The metal to insulator (MIT) transition is accompanied by huge changes in physical responses by the control and tuning of experimental parameters like doping, pressure, chemical composition, and magnetic field. Here, we study the magnetic field-driven MIT for two pnictides in their elemental form, namely Arsenic and Bismuth. At low temperatures, Bismuth shows an unusual behaviour of a re-entrant IMT at high fields in addition to a higher temperature MIT at smaller fields. However, Arsenic shows the commonly observed single MIT. The Shubnikov de Haas (SdH) oscillations are observed for both As and Bi below 10 K. Giant magneto-resistance of the order of ~105 (MR%) is observed for both crystals at 2 K and 14 Tesla transverse magnetic field. The unusual Kohler scaling behaviour of MR at low temperature indicate the presence of increased carrier density attributed to the melting of excitons. Based on a microscopic model, the microscopic processes underpinning the unusual features of a field-driven MIT and re-entrant IMT, along with the relevance of both excitonic and Bose metal correlations near these incipient instabilities, are qualitatively described in the framework of field-driven excitonic condensate and Das-Doniach preformed pair scenarios in one single picture.

We report the existence of a large set of ferromagnetic scarred states in the one-dimensional transverse-field Ising model with long-range interactions, in a regime with no ferromagnetic phase at finite temperature. These scarred states are distributed over different spectral regions, surrounded by paramagnetic states. We show that simple initial conditions, consisting in a few small magnetic domains, selectively populate these scarred states. This leads to the appearance of a special dynamical phase, which we call scarred ferromagnetic phase. As a consequence, initial states with a small number of small magnetic domains evolve towards ferromagnetic equilibrium states, whereas initial states with larger domains or no magnetic structure relax to the expected thermal paramagnetic equilibrium state.

We study the complementation (in $\ell_\infty$) of the Banach space $c_{0,\mathcal{I}}$, consisting of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the supremum norm, where $\mathcal{I}$ is an ideal of subsets of $\mathbb{N}$. We show that the complementation of these spaces is related to a condition requiring that the ideal is the intersection of a countable family of maximal ideals, which we refer to as $ω$-maximal ideals. We prove that if $c_{0,\mathcal{I}}$ admits a projection satisfying a certain condition, then $\mathcal{I}$ must be a special type of $ω$-maximal ideal. Additionally, we characterize when the quotient space $c_{0,\mathcal{J}} / c_{0,\mathcal{I}}$ is finite-dimensional for two ideals $\mathcal{I} \subsetneq \mathcal{J}$.

Dissipative quantum error correction (QEC) autonomously protects quantum information using engineered dissipation and offers a promising alternative to error correction via measurement and feedback. However, scalability remains a challenge, as correcting high-weight errors typically requires increasing dissipation rates and exponentially many correction operators. Here, we present a scalable dissipative QEC protocol for discrete-variable codes, correcting multi-qubit errors via a trickle-down mechanism that sequentially reduces errors weight. Our construction exploits redundancy in the Knill-Laflamme conditions to design correction operators that act on multiple error subspaces simultaneously, thereby reducing the overhead from exponential to polynomial in the number of required operators. We illustrate our approach with repetition codes under biased noise, showing a fourfold improvement in the exponential suppression factor at realistic physical error rates. Our approach connects autonomous QEC for discrete-variable codes with demonstrated dissipative protocols for bosonic codes and opens up new avenues for traditional measurement-feedback QEC and fault-tolerant quantum operations.

Long Gamma-Ray Bursts (LGRBs) are often associated with the collapse of stripped-envelope massive stars. Powerful relativistic jets drill through the stellar envelope before the gamma emission. Previous hydrodynamical studies imposed jets artificially, neglecting accretion dynamics, while the central engine simulations have reproduced jet launching via the Blandford-Znajek mechanism focusing on the inner core regions. However, both the central engine and the progenitor structure are crucial to determining the jet's evolution. In this study, we present axisymmetric (2.5-D) GRMHD simulations that self-consistently follow jet formation from the black-hole horizon to breakout at the stellar surface ($R_\star \sim 10^{10}$~cm). The setup assumes a Kerr black hole with spin $a \sim 0.9$ in the centre of three progenitor models, varying the magnetic-field strength and geometry. Relativistic jets are successfully launched by a strong dipolar magnetic field ($B_0 \gtrsim 10^{12}$-$10^{14}$~G) from magnetically arrested disks. These jets, initially magnetically dominated, convert energy into thermal and kinetic during their propagation. We found breakout times within $1.8 \lesssim t_{\rm bo} \lesssim 3.5$~s and luminosities $L_j \sim 5\times10^{49}-7\times10^{52}$~erg\,s$^{-1}$. Our results highlight the role of the initial magnetic field strength and its geometry, emphasizing the progenitor's density distribution as a key factor impacting the final structure and dynamics of LGRB jets.

We develop a finite-cutoff formulation of BTZ black-hole thermodynamics in which a circular cavity at radius $R$ is treated as a genuine holographic screen. The resulting ensemble is simultaneously a quasilocal gravitational system in the Brown--York sense and a finite-radius realization of holography in AdS$_3$. This viewpoint organizes the thermodynamics of static and rotating BTZ geometries in terms of local intensive data measured at the wall, the corresponding quasilocal stress tensor, and the radial Hamilton--Jacobi flow of the on-shell action. We show how the cavity description naturally connects to finite-cutoff AdS/CFT and to the thermodynamics of $T\bar T$-deformed two-dimensional theories. The program developed in this paper includes a grand-canonical treatment of rotating BTZ black holes, an off-shell free-energy landscape for the cavity ensemble, a finite-size Hawking--Page transition with a holographic interpretation, radial flow equations for energy and pressure, and a microscopic density-of-states analysis adapted to finite cutoff. The cavity radius therefore plays a dual role: it is a thermodynamic control parameter in the bulk and an RG scale in the dual description.

The source binary of the gravitational-wave (GW) event GW230529, detected at the beginning of the fourth LIGO-Virgo-KAGRA observing run, was inferred to consist of a NS and a compact object in the lower mass gap, a purported gap between the most massive NSs ($\sim 3\,M_\odot$) and least massive black holes (BHs; $\sim 5\,M_\odot$) based on compact-object observations in the Milky Way. While the nature of the mass-gap object could not be determined from the GW data alone for this event, definitively distinguishing whether this object is a NS or BH would have profound implications for the NS equation of state, supernova physics, and multimessenger astronomy. In this work, we perform parameter estimation on a suite of simulated GW systems with parameters similar to those of the GW230529 source binary to investigate whether the ambiguity in the physical nature of the source is a generic result for such systems. We vary the intrinsic properties of the simulated systems, the detector noise properties, the signal-to-noise ratios (SNRs), and the waveform model used in recovery. We find that the low SNR of GW230529 is the key reason for the ambiguity in determining whether the mass of the primary object in the binary is consistent with a low-mass BH or a high-mass NS, and thus the priors used for the masses and spins have a significant impact on the posterior distribution, which is a generic result for low-SNR events. The inclusion of tidal effects in the waveform model also contributes to the observed degeneracies in the posteriors, since the statistical uncertainties in analyses of GW events like GW230529 are larger for waveform models including tidal effects. We show that the future observation of such a system with a higher SNR ($\sim 30$) would increase the precision of the mass measurements sufficiently to allow us to determine the nature of the mass-gap object.

We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph $K_{2n}$. We show that they are special subsets of the association scheme obtained from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra $\mathbb{C}[S_{2n}]$ and using the representation theory of $S_{2n}$.

The search for chiral Majorana fermions in quantum anomalous Hall insulator/\textit{s}-wave superconductor heterostructures has attracted intense interest, yet remains controversial due to the lack of conclusive evidence. A key issue is that the heterostructure's metallization can produce half-integer conductance signatures resembling those of chiral Majorana fermions, thereby complicating their identification. In this Letter, we investigate how the competition between metallization and chiral Majorana fermions depends on superconductor thickness, revealing its critical role through three distinct regimes: (i) For thin superconductors ($\sim$10 nm), metallization shows periodic oscillations with thickness, matching the Fermi wavelength. (ii) Intermediate thicknesses ($\sim$100 nm) exhibit periodic windows for observing chiral Majorana fermions. (iii) Thick superconductors ($\sim$1000 nm) sustain stable chiral Majorana fermions that are insensitive to thickness variations. These results suggest that superconductor thickness is a key control parameter for advancing efforts to conclusively identify chiral Majorana fermions.

Reading is foundational for educational, employment, and economic outcomes, but a persistent proportion of students globally struggle to develop adequate reading skills. Some countries promote digital tools to support reading development, alongside regular classroom instruction. Such tools generate rich log data capturing students' behaviour and performance. This study proposes a dynamic cognitive diagnostic modeling (CDM) framework based on restricted latent class models to trace students' time-varying skills mastery using log files from digital tools. Unlike traditional CDMs that require expert-defined skill-item mappings (Q-matrix), our approach jointly estimates the Q-matrix and latent skill profiles, integrates log-derived covariates (e.g., reattempts, response times, counts of mastered items) and individual characteristics, and models transitions in mastery using a Bayesian estimation approach. Applied to real-world data, the model demonstrates practical value in educational settings by effectively uncovering individual skill profiles and the skill-item mappings. Simulation studies confirm robust recovery of Q-matrix structures and latent profiles with high accuracy under varied sample sizes, item counts and different sparsity of Q-matrices. The framework offers a data-driven, time-dependent restricted latent class modeling approach to understanding early reading development.

Where in the present-day Milky Way should we search for the remnants of its earliest stars? We address this question using the DORCHA (Gaelic for Dark; DUR-uh-khuh) suite: a set of 25 high-resolution, dark-matter-only cosmological zoom-in simulations of Milky Way analogue (MWA) haloes evolved to $z=0$. Of these, 15 are isolated and the rest are in pairs, similar to the MW and M31. By identifying and tagging the most bound material in high-redshift ($z\geq5$) progenitor haloes -- those likely to host early star formation -- we track the present-day phase-space distribution of this ancient component. We find that this material is highly centrally concentrated at $z=0$, with 90 -- 100 per cent residing within $r \lesssim 15\,h^{-1}\,\mathrm{kpc}$. It exhibits steep density profiles ($ρ\propto\,r^{-4}$), low velocity dispersions ($σ_r / σ_{\rm max} \lesssim 0.6$), and radially biased orbits ($β\gtrsim 0.5$ for $r \gtrsim 0.1\,R_{200}$), consistent with a relaxed, centrally embedded population. These results hold across haloes with diverse formation histories and environments, suggesting that the dynamical signature of early progenitors is robust to later mergers and interactions. Our findings imply that the fossil record of the first generations of stars -- including Population III and extremely metal-poor stars -- should be sought in the innermost regions of the Milky Way, where they retain distinctive kinematic imprints. While these stellar populations may overlap, we caution that low metallicity does not uniquely identify ancient stars, nor vice versa. The DORCHA suite thus provides a physically motivated baseline for interpreting observations from Galactic Archaeology surveys targeting the bulge and inner halo.

In this work, we investigate the emergence of Weyl points in an inversion symmetry-breaking 1T-NiTe$_2$ system. Through first-principles calculations based on the density functional theory combined with tight-binding methods, we find three distinct sets of Weyl crossings under an appropriate symmetry breaking. The first set, composed of four Weyl points, emerges from the Dirac semimetal. Surprisingly, the other two sets result in additional twenty-four Weyl crossings, depending on the weight of the symmetry breaking. We investigate the topological characteristics of the Weyl semimetals by computing the Weyl chirality, Berry curvature, and the evolution of Wannier charge centers. Additionally, the bulk-boundary correspondence has been shown by computing the Fermi arcs. Our results provide a way for creating and manipulating distinct sets of Weyl points with appropriate external control, which can be valuable for applications in Weyltronics.

We prove that a local trilinear extension inequality on the paraboloid in three dimensions is equivalent to the Fourier restriction conjecture, and then we prove a variant involving smooth Alpert wavelets that represents the weakest such inequality the authors could find that characterizes the Fourier extension conjecture.

We introduce a family of particle systems on sparse graphs where local interactions occur via hitting times, providing a dynamic and tractable model for default cascades in large sparsely-connected financial networks. Building on the framework of Lacker, Ramanan and Wu (2023), we extend convergence theory to systems with singular interactions, capturing the abrupt and discontinuous nature of systemic events. We establish conditions for well-posedness through a minimality principle and connect fragility to dynamic percolation thresholds. Our analysis demonstrates continuity of the joint law of defaults with respect to local graph convergence, establishes convergence of empirical distributions, and characterizes the default time distribution in tree-like networks. This framework offers a rigorous and flexible foundation for modeling systemic risk in evolving financial systems, featuring continuous-time dynamics, heterogeneous and local interactions, and instantaneous default cascades.

Protein fitness landscapes frequently exhibit epistasis, where the effect of a mutation depends on the genetic context in which it occurs, i.e., the rest of the protein sequence. Epistasis increases landscape complexity, often resulting in multiple fitness peaks. In its simplest form, known as global epistasis, fitness is modeled as a non-linear function of an underlying additive trait. In contrast, more complex epistasis arises from a network of (pairwise or many-body) interactions between residues, which cannot be removed by a single non-linear transformation. Recent studies have explored how global and network epistasis contribute to the emergence of functional bottlenecks - fitness landscape topologies where two broad high-fitness basins, representing distinct phenotypes, are separated by a bottleneck that can only be crossed via one or a few mutational paths. Here, we introduce and analyze a stylized model of global epistasis with an additive underlying trait. We demonstrate that functional bottlenecks arise with high probability if the model is properly calibrated. Furthermore, our results underscore that a proper balance between neutral and non-neutral mutations is needed for the emergence of functional bottlenecks.

In this paper, we study the asymptotic behavior of energy and action ground states to the following pseudo-relativistic Hartree equation \[ \left(\sqrt{-c^2Δ+m^2c^4}-mc^2\right)u + λu = \left(|x|^{-1}*|u|^2\right)u \] as the speed of light $c\to\infty$. We obtain an asymptotic expansion of the ground state as $c \to \infty,$ which is new in the case of the energy ground state and generalizes the results of Choi, Hong, and Seok (2018) for the action ground state.

We study concentration operators acting on the Fourier symmetric Sobolev space $H$ consisting of functions $f$ such that $\int_{\mathbb{R}} |f(x)|^2(1+x^2) dx + \int_{\mathbb{R}} |\hat{f}(ξ)|^2(1+ξ^2) dξ< \infty $. We find that the Bargmann transform is a unitary operator from $H$ to a weighted Fock space. After identifying the reproducing kernel of $H$, we discover an unexpected phenomenon about the decay of the eigenvalues of a two-sided concentration operator, namely that the plunge region is of the same order of magnitude as the region where the eigenvalues are close to 1, contrasting the classical case of Paley--Wiener spaces.

Unlike traditional model-based reinforcement learning approaches that estimate system parameters from data, non-model-based data-driven control learns the optimal policy directly from input-state data without any intermediate model identification. Although this direct reinforcement learning approach offers increased adaptability and resilience to model misspecification, its reliance on raw data leaves it vulnerable to system noise and disturbances that may undermine convergence, robustness, and stability. In this article, we establish the convergence, robustness, and stability of value iteration (VI) for data-driven control of stochastic linear quadratic (LQ) systems in discrete-time with entirely unknown dynamics and cost. Our contributions are three-fold. First, we prove that VI is globally exponentially stable for any positive semidefinite initial value matrix in noise-free settings, thereby significantly relaxing restrictive assumptions on initial value functions in existing literature. Second, we extend our analysis to settings with external disturbances, proving that VI maintains small-disturbance input-to-state stability (ISS) and converges within a small neighborhood of the optimal solution when disturbances are sufficiently small. Third, we propose a new non-model-based robust adaptive dynamic programming (ADP) algorithm for adaptive optimal controller design, which, unlike existing procedures, requires no prior knowledge of an initial admissible control policy. Numerical experiments on a ``data center cooling'' problem demonstrate the convergence and stability of the algorithm compared to established methods, highlighting its robustness and adaptability for data-driven control in noisy environments. Finally, we apply the method to dynamic portfolio allocation, demonstrating its practical relevance outside traditional control tasks.

Given an action $G \curvearrowright X$ of a discrete and countable infinite group $G$ on a compact and Hausdorff space $X$, we regard $\ell^1(G\curvearrowright X)$ as the Banach *-algebra crossed product associated to the action. We characterize topological freeness of the action by showing that it is equivalent to every nontrivial closed ideal of $\ell^1(G\curvearrowright X)$ intersecting $C(X)$ nontrivially. Most surprisingly, we show that when $G$ is torsion-free and abelian, $\ell^1(G\curvearrowright X)$ can detect freeness of $G \curvearrowright X$: indeed, we show that $G\curvearrowright X$ is free if and only if every closed ideal of $\ell^1(G\curvearrowright X)$ is self-adjoint, a property that is automatic in $C^*$-algebras. We also show with an example that this result does not hold beyond the torsion-free abelian case.

A three-dimensional quasi-Fuchsian Lorentzian manifold $M$ is a globally hyperbolic spacetime diffeomorphic to $Σ\times (-1,1)$ for a closed orientable surface $Σ$ of genus $\geq 2$. It is the quotient $M=Γ\backslash Ω_Γ$ of an open set $Ω_Γ\subset {\rm AdS}_3$ by a discrete group $Γ$ of isometries of ${\rm AdS}_3$ which is a particular example of an Anosov representation of $π_1(Σ)$. We first show that the spacelike geodesic flow of $M$ is Axiom A, has a discrete Ruelle resonance spectrum with associated (co-)resonant states, and that the Poincaré series for $Γ$ extend meromorphically to $\mathbb{C}$. This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian $\Box$ of $M$, which is meromorphic on $\mathbb{C}$ with poles of finite rank, defining a notion of quantum resonances and quantum resonant states related to the Ruelle resonances and (co-)resonant states by a quantum-classical correspondence. This initiates the spectral study of convex co-compact pseudo-Riemannian locally symmetric spaces.

We present a systematic comparison of statistical approaches to Baryon Acoustic Oscillation (BAO) analysis using DESI DR2 data. We evaluate four methods for handling the nuisance parameter $β=1/(H_0 r_d)$: marginalization, profiling, Taylor expansion, and full likelihood analysis across multiple cosmological models. Our results demonstrate that while these methods yield consistent constraints for $Λ$CDM and $Ω_K$CDM models, they produce notable differences for models with dynamical dark energy parameters. Through eigenvalue decomposition of Fisher matrices, we identify extreme parameter degeneracies in $ww_a$CDM and $Ω_Kww_a$CDM models that explain these statistical sensitivities. Surprisingly, $Ω_K$CDM shows the highest information content across datasets, suggesting BAO measurements are particularly informative about spatial curvature. We further use skewness and kurtosis analysis to identify deviations from Gaussianity, highlighting limitations in Fisher approximations in the dark energy models. Our analysis demonstrates the importance of careful statistical treatment when extracting cosmological constraints from increasingly precise measurements.

In this paper, we use geometric methods to study the relations between admissible representations of $\mathbf{GL}_n(\mathbb{C})$ and unramified representations of $\mathbf{GL}_m(\mathbb{Q}_p)$. We show that the geometric relationship between Langlands parameter spaces of $\mathbf{GL}_n(\mathbb{C})$ and $\mathbf{GL}_m(\mathbb{Q}_p)$ constructed by the first named author is compatible with the functor recently defined algebraically by Chan-Wong. We then show that the said relationship intertwines translation functors on representations of $\mathbf{GL}_n(\mathbb{C})$ and partial Bernstein-Zelevinskii derivatives on representations of $\mathbf{GL}_m(\mathbb{Q}_p)$, providing purely geometric counterparts to some results of Chan-Wong. In the sequels, the techniques of this work will be extended to real and $p$-adic classical groups and used to study their Arthur packets.

Entanglement of fundamental degrees of freedom in particle physics is generated ab initio in scattering processes. In the case of a pure $SU(N)$ gauge theory, two gluons in a product state can be entangled in their polarizations as the result of three- and four-gluon vertex interactions. We find that entanglement is only produced when the initial product state have opposite polarizations, and that maximal entanglement is generated when the outgoing particles form an angle of $\fracπ{2}$ with respect to the incoming ones. A surprising result is that the amount of entanglement among gluon polarizations is independent of the gauge group at any angle and, thus, of the color degree of freedom. Universality of entanglement for any gauge group emerges. It is also found that a small deviation of the relative weight between three- and four-gluon vertices would prevent the generation of maximal entanglement. This can be seen as a small piece of a possible it from qubit principle underlying fundamental interactions, that is a principle that requires physics to be quantum in origin.

An element of the algebra $M_n(\mathbb{F})$ of $n \times n$ matrices over a field $\mathbb{F}$ is called an involution if its square equals the identity matrix. Gustafson, Halmos, and Radjavi proved that any product of involutions in $M_n(\mathbb{F})$ can be expressed as a product of at most four involutions. In this article, we investigate the bijective linear preservers of the sets of products of two, three, or four involutions in $M_n(\mathbb{F})$.

We derive an analytical solution for the one-point distribution of a passive scalar in decaying homogeneous turbulence, in the limit of strong turbulence (high Re, fixed Schmidt number). Velocity statistics are governed by the Euler ensemble, a spontaneously stochastic exact solution of the loop equation from the Navier-Stokes equations in the strongly turbulent regime. The scalar's advection-diffusion problem is also recast as a solvable linear loop equation. For a localized initial condition, the solution consists of expanding concentric shells: the radial scalar profile is quantized and piecewise parabolic, with gaps organized by Euler totients - an arithmetic structure distinct from conventional scaling. This shell pattern is the unique solution in the Euler ensemble, smoothed by any finite diffusivity. The result provides the underlying geometric structure for scalar transport in decaying strong turbulence, relevant in astrophysical or quantum-fluid regimes where dissipation is negligible. This may explain the "ramp-cliff" structures observed in turbulent mixing half a century ago. While this shell structure is hard to resolve in DNS, its statistical signature is robustly captured by the volume-averaged scalar density, a measurable quantity.

In 1996, Hatano and Nelson proposed a non-Hermitian lattice model containing an imaginary Peierls phase [Phys. Rev. Lett. 77 570-573 (1996)], which subsequent analyses revealed to be an instance of a new class of topological systems. Here, we experimentally realize a continuum analog to this model containing an imaginary gauge potential using a homogeneous spin-orbit coupled Bose-Einstein condensate (BEC). Non-Hermiticity is introduced by adding tunable spin-dependent loss via microwave coupling to a subspace with spontaneous emission. We demonstrate that the resulting Heisenberg equations of motion for position and momentum depend explicitly on the system's phase-space distribution. First, we observe collective nonreciprocal transport in real space, with a "self-acceleration" that decreases with the BEC's spatial extent, consistent with non-Hermitian Gross-Pitaevskii simulations. We then examine localized edge states: the relatively strong interactions in our BEC suppress the formation of topological edge states, yielding instead highly excited states localized by an interplay between self-acceleration and wavefunction spreading. Finally, we confirm that our non-Hermitian description remains valid at all times by comparing to a multi-level master-equation treatment.

Internal activity can fundamentally reshape the mechanical behavior of solids, yet its role in softening and failure remains incompletely understood. In this study, we investigate spontaneous deformations in activated solids via non-affine fluctuations that quantify local rearrangements relative to global strain. Using scaling analysis and numerical simulations, we show that non-affinity in crystalline solids grows quadratically with active speed, increases linearly with persistence time before saturating, and scales inversely with the distance to the melting density. Spatial correlations reveal an activity-dependent growing correlation length, while relaxation dynamics are governed by the active persistence time. With increasing activity, the distributions of local non-affinity broaden, become more skewed, and develop heavy tails, eventually forming a secondary maximum that signals coexisting small and large non-affinities; this heterogeneity precedes defect formation and two-step melting from solid to hexatic and ultimately to fluid. Finally, we demonstrate that spatially patterned activation provides a simple route to locally induce non-affinity and mechanical softening. Our predictions are experimentally testable and suggest a pathway to tunable mechanics in adaptive metamaterials, with implications for mechanical regulation in biological systems.

In this paper we derive a novel characterization result for time-consistent stochastic control problems with higher-order moments, originally formulated by Wang et al. [SIAM J. Control. Optim., 63 (2025), 1560--1589], and newly explore many solvable instances including a mean-variance-excess kurtosis portfolio selection problem. By improving an asymptotic result of the variational process for the uniform boundedness and integrability properties, we obtain both the sufficiency and necessity of an equilibrium condition for an open-loop Nash equilibrium control (ONEC). This condition is simply formulated by the diagonal processes of a flow of backward stochastic differential equations (BSDEs) whose data do not necessarily satisfy the usual square-integrability condition. In particular, for linear controlled dynamics with deterministic parameters, we show that the ONEC can be derived by solving a polynomial algebraic equation under a class of nonlinear objective functions. Interestingly, the mean-variance equilibrium strategy is an ONEC for our general higher-order moment problem if and only if a homogeneity condition holds. Additionally, in the case with random parameters, we characterize the ONEC by finitely many BSDEs with a recurrence relation. As an intuitive illustration, the solution to the mean-variance-skewness problems is given by a quadratic BSDE.

Galaxy and halo scaling relations, connecting a broad range of parameters, are well established from observations. The origin of many of these relations and their scatter is still a matter of debate. It remains a sizable challenge for models to simultaneously and self-consistently reproduce as many scaling relations as possible. We introduce the Magneticum Pathfinder hydrodynamical cosmological simulation suite, to date the suite that self-consistently covers the largest range in box volumes and resolutions. It is the only cosmological simulation suite that is tuned on the hot gas content of galaxy clusters instead of the stellar mass function. By assessing the successes and shortcomings of tuning to the hot gas component of galaxy clusters, we aim to further our understanding of the physical processes shaping the Universe. We analyze the importance of the hot and cold gas components for galaxy and structure evolution. We analyze 28 scaling relations, covering large-scale global parameters as well as internal properties for halos ranging from massive galaxy clusters down to galaxies, and show their predicted evolution from z=4 to z=0 in comparison with observations. These include the halo-to-stellar-mass and Kennicutt--Schmidt relations, the cosmic star formation rate density as well as the Fundamental Plane. Magneticum Pathfinder matches a remarkable number of the observed scaling relations from z=4 to z=0, including challenging relations like the number density of quiescent galaxies at cosmic dawn, the mass--size evolution, the mass--metallicity relation, the Magorrian relation, and the temperature--mass relation. We compile our data to allow for straightforward future comparisons. Galaxy properties and scaling relations arise naturally and the large scatter in observables at high redshift is crucial to distinguish the various galaxy formation models reproducing the z=0 relations.

Let $E$ be a vector space over a countable field of dimension $\aleph_0$. Two infinite-dimensional subspaces $V,W \subseteq E$ are almost disjoint if $V \cap W$ is finite-dimensional. This paper provides some improvements on results about the definability of maximal almost disjoint families (mad families) of subspaces in [18]. We construct a full mad family of block subspaces in ZFC, answering a problem by Smythe in the positive. A variant of this construction shows that there exists a completely separable mad family of block subspaces in ZFC. We also discuss the abstract Mathias forcing introduced by Di Prisco-Mijares-Nieto in [12], and apply it to show that in the Solovay's model obtained by the collapse of a Mahlo cardinal, there are no full mad families of block subspaces over $\mathbb{F}_2$.