of abstract quantum cluster structures, with some analysis of the difference between the category of these and that of the unquantized version. In order to show the relationship between abstract quantum cluster structures and quantum cluster algebras, we reformulate the usual construction of the latter in a way that is more amenable to our purposes and which we expect will be of independent interest and use.

To address the common problem of high dimensionality in tensor regressions, we introduce a generalized tensor random projection method that embeds high-dimensional tensor-valued covariates into low-dimensional subspaces with minimal loss of information about the responses. The method is flexible, allowing for tensor-wise, mode-wise, or combined random projections as special cases. A Bayesian inference framework is provided featuring the use of a hierarchical prior distribution and a low-rank representation of the parameter. Strong theoretical support is provided for the concentration properties of the random projection and posterior consistency of the Bayesian inference. An efficient Gibbs sampler is developed to perform inference on the compressed data. To mitigate the sensitivity introduced by random projections, Bayesian model averaging is employed, with normalising constants estimated using reverse logistic regression. An extensive simulation study is conducted to examine the effects of different tuning parameters. Simulations indicate, and the real data application confirms, that compressed Bayesian tensor regression can achieve better out-of-sample prediction while significantly reducing computational cost compared to standard Bayesian tensor regression.

We show that in a vector space over Z_3, the union of any four linear bases is an additive basis, thus proving the Additive Basis Conjecture for Z_3, and providing an alternative proof of the weak 3-flow conjecture.