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This paper aims at developing model-theoretic tools to study interpretable fields and definably amenable groups, mainly in $\mathrm{NIP}$ or $\mathrm{NTP_2}$ settings. An abstract theorem constructing definable group homomorphisms from generic data is proved. It relies heavily on a stabilizer theorem of Montenegro, Onshuus and Simon. The main application is a structure theorem for definably amenable groups that are interpretable in algebraically bounded perfect $\mathrm{NTP_2}$ fields with bounded Galois group (under some mild assumption on the imaginaries involved), or in algebraically bounded theories of (differential) NIP fields. These imply a classification of the fields interpretable in differentially closed valued fields, and structure theorems for fields interpretable in finitely ramified henselian valued fields of characteristic $0$, or in NIP algebraically bounded differential fields.
Laser cooling of alkali atoms typically requires time-varying magnetic fields, introducing unwanted coupling between atom preparation and coherent operations. Here we demonstrate sub-Doppler laser cooling and optical transport of alkali atoms in a fully static magnetic-field configuration. Using a blue-detuned Type-II magneto-optical trap (MOT) operating on the closed $F=3 \rightarrow F'=2$ transition of the D2 line in cesium, we achieve temperatures of 17(1) $μ$K without changing the magnetic-field gradient between cooling stages. This enables direct loading into a shallow optical lattice and transport over 17 cm within the same static-field environment. In contrast to conventional alkali cooling schemes with dynamic fields, our approach establishes a continuous cooling and transport protocol compatible with static-field platforms. These results validate Type-II cooling as a practical technique for alkali atoms and provide a new route toward continuous-operation architectures in sensing and quantum computing.
Totally symmetric arbitrary spin conformal fields propagating in the flat space of even dimension greater than or equal to four are studied. For such fields, we develop a general ordinary-derivative light-cone gauge formalism and obtain restrictions imposed by the conformal algebra symmetries on interaction vertices. We apply our formalism for the detailed study of conformal scalar and vector fields. For such fields, all parity-even cubic interaction vertices are obtained. The cubic vertices obtained are presented in terms of dressing operators and undressed vertices. We show that the undressed vertices of the conformal scalar and vector fields are equal, up to overall factor, to the cubic vertices of massless scalar and vector fields. Various conjectures about interrelations between the cubic vertices for conformal fields in conformal invariant theories and the cubic vertices for massless fields in Poincare invariant theories are proposed.
We construct parameterized families of imaginary (resp. real) quadratic fields whose class groups have $n$-rank at least $2$.
Motivated by the well-known Paley graphs over finite fields and their generalizations, in this paper we explore a natural multiplicative-additive analogue of such graphs arising from vector spaces over finite fields. Namely, if $n\ge 2$ and $U\subsetneq \mathbb F_{q^n}$ is an $\mathbb F_q$-vector space, $G_{U}$ is the (undirected) graph with vertex set $V(G_U)=\mathbb F_{q^n}$ and edge set $E(G_U)=\{(a, b)\in \mathbb F_{q^n}^2\,|\, a e b, ab\in U\}$. We describe the structure of an arbitrary maximal clique in $G_U$ and provide bounds on the clique number $ω(G_U)$ of $G_U$. In particular, we compute the largest possible value of $ω(G_U)$ for arbitrary $q$ and $n$. Moreover, we obtain the exact value of $ω(G_U)$ when $U\subsetneq \mathbb F_{q^n}$ is any $\mathbb F_q$-vector space of dimension $d_U\in \{1, 2, n-1\}$.
We prove elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if $(K,v)$ is strongly dependent then so is its henselization.
Let $\ell$ be a prime number, $F$ be a global function field of characteristic $\ell$. Assume that there is a prime $P_\infty$ of degree $1$. Let $\mathcal{O}_F$ be the ring of functions in $F$ with no poles outside of $\{P_\infty\}$. We study solutions to Catalan's equation $X^m-Y^n=1$ over $\mathcal{O}_F$ and show that under certain additional conditions, there are no non-constant solutions which lie in $\mathcal{O}_F$, when $m,n>1$.
Since the development of higher local class field theory, several explicit reciprocity laws have been constructed. In particular, there are formulas describing the higher-dimensional Hilbert symbol given, among others, by M. Kurihara, A. Zinoviev and S. Vostokov. K. Kato also has explicit formulas for the higher-dimensional Kummer pairing associated to certain (one-dimensional) $p$-divisible groups. In this paper we construct an explicit reciprocity law describing the Kummer pairing associated to any (one-dimensional) formal group. The formulas are a generalization to higher-dimensional local fields of Kolyvagin's reciprocity laws. The formulas obtained describe the values of the pairing in terms of multidimensional $p$-adic differentiation, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups. In the second part of this paper, we will apply the results obtained here to give explicit formulas for the generalized Hilbert symbol and the Kummer pairing associated to a Lubin-Tate formal group. The results obtained in the second paper constitute a generalization to higher local fields, of the formulas of Artin-Hasse, K. Iwasawa and A. Wiles.
Let $K$ be a finitely generated field. We construct an $n$-dimensional linear system $\mathcal{L}$ of hypersurfaces of degree $d$ in $\mathbb{P}^n$ defined over $K$ such that each member of $\mathcal{L}$ defined over $K$ is smooth, under the hypothesis that the characteristic $p$ does not divide $\gcd(d, n+1)$ (in particular, there is no restriction when $K$ has characteristic $0$). Moreover, we exhibit a counterexample when $p$ divides $\gcd(d, n+1)$.
Our goal is to discuss in detail the calculation of the mean number of stationary points and minima for random isotropic Gaussian fields on a sphere as well as for stationary Gaussian random fields in a background parabolic confinement. After developing the general formalism based on the high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory (RMT) techniques to perform analysis of the random energy landscape of $p-$spin spherical spinglasses and a related glass model, both displaying a zero-temperature one-step replica symmetry breaking glass transition as a function of control parameters (e.g. a magnetic field or curvature of the confining potential). A particular emphasis of the presented analysis is on understanding in detail the picture of "topology trivialization" (in the sense of drastic reduction of the number of stationary points) of the landscape which takes place in the vicinity of the zero-temperature glass transition in both models. We will reveal the important role of the GOE "edge scaling" spectral region and the Tracy-Widom distribution of the maximal eigenvalue of GOE matrices for providing an accurate quantitative description of the universa
In this paper, we generalise the construction of the Bloch-Kato exponential map to complete discrete valuation fields of mixed characteristic (0,p) whose residue fields have a finite p-basis. As an application we prove an explicit reciprocity law, extending a result of Cherbonnier and Colmez in the classical case. This result relies on the calculation of the Galois cohomology of a p-adic representation V in terms of its (phi,G)-module.
In this paper, we show an analogue of Kural, McDonald and Sah's result on Alladi's formula for global function fields. Explicitly, we show that for a global function field $K$, if a set $S$ of prime divisors has a natural density $δ(S)$ within prime divisors, then $$-\lim_{n\to\infty} \sum_{\substack{1\le °D\le n\\ D\in \mathfrak{D}(K,S)}}\frac{μ(D)}{|D|}=δ(S),$$ where $μ(D)$ is the Möbius function on divisors and $\mathfrak{D}(K,S)$ is the set of all effective distinguishable divisors whose smallest prime factors are in $S$. As applications, we get the analogue of Dawsey's and Sweeting and Woo's results to the Chebotarev Density Theorem for function fields, and the analogue of Alladi's result to the Prime Polynomial Theorem for arithmetic progressions. We also display a connection between the Möbius function and the Fourier coefficients of modular form associated to elliptic curves. The proof of our main theorem is similar to the approach in Kural et al.'s article.
Binary neutron star mergers are expected to generate intense magnetic fields that power relativistic and non-relativistic outflows and shape their multimessenger signatures. These fields likely arise from the turbulent amplification of initially weak magnetic fields during the merger, particularly via the Kelvin-Helmholtz instability at the collisional interface between the stars. While previous studies have shown efficient amplification to magnetar-level strengths, the degree of large-scale coherence of the resulting field remains uncertain. We present general-relativistic, dynamical spacetime, magnetohydrodynamic simulations following the evolution of initially weak, pulsar-like magnetic fields in a binary neutron star merger. We find rapid magnetic field growth at small scales with clear signatures of small-scale turbulent dynamo action. At the highest resolutions, we additionally observe the emergence of coherent magnetic structures on larger scales. Our results imply that strong, ordered magnetic fields may be present immediately after merger, with important implications for the subsequent evolution of the remnant and its observable electromagnetic and gravitational-wave signa
Neutron stars can harbor extremely strong magnetic fields, yet the structure and stability of their magnetic field configuration remain poorly understood. Observations of pulsars indicate that the large-scale external field is predominantly dipolar far from the star, while the internal magnetic configurations are largely unconstrained. We investigate the dynamical stability of magnetized neutron stars through long-term numerical-relativity simulations. We explore a range of models with an initial external dipole field and mixed poloidal-toroidal internal field where the energy of the toroidal component varies up to $90\%$ of the magnetic energy. We find that the internal magnetic field relaxes toward a dynamically stable mixed poloidal-toroidal geometry, in which the toroidal component contributes to $\lesssim10\%$ of the total magnetic energy both in the exterior and in the interior. This configuration emerges within one Alfvén time following the saturation of the Tayler instabilities and also aided by gravitational-wave emission. These results suggest that long-lived neutron star magnetic fields are strongly constrained toward stable mixed configurations, with important implicati
We perform a series of simulations of magnetised Binary Neutron Star mergers, with varying magnetic field topologies in the initial data, as well as varying Equations of State, and mass ratios. In this paper, a companion paper to arXiv:2506.18995, we analyse the impact of the initial field configuration on the gravitational wave signal, the amplification of the magnetic field, and the ejected material. We investigate the dependence of the phase evolution of the gravitational wave in the post-merger on the initial magnetic field, finding that dephasing between the $(\ell=2,m=2)$ mode of the gravitational wave, and the $(2,1)$ and $(3,3)$ modes may be strongly impacted by the numerical reconstruction scheme. The magnetic field amplification during the Kelvin-Helmholtz dominated phase may be considerably enhanced by anti-aligned fields, or suppressed by toroidal fields. The post-merger amplification of the field due to winding may be suppressed by toroidal fields, and enhanced by asymmetries or mixtures of poloidal and toroidal fields. The field strength in the ejecta may be impacted by the initial magnetic field, with configurations which lead to large amplifications and those with m
Using high-resolution AthenaK simulations of a twisted toroidal flux tube, we study the flux emergence of magnetic structures in the shear layer of a hot massive neutron star typical of a binary neutron star remnant. High-resolution simulations demonstrate that magnetic buoyant instabilities allow for emergence only for extremely large magnetic fields significantly exceeding $10^{17}~\mathrm{G}$, and more typical fields around $10^{16}~\mathrm{G}$ are instead dominated by hydrodynamic effects. Because merger remnants tend to be stable against hydrodynamic convection, our work places strong limitations on the mechanisms by which massive binary neutron star remnants can produce the magnetically-driven outflows needed to power jets.
The emerging field of diverse intelligence seeks an integrated view of problem-solving in agents of very different provenance, composition, and substrates. From subcellular chemical networks to swarms of organisms, and across evolved, engineered, and chimeric systems, it is hypothesized that scale-invariant principles of decision-making can be discovered. We propose that cognition in both natural and synthetic systems can be characterized and understood by the interplay between two equally important invariants: (1) the remapping of embedding spaces, and (2) the navigation within these spaces. Biological collectives, from single cells to entire organisms (and beyond), remap transcriptional, morphological, physiological, or 3D spaces to maintain homeostasis and regenerate structure, while navigating these spaces through distributed error correction. Modern Artificial Intelligence (AI) systems, including transformers, diffusion models, and neural cellular automata enact analogous processes by remapping data into latent embeddings and refining them iteratively through contextualization. We argue that this dual principle - remapping and navigation of embedding spaces via iterative error
We present new numerical-relativity simulations of a magnetized binary neutron star merger performed with AthenaK. The simulations employ a temperature- and composition-dependent tabulated nuclear equation of state, with initially dipolar fields with a maximum initial strength of ${\sim}10^{16}\ {\rm G}$ which extend outside the stars. We employ adaptive mesh refinement and consider three grid resolutions, with grid spacing down to $Δx_{\rm min} \simeq 92\ {\rm m}$ in the most refined region. When comparing the two highest resolution simulations, we find orbital dephasing of over 7 orbits until merger of only $0.06$ radians. The magnetic field is amplified during the merger and we observe the formation of a magnetized funnel in the polar region of the remnant. Simulations are continued until about $30$ milliseconds after merger. However, due to significant baryonic pollution, the binary fails to produce a magnetically-dominated outflow. Finally, we discuss possible numerical and physical effects that might alter this outcome.
The magnetic field configuration in the interior of neutron stars and its stability are open problems and may be impacted by the influence of a turbulent cascade within the star. Assessing the impact of turbulent flow with numerical simulations requires incredibly high resolution as well as long lived simulations covering multiple Alfven times. We present a series of simulations of magnetised neutron stars with resolution up to 29m and lasting at their longest 1.2s to assess this issue, the longest lasting and highest resolution such simulations to date. At the highest resolution we find evidence for a turbulent cascade absent in an unmagnetised star which cannot be captured with lower resolution simulations, consistent with Kolmogorov power law scaling. The presence of turbulence triggers an inverse cascade of helicity, while at late times the net helicity appears to vanish, suggesting that a twisted-torus is not formed in the magnetic field. We find that the presence of the magnetic field excites a characteristic quadrupolar oscillation of the density profile at 145 Hz, consistent with Alfvenic modes proposed as the source of quasi-periodic oscillations observed in magnetars.
Given the compact binary evolution problem of numerical relativity, in the finite-difference, block-based, adaptive mesh refinement context, choices must be made on how evolved fields are to be discretized. In GR-Athena++, the space-time solver was previously fixed to be vertex-centered. Here, our recent extensions to a cell-centered treatment, are described. Simplifications in the handling of variables during the treatment of general relativistic magneto-hydrodynamical (GRMHD) evolution are found. A novelty is that performance comparison for the two choices of grid sampling is made within a single code-base. In the case of a binary black hole inspiral-merger problem, by evolving geometric fields on vertex-centers, an average $\sim 20\%$ speed increase is observed, when compared against cell-centered sampling. The opposite occurs in the GRMHD setting. A binary neutron star inspiral-merger-collapse problem, representative of typical production simulations is considered. We find that cell-centered sampling for the space-time solver improves performance, by a similar factor.