Papers

We construct a sequence of finite Rodriguez-Villegas transforms converging locally uniformly to the Riemann $\xi$-function in the critical strip. The input is a positive symmetric profile on the unit interval obtained from the Riemann theta kernel through convolution with the hyperbolic-secant kernel and the logistic coordinate. The profile is a Stieltjes function of $x(1-x)$. Its Bernstein polynomials produce reciprocal numerators and exact finite functional equations. The same numerators admit an exact realization as fermionic supertraces, while the Bernstein polynomials are normalized Gibbs traces.

We introduce a bi-oriented interlaced matrix product on $2\times 2$ matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley–Dickson doubling. The opposite orientations $X_R$ and $X_L$ coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators $Q,P$ on $L^2(\mathbb{R})$, a matrix $D$ and its companion $\widetilde{D}$ satisfy, on the common Schwartz domain, $\widetilde{D} X_R D =\widetilde{D} X_L D =2H Id_2$, where $H=(P^2+Q^2)/2$. We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations $\Phi=A+iB$ and $\Psi=A-iB$ decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space $S(\mathbb{R}^2)\subset L^2(\mathbb{R}^2) \subset S'(\mathbb{R}^2)$; the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.

Inspired by black hole thermodynamics and other results suggesting that general relativity has some intrinsic connection with thermodynamics, this work aims to establish a quasi-local thermodynamic formalism in general spacetimes. With this purpose, we generalize the Bondi-Sachs formalism to past light cones associated with an arbitrary worldline. From this generalization, we demonstrate the emission of Hawking radiation in a quasi-local context through a few examples, suggesting that it may occur under mild conditions. In addition, we argue that an "irreversible inequality" is satisfied in typical gravitational systems, indicating that an analogue of the second law may be established. Together with a definition of internal energy derived from the Landau-Lifshitz pseudotensor, these results are used to construct a postulational framework that generalizes classical thermodynamics in order to include the dynamics of spacetime. As a consequence, it is possible to define the rest-mass density from thermodynamic variables, decoupling it from the dynamics of constituent numbers.

We consider the 2D Maxwell–Lorentz equations with an extended charged rotating particle. The system admits soliton solutions corresponding to a particle moving with a constant velocity and rotating with a constant angular velocity. Our main result is the asymptotic stability of the solitons.

We prove that the subcritical mass gap is comparable to the Lee-Yang gap over free-boundary square boxes raised to the power $8/15$. Random-current moment bounds, a correlation inequality and a finite block expansion give the upper bound. The lower bound follows from the Lee-Yang product, the susceptibility and the critical magnetization. We also obtain uniform factorial bounds on the field derivatives of the pressure.

We establish an upper bound for the ground state energy density of a dilute Bose gas, interacting through a general short-range, repulsive potential. To reach this goal, we extend the approach recently introduced in [1] for hard spheres.

In this paper, we study the deformed Hermitian-Yang-Mills (dHYM) equation on the non-compact Calabi-Yau manifold $Z = {\rm{Tot}}({\bf{K}}_{X})$, where $X$ is a rational homogeneous variety. Since $X$ is a Fano variety, $Z$ is a resolution of the singularity at the vertex of the affine cone ${\rm{Aff}}(X)$, provided the cone is built using the anticanonical polarization ${\bf{L}} = {\bf{K}}_{X}^{-1}$. Using the cohomogeneity-one symmetry of the Ricci-flat Kähler metric obtained via the Calabi ansatz on $Z$, we reduce the fully nonlinear PDE underlying the dHYM equation to a scalar, asymptotically autonomous ordinary differential equation (ODE). From this, we determine the exact condition on the topological phase that guarantees global existence of solutions. As an application, we show that every holomorphic line bundle over $Z$ admits a smooth, globally defined Hermitian connection solving the dHYM equation, provided the total phase lies in an explicit open interval determined by the Lie-theoretic data. Also, we prove a rigidity result classifying the exact geometric conditions under which the dHYM solution collapses into a classical Hermitian-Yang-Mills (HYM) connection. The results established generalize previous constructions and provide a substantial new class of examples. Furthermore, the approach presented allows one to study the behavior of the dHYM solutions through ODE methods. Using this approach, we construct the first explicit non-trivial example of a Hermitian-Einstein connection on a line bundle over a non-toric Calabi-Yau manifold which is not dHYM.

We study determinantal planar Coulomb gases with a smooth radial potential, a smooth radial perturbation, jump-type singularities (including the bulk, semi-hard, hard, and soft-edge regimes), and a hard wall region given by a centered disk, a disk complement, or an annulus. We obtain explicit asymptotic expansions of the partition functions up to and including the term of order $1$. Our results unify and generalize many of the earlier works.

This paper establishes a mesoscopic central limit theorem for linear statistics of $\beta$-ensembles or log-gas, as the dimension $N\to\infty$, in the temperature regime $1/N\ll\beta(N)\le 1$. For simplicity, we assume that the potential is one-cut regular and analytic. In this regime, the size of the fluctuations depends on $\beta(N)$ and the mesoscopic scale. We show that there is a transition at a critical $\eta\asymp 1/ N\beta(N)$ between a Random Matrix regime, where the limiting variance is given by the $\mathsf{H}^{1/2}$-norm and a Poisson regime where the limiting variance is given by the $L^2$-norm. We also describe the critical regime. The proof of the CLT relies on optimal local laws at intermediate temperatures and Stein's method for $\beta$-ensembles. In particular, in this regime, it is necessary to construct new correction terms to the classical equilibrium measure to obtain a suitable re-centring of linear statistics and describe their fluctuations. We also obtain a free energy expansion.

A KZ-type connection in characteristic $p$ has $p$-curvature operators. These operators are commuting endomorphisms of the connection. We present a Bethe-ansatz-type construction of their eigenvectors and eigenvalues. The eigenvector is constructed as the value of a characteristic $p$ hypergeometric integral and extends to a flat eigensection over the Frobenius neighborhood of the corresponding Bethe point.

We introduce and develop Functional Dimensional Regularization (FDR), a novel functional renormalization group (RG) framework that extends the principles of dimensional regularization beyond the perturbative $\varepsilon$-expansion. The key insight is to define the FDR beta functions, valid in any continuous dimension $d$ and characterized by threshold functions, as a sum over all critical dimensions of the corresponding scheme-independent beta functions computed in DR. We demonstrate that this formalism exhibits all the hallmarks of a fully-fledged functional RG. Within the local potential approximation, we perform a general fixed point analysis, recovering the $\varepsilon$-expansion for all multi-critical models and establishing a direct connection to functional scaling solutions, spike plots, and eigen-perturbation spectra. Extending the framework to the second order of the derivative expansion, we compute the critical exponents for the Ising universality class in $d=3$, providing estimates for the anomalous dimension $\eta$ and RG spectrum that converge rapidly and show favorable agreement with state-of-the-art results. We present a detailed comparison with non-perturbative and proper-time RG approaches, highlighting structural similarities and key differences. In particular, we introduce a “distillation” procedure that allows one to systematically derive the FDR flow from any other one-loop-exact RG scheme. Our results establish FDR as a self-contained and competitive RG framework that combines the technical simplicity of dimensional regularization with the versatility of the functional approach.

We present bulk-primary operator-product coefficients for every Virasoro minimal model with an $E_6$, $E_7$, or $E_8$ modular invariant. Dividing by a universal product of chiral generalized-minimal-model OPE coefficients reduces the conformal bootstrap problem to a finite algebraic problem governed by root-of-unity quantum-group $6j$ symbols. The exceptional factors of the OPE coefficients can be chosen independent of the second Kac indices $s$, and their dependence on the central charge reduces to finitely many reference cases and explicit phases. A finite computer-assisted calculation yields candidate algebraic $s=1$ seed data and high-precision numerical evidence for existence and uniqueness modulo primary-field signs. We prove that exact existence and uniqueness for these seeds imply the corresponding result for all Kac labels within the genus-zero bulk locality and crossing equations; the finite seed step remains uncertified. The three families require $18$, $32$, and $107$ representative reduced OPE coefficients for $E_6$, $E_7$, and $E_8$, respectively. We provide reconstruction conventions and coefficient tables for both unitary and nonunitary models. Our calculation assumes only Virasoro symmetry, bulk locality (permutation symmetry) and OPE associativity (crossing equations).

Asymmetric orbifolds provide concrete examples of non-geometric constructions dubbed T-folds. All Z2 point groups of six dimensional asymmetric order-two orbifolds are identified. The order-two T-fold configurations on the SO(12) lattice are classified at the fermionic point of type II string theories for all Z2 point groups. The spectra of the models on these configurations are presented parametrically, in terms of certain generalised GSO phases. The minimal effective Hodge numbers were found to be (h_{11},h_{12})=(1,1) for six T-fold configurations. Asymmetric generalisations of the mirror symmetry map are conjectured. The orientifoldable configurations using the basic worldsheet parity were identified within the classification. Twisted sectors corresponding to pure asymmetric twists may contain Rarita-Schwinger multiplets with spin-3/2 states. Throughout the paper, generalised GSO projections are chosen to preserve the maximal amount of supersymmetry possible. Relaxing this, the order-two point groups were identified for which non–supersymmetric T-folds can be constructed. Since some of them may enhance to N=1 or, even, N=2 supergravities, we argue that appearance of spin-3/2 states necessarily implies that the spectrum has reorder itself in a supersymmetric fashion and hence that the vacuum energy vanishes.

We introduce causal graph rewriting, a model of computation in which local rules are applied on directed acyclic graphs in an asynchronous manner. The non-determinism arising from asynchrony is disciplined by the oriented edges, which must be understood as both computational dependencies and locality constraints—and are themselves subject to the rewriting. We illustrate the model through two examples: a particle system, and a time-dilation example—reminiscent of general relativity. We study the well-definedness and properties of induced subgraphs and graph composition, which isolate and recombine the region affected by a rewrite. We then study locality with respect to these constructions, showing how a local rewrite preserves positions, borders, and context. Our main result concerns sequential composition: locality extends from single rule applications to arbitrary valid sequences, as any local rule is automatically $*$-local and $*$-extensive. We also formalise and prove the simulation of any one-dimensional cellular automaton.

We study the edge spectrum of two-dimensional lattice Hamiltonians and magnetic Schrödinger operators in a mobility gap, an energy interval in which the bulk spectrum is potentially localized rather than necessarily absent. Under suitable localization and boundary assumptions, we prove that a constant nonzero bulk topological index implies that the entire interval belongs to the absolutely continuous spectrum of the corresponding half-space operator. The absolutely continuous spectral multiplicity is bounded below by the absolute value of the index at almost every energy in the interval. The first version of the auxiliary-space cut construction in Section 2.3 was generated using ChatGPT 6.

The constraints for the effective local gauge groups stemming from the causal action principle for causal fermion systems are reviewed. The constraint which are quadratic in the gauge potentials (coming from the so-called bilinear logarithmic terms) are shown to make a connection between the structures of Lie algebras and Clifford algebras. This connection is worked out systematically starting from general chiral potentials corresponding to the gauge group $\mathrm{U}(N) \times \mathrm{U}(N)$. The general results are illustrated in various examples.

A Ševera bicomplex is a bicomplex $(M,d_{0},d_{1})$ whose differentials $d_{i}$ are Grothendieck differential operators of order $\leq i$ with respect to a given $A$-module structure on $M$, where $A$ is a graded commutative algebra. One also requires that $(M,d_{0})$ admits a contracting homotopy $K$ that is a differential operator of the same order as $d_{0}$. Under suitable assumptions, the differential $d_{2}$ of the second page of the spectral sequence associated with $(M,d_{0},d_{1})$ is a Batalin-Vilkovisky operator on $E_{2}$, so that when $E_{2}$ is a free rank 1 $A$-module, the graded commutative algebra structure of $A$ is enhanced to a Gerstenhaber algebra structure, and the choice of a $d_{2}$-closed basis element for $E_{2}$ further enhances this to a Batalin-Vilkovisky algebra structure. The prototypical example of this construction is Ševera's description of the Batalin–Vilkovisky algebra structure on the algebra of smooth functions of an odd symplectic manifold. Throughout the whole article we look at Batalin-Vilkovisky algebras through the lenses of Cartan calculus. Indications of a generalization to derived Cartan calculus are briefly discussed in the concluding section.

For each sufficiently large $N$, we construct smooth solutions of the magnetohydrostatic equations on embedded solid tori whose regular pressure levels are nested tori, whose magnetic field vanishes exactly on a round magnetic axis, and whose group of Euclidean symmetries, even when isometries reversing the sign of the field are admitted, is exactly the cyclic group $C_N$ of rotations through multiples of $2\pi/N$. For each such $N$, these equilibria form a smooth one-parameter family which is locally nontrivial in the moduli space of embedded configurations and have none of the plane-reflection, axial, or helical symmetries conjectured by Grad. A Lean 4 certification of the above results is also provided.

We report the first calculation of the lowest-known energy configuration for $N=100\,000$ classical point charges confined to a disk and interacting via the $1/r$ Coulomb potential - a system size never before achieved for the Thomson problem in a disk, more than doubling the previous record of $N=40\,886$ reported by Amore and Zarate. An adaptive defect-targeting subdomain optimization strategy yields an approximately linear growth of the wall-clock time per cycle with $N$; for $N=100\,000$ a single $24$-core CPU workstation without GPU acceleration required $\approx31$ hours. The energy $E_{\min}=7.80466624157\times10^{9}$ deviates from the standard empirical asymptotic expansion by only $2.525\times10^{-7}$. We demonstrate that this remarkable agreement provides a stringent test of the rigorous next-order asymptotic theory for Riesz interactions, while the residual deviation is consistent with an $O(N^{4/3})$ boundary-layer correction whose absence from the standard fitting basis biases the empirical $N^{3/2}$ coefficient away from its exact crystalline value, thereby tying the deviation to the two-dimensional crystallization conjecture. Bond-orientational order analysis reveals a polycrystalline bulk threaded by radial grain boundaries, establishing a new benchmark for two-dimensional Coulomb systems.

Discrete gradient methods provide a structure-preserving approach to energy-consistent time integration of dissipative Hamiltonian systems. In their conventional formulation, however, they generally attain at most second-order convergence. Extending the perturbation principle previously developed for Poisson systems to the dissipative setting, we derive fourth-order average vector field discrete gradient schemes. A congruence-based completion preserves the positive semi-definiteness of the dissipation matrix, thereby guaranteeing unconditional energy dissipativity for all positive step sizes. This completion concept provides a systematic route to higher-order energy-consistent schemes. For purely dissipative systems, an alternative exponential completion is derived, together with structure-preserving rational approximations. In particular, we characterize Padé approximants that preserve positive semi-definiteness and can be incorporated into energy-associated splitting schemes. Numerical experiments on a damped physical pendulum and a dissipative Fermi–Pasta–Ulam system demonstrate the importance of the positive semi-definite completion for avoiding nonphysical numerical instabilities and capturing the correct qualitative dynamics. They further show that the proposed methods achieve the desired accuracy and energy consistency at a computational cost competitive with existing Galerkin-type approaches. Moreover, the fourth-order schemes outperform the conventional second-order discrete gradient method in computational efficiency over the considered accuracy range, despite their higher per-step cost.

We study strict ground-level gaps for autonomous nonlinear Dirac equations on noncompact metric graphs. The relevant level at infinity is the autonomous full-line level \(d_\lambda^\infty=d_{\lambda,\mathbb R}\), since a concentrating sequence escaping along a half-line sees the real line after recentering far from all vertices. We prove that, uniformly for \(\lambda\) in a compact subinterval \(I\Subset(-mc^2,mc^2)\), the presence of a sufficiently long pendant path in the essential reduction of the graph implies \[ d_\lambda(\mathcal G)<d_\lambda^\infty. \] The proof uses a half-line restriction of a symmetric full-line ground state, uniform exponential decay, resolvent localization for massive Dirac-Kirchhoff spectral projections, and a compactness argument for generalized Nehari fibres. We also identify elementary graph classes for which no such gap can hold: pure transmission graphs whose essential reduction is isometric to the full line satisfy \(d_\lambda(\mathcal G)=d_\lambda^\infty\).

Dirac-harmonic maps are a mathematical version of the supersymmetric non-linear sigma model of quantum field theory. Since they arise as critical points of an unbounded energy functional, it is a challenging task to establish general existence results. Within this manuscript we mainly prove a general existence result based on index theory for manifolds with boundary and apply it in various situations. Due to the presence of a boundary there are additional contributions in the Atiyah-Patodi-Singer index formula allowing for a non-vanishing index, which is the key argument in our proofs. Further explicit examples based on twistor spinors on surfaces are also presented.

In environmental feedback systems, self-interested behaviors of rational agents often undermine cooperation and environmental sustainability. Although punitive incentive mechanisms are widely recognized as effective in addressing such social dilemmas, the timing of their implementation under different environmental conditions remains insufficiently understood. In this paper, we develop a multi-agent environmental feedback game framework with coupled incentive intensities across resource states. We investigate how system dynamics vary with incentive intensity under both resource-abundant and resource-scarce conditions. Theoretical analysis shows that imposing penalties in resource-abundant states can transform the system from a tragedy of the commons into a bistable regime that supports full cooperation. In contrast, imposing penalties in resource-scarce states leads to complex dynamical behaviors, including interior stability, heteroclinic cycles, and Hopf bifurcations, which hinder the emergence of stable cooperation. These results highlight the importance of timing in incentive regulation. Proactive intervention under favorable environmental conditions is more effective in sustaining cooperation than delayed enforcement after environmental degradation.

Two-point correlation functions are a standard summary statistic in cosmic shear and large-scale-structure analyses. Their values are, however, constrained: non-negativity of the underlying power spectrum restricts any admissible sequence of correlation coefficients $(r_1,\ldots,r_N)$ to a bounded convex region, described in the one-dimensional case by the recursive interval geometry of Schneider & Hartlap (2009). Their formalism introduces an affine variable $x_n$ that maps the admissible interval for $r_n$, given $r_1,\ldots,r_{n-1}$, to $[-1,+1]$. We identify $x_n$ with the partial autocorrelation coefficient $\alpha_n$ of the associated positive Toeplitz correlation matrix, a classical quantity in time-series analysis. The partial autocorrelations thus provide natural coordinates on the admissible region, each $\alpha_n$ varying independently in $[-1,+1]$. This identification makes the standard partial-autocorrelation toolbox directly applicable. The admissible intervals for $r_n$, cumbersome to obtain at higher order in the original determinant-based formalism, now follow from an $\mathcal{O}(N^2)$ recursion at arbitrary order. The inverse hyperbolic tangent used in the quasi-Gaussian likelihood construction of Wilking & Schneider (2013) becomes Fisher's $z$-transformation of partial autocorrelations, providing a classical statistical interpretation of its empirical Gaussianising effect. Numerical experiments illustrate the practical use of these natural coordinates in the tested one-dimensional settings. Higher-dimensional isotropic constraints require additional geometric input and lie outside the direct scope of this identification.

Trust develops through learning, while collective behavior can alter the environment in which later decisions are made. Reinforcement-learning models describe adaptation, and eco-evolutionary models describe behavior-environment feedback, but how an endogenous environment changes trust through material incentives and perceived states remains unclear. We couple a fixed-role, two-population hierarchical trust game populated by heterogeneous tabular $Q$-learning agents to a centered endogenous environment. For the main feedback comparisons, we initialize the fixed-payoff baseline and feedback branches from the same learned state to control for prior learning. We then compare responses when payoff parameters or the observed environmental tier vary. As the payoff multiplication factor increased, successful trust in the fixed-payoff game crossed from a low to a higher finite-horizon level, led by risk-seeking investors. Environmental feedback promoted trust near perception-tier boundaries and reinforced its decline at the high-return operating point. At the high-return operating point, payoff feedback reinforced the learning-driven decline in trust. Stronger drive and longer horizons deepened this negative effect. Compared with a fixed-payoff trust game, environmental feedback carries past collective outcomes into future payoffs and observed states.

A two-component Bose–Einstein condensate can host a pseudospin Hopf texture coupled to the viscous normal flow of its thermal cloud. Motivated by OpenAI's reported finite-time Navier–Stokes singular solution, we evolve such a texture in a smooth flow reproducing the geometry and similarity scaling of the collapsing axisymmetric core. The evolution creates and annihilates preimage pairs while preserving the Hopf charge. The first one-to-three transition exhibits the square-root opening characteristic of a Thom $A_2$ fold. A subsequent three-to-five transition precedes a burst-like regime of recurrent fold catastrophes and near-caustic activity.

On a $d$-dimensional, $d\geq 2$, globally hyperbolic spacetime with a timelike boundary $(\mathcal{M},g)$, we investigate the Klein-Gordon equation with Robin boundary conditions. First of all, we prove existence and uniqueness of the advanced and retarded fundamental solutions, discussing in addition their structural properties. Secondly, under the hypothesis of infinitesimal convexity of $\partial\mathcal{M}$ and of local finiteness of boundary reflections for broken null bicharacteristics, we characterize their wavefront set in the interior $\mathring{\mathcal{M}}$, using the propagation of singularities theorem of Melrose and Sjöstrand. In the second part of the paper, we introduce a notion of Robin-Hadamard two-point correlation function as a positive distribution on $\mathcal{M}\times\mathcal{M}$ with prescribed antisymmetric part and singular structure, the latter being characterized in terms of its wavefront set. We complement this definition with a local Hadamard form. Up to a smooth remainder, this consists of the standard directed Hadamard parametrix together with additional contributions associated with the reflected null rays. Their coefficients are determined by transport equations supplemented by Robin matching conditions at the boundary. The construction applies to an arbitrary, but finite number of reflections. We prove that this local formulation is equivalent to the wavefront set condition, thereby establishing a counterpart of Radzikowski's theorem in the presence of a timelike boundary. Finally, starting from the ground state on suitable static backgrounds and using a deformation argument, we prove existence of Robin-Hadamard two-point correlation functions on the class of spacetimes considered.

We consider $N \times N$ random band matrices $H = (H_{xy})$ with centered complex Gaussian entries, indexed by points $x,y$ on the two-dimensional discrete torus $(\mathbb{Z} / \sqrt{N} \mathbb{Z})^2$. The matrix entries $H_{xy}$ vanish whenever the distance between $x$ and $y$ exceeds the bandwidth parameter $W$. We prove that if $W \geq (\log N)^{7}$, then, with high probability, all bulk eigenvectors are delocalized. Equivalently, this yields a lower bound of order $\exp(W^{1/7})$ for the localization lengths of two-dimensional random band matrices, improving the superpolynomial lower bound $W^C$ for every fixed constant $C>0$ established in arXiv:2503.07606.

We prove a recursive formula for construction of generalized spectral basis. Example of application is presented for symbolic computation with non-diagonalizable matrix that describes electromagnetic wave propagation in anisotropic media. The method is general enough and can be applied, for example, to multivectors of the Clifford algebra.

The exact behaviour of the enveloppes of the single-time and two-time correlators is found for the voter model on a fractal substrate, with nearest-neighbour interactions. Herein the geometry of the fractal substrate is described by its non-integer geometric fractal dimension $d_f$, and its topology and diffusive transport by the distinct spectral dimension $d_s$. On the level of the equations of motion of the correlators this can be modelled by considering a space-dependent diffusion constant ${\cal D}(r)\sim r^{-\theta}$ which implies the spectral index $\theta$, itself a function of $d_f$ and $d_s$. With a scaling ansatz, the generic phenomenology of ageing is confirmed and the dynamic exponent ${z}=2+\theta$ and the autocorrelation exponent $\lambda=d_f$ are derived. The explicitly found dynamic scaling functions are shown to depend only on the spectral dimension $d_s$. The decay of the enveloppe of the density of active interfaces with time is described by the exponent $\alpha=1-d_s/2$ for $d_s<2$, confirming the results of preexisting numerical simulations.