Papers

We show that for any positive operator monotone function $f$ the inequality $\int_{[0,\infty)} f(t) \, \mathrm{d} \left\lVert E^{\Delta_{\varphi,\psi}}(t) \xi_\psi \right\rVert^2 + f'_\infty \varphi(1-s(\psi)) \geq \frac{f(1)}{2} \Bigl( \varphi(1) + \psi(1) - \lVert \varphi - \psi \rVert \Bigr)$ holds, where $\varphi, \psi \in \mathcal{M}_{*,+}$ are two normal positive linear functionals on a von Neumann algebra $\mathcal{M}$, and $\Delta_{\varphi,\psi}$ is the associated relative modular operator. Choosing $f(t) = t^s$ with $s \in [0,1]$, the Powers-Størmer-Ogata inequality is recovered.

The super Schottky ideal defines the image of the superperiod map, which takes a family of genus $g$ supercurves to its Jacobian. We prove that if $g \geq 5$, then $d=g$ is the minimal number such that $I_{Sch,g}^d \subset I_{s-Sch,g}$, where $I_{Sch,g}$ is the Schottky ideal and $I_{s-Sch,g}$ is the super Schottky ideal.

We study the Heisenberg XXX spin chain with twisted periodic boundary conditions with a defective twist matrix. Using quantum separation of variables, we show that the generalised eigenvectors of this system can be characterised in terms of a modified TQ equation that we have named "Jordanian TQ equation". We also show that the wavefunction in the separated basis is only separable for true eigenvectors, while the wavefunction for generalised eigenvectors of higher rank is given instead by a sum over weak compositions.

The study of super Riemann surfaces and their moduli led Witten and Felder, Kazhdan and Polishchuk to ask what is the smallest $d$ such that the $d$-th power of the Schottky ideal is contained in the super Schottky ideal. Felder, Kazhdan and Polishchuk proved that $d=g$ in odd genus $g\geq5$ and that $d\in\{g-1,g\}$ in even genus $g\geq4$; Y.~Shen subsequently proved that $d=g$ in even genus $g\geq6$. Thus the only remaining open case was $g=4$. Here we close this gap by showing that in genus $4$ too, the smallest power of the Schottky ideal contained in the super Schottky ideal is $d=g=4$. The question is equivalent to one concerning the codifferential of the super period map. The quadratic odd contribution sends a conormal vector to a bivector, represented by a skew-symmetric $(2g-2)\times(2g-2)$ matrix. The question is to find such a vector for which, at a generic point, this matrix has maximal rank $2g-2$, or $6$ in our case. Our computation uses the standard exact sequences on $C\times C$ attached to the sheaves $\OO(a,b,c)$, together with a degeneration to a vanishing theta-null and some facts about the Szegő kernel. At a curve with a vanishing theta-null, a regularized version of the codifferential can be described by an explicit multiplication map, allowing an easy computation of the rank. This rank turns out to be $4$. A first-order calculation on an explicit deformation of a cyclic trigonal vanishing theta-null differentiates the global Gaussian-map identity for the regularized conormal bivector. Including the variation of the Gaussian map gives a nonzero first normal symbol. The resulting first-order form is nonzero on the null space of the limiting skew-symmetric matrix, so the rank jumps to $6$ on nearby curves, showing that $d=4$.

We completely classify unitary 2d CFTs with central charge $c=1$ and discrete spectrum. We confirm the folklore that the only such theories are the compact free boson, its $\mathbb{Z}_2$-orbifold, and three exceptional theories obtained as orbifolds of the SU(2)$_1$ WZW model. The proof is by recasting the $c=1$ modular bootstrap as a crystalline measure problem for the density of states, and analytically solving it. We also show that, once the density of states is specified, the full CFT can be uniquely reconstructed.

We follow a $D$-module approach to the construction of canonical differential equations for Feynman integrals directly from their Gel'fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems. Starting from the Schwinger representation, we identify the associated generalized Euler integral and its GKZ system, as an alternative to the Lee-Pomeransky representation. The generalized Euler formulation allows reductions associated with facets of the Newton polytope to be identified directly, reducing the number of variables in the resulting differential systems. We construct the associated Pfaffian systems using Frobenius bases, which also provide direct access to the singular loci. After a suitable rationalizing change of variables, the systems are brought into canonical $\epsilon$-form. A key feature of the construction is that the dimension of the canonical basis is determined by the holonomic rank of the GKZ system and, for the examples considered, it is smaller than or equal to the number of master integrals obtained from integration-by-parts, thus providing a reduced description of the differential system.

We study eigenvalues of discrete Schrödinger operators $H=\Delta+V$ on $\mathbb{Z}^2$, where $\Delta$ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of $\mathbb{Z}^2$ and $V$ decays at infinity. By Weyl's theorem, the essential spectrum of $H$ is $[-4,4]$. We determine the sharp decay thresholds for existence of eigenvalues in three distinct spectral regimes. While it is natural to expect different behavior at the spectral edge $\lambda=\pm4$, and the bulk, there is a further distinction between the regular energies $0<|\lambda|<4$ and the interior critical energy $\lambda=0$ stemming from the reducibility of the corresponding Fermi surface in the latter case. For every $0<|\lambda|<4$, we construct potentials $V$ satisfying $|V(n)|\leq C|n|^{-1}$ for which $\lambda$ is an eigenvalue of $\Delta+V$, and prove absence of eigenvalues when $|V(n)|\leq C|n|^{-1-\varepsilon}$ for some $\varepsilon>0$. At $\lambda=0$, the critical power changes and we construct potentials $V$ satisfying $|V(n)|\leq C|n| ^{-2}$ for which $0$ is an eigenvalue of $\Delta+V$, as well as prove absence when $|V(n)|\leq C|n|^{-2-\varepsilon}$ for any $\varepsilon>0$. Finally, at each spectral edge, we show that, for every $K\geq3$, an eigenvalue can be created by potentials supported on exactly $K$ sites, whereas a potential supported on at most two sites cannot create an edge eigenvalue. The proofs combine Green-function expansions and moment cancellation, Hilbert-space-valued iterations, discrete Carleman estimates, and a uniform Green-kernel estimate.

We solve the robust ten martini problem: Cantor spectrum with no arithmetic restriction on the irrational frequency for Type I analytic one-frequency quasiperiodic Schrodinger operators. We introduce Type I through a new acceleration-type quantity with stability under analytic perturbations. This yields a global open class of analytic potentials whose associated operators have Cantor spectrum for every irrational frequency.

Entanglement can reduce the communication required for coding tasks, but establishing the minimum achievable cost is essential to understanding its limits. We address this question in a zero-error source-coding task where Alice receives a word and Bob knows an unordered pair of candidates containing it. Alice does not know the pair and must enable Bob to identify her word without error using shared entanglement and one classical message. The candidates satisfy a balanced-difference promise: for words in $\mathbb{Z}_q^n$ with $n=q\ell$, each residue modulo $q$ occurs exactly $\ell$ times in their coordinatewise difference. For all integers $q\geq2$ and $\ell\geq1$, we prove that the task requires exactly $n$ messages when $(q-1)\ell$ is even and two messages when it is odd. These minima allow arbitrary finite-dimensional shared states independent of the inputs and arbitrary local measurements. In even parity, this establishes optimality of an existing entanglement-assisted protocol. In odd parity, an explicit deterministic protocol achieves the optimum of one bit without entanglement. Our proof combines Fourier analysis with a combinatorial counting argument to determine the smallest eigenvalue of the associated graphs. In even parity, this resolves the spectral assertion of Cao et al.'s Conjecture 6.3 for balanced cyclic generalized Hadamard graphs. Together with an explicit odd-parity bipartition, this determines the quantum chromatic number as $n$ in even parity and $2$ in odd parity, where the classical chromatic number is also $2$. All lemmas, theorems, and corollaries are formalized and verified in Lean.

After proving the moonshine conjecture, Borcherds asked in 1995 whether there exist only finitely many holomorphic automorphic products of singular weight. We show that, up to conjugation, there are exactly 27 such products on lattices of the form $2U\oplus L$. The classification proceeds by identifying each product at a standard $1$-dimensional cusp with the superdenominator of a BKM superalgebra that carries a graded module structure over an affine Lie superalgebra $\hat{\mathfrak{g}}$. The auxiliary algebras $\hat{\mathfrak{g}}$ are shown to satisfy strong conditions that reduce the possibilities to an initial list of 1372 candidates. An elimination argument leaves exactly 85 algebras $\hat{\mathfrak{g}}$ that correspond to the 27 products. This classification is closely connected with certain distinguished families of vertex operator superalgebras. Our results provide a systematic framework for extending affine Lie superalgebras to hyperbolic ones with modularity, generalizing the work of Feingold and Frenkel (1983).

We study the eigenvalues and eigenvectors of the adjacency matrix $A$ of the Erdős-Rényi graph $\mathbb G(N,p)$ in the regime $Np \gg (\log N)^2$. We establish optimal isotropic delocalization for the bulk eigenvectors $\boldsymbol u$, meaning that $\langle \boldsymbol v, \boldsymbol u\rangle^2 \leq \frac{C \log N}{N}$ with very high probability for any deterministic normalized $\boldsymbol v$. In addition, we prove local spectral universality in the bulk by showing that the local spectral statistics coincide with those of the GOE. The main tool of our proof is a local law for the resolvent of $A$, down to optimal spectral scales and with optimal error bounds. Its proof relies on a new approach to local laws that we call the Bernoulli flow. It is a characteristic flow method in which the usual Brownian process is replaced by a Bernoulli process, where each edge of the graph forms an independent Markov process that jumps at unit rate from closed to open. The spectral parameter flows according to a suitably constructed matrix-valued Bernoulli characteristic flow.

We discover an exponent-dependent phase transition phenomenon for optimal hypercontractivity: for every prescribed $q_0>2$, there exists a reversible continuous-time Markov chain on three state space with normalized spectral gap whose $(2,q)$-optimal hypercontractivity time satisfies $$ \text{$t_{\mathrm{opt}}(2,q)=\frac12\log(q-1)$ if and only if $q\ge q_0$},$$ whereas the strict inequality $t_{\mathrm{opt}}(2,q)>\frac12\log(q-1)$ holds for $2<q<q_0$.

By exploiting complex coordinates and elementwise matrix products, a single and simple differential equation is derived that describes the motion of every multi-arm planar pendulum.

Image reconstruction in optical interferometry remains a challenging inverse problem due to sparse Fourier sampling and the strong dependence of the reconstructed images on user-defined parameters. We present PYRA (Python for MiRA) and MYTHRA (Mean Astrophysical Images with PYRA), two tools designed to improve the robustness and reproducibility of interferometric imaging. PYRA automatically generates large ensembles of MiRA reconstructions by exploring a wide range of reconstruction parameters, while MYTHRA statistically analyzes the resulting images to identify the most consistent solutions and derive a final averaged image with associated uncertainty estimates. The methodology is validated using the blind datasets of the 2024 Interferometric Imaging Contest, where our reconstruction strategy achieved the best overall score. We further apply the method to VLTI/MATISSE observations of the triple AGB system {\pi}1 Gru and the A[e] supergiant 3 Pup. These results demonstrate that statistical analysis of large reconstruction ensembles provides a powerful framework for assessing image reliability and reducing reconstruction artifacts in optical interferometry.

We study the quantised version of the Kadomtsev-Petviashvili equation in the formulation proposed in [4]. We analyse two aspects. One is the conservation of the first higher charge $H_3$ (besides the total mass $H_0$, the momentum $H_1$ and the Hamiltonian $H_2$). We find the expression of $H_3$ and prove commutativity, provided we regularise a specific ultraviolet divergence. The result then does not depend on the choice of a regulator, under fairly general assumptions. We then study the Bethe ansatz in finite volume and obtain a general solution to the two-body problem. We have used AI supervised by us to aid the proofs.

Universality relies on combining operators with different algebraic properties, which depend on the physical structure of the Hilbert space encoding the information. Systems of identical symmetric particles—including bosons and symmetric collections of spin-$1/2$ particles—can be mapped to angular-momentum systems, where rotations supplemented by quadratic, spin-squeezing interactions provide universal control. We revisit universality in this framework and confirm that such linear and quadratic operations provide universal control on every fixed-particle-number subspace. At the abstract level, however, we show that the dynamical Lie algebra generated by all degree-at-most-two elements does not exhaust the full universal enveloping algebra of $\mathfrak{su}_{\mathbb{R}}(2)$, contrary to a stronger claim made previously: higher powers of the quadratic Casimir element cannot be generated by commutators and linear combinations. This obstruction does not conflict with fixed-sector universality because, under a fixed irreducible spin-$n/2$ representation, all powers of the Casimir element act as scalar multiples of the identity. We determine the resulting abstract dynamical Lie algebras exactly and show that the central-free and full degree-at-most-two algebras map, respectively, onto $\mathfrak{su}_{\mathbb{R}}(n+1)$ and $\mathfrak{u}_{\mathbb{R}}(n+1)$, establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.

We propose a dissipative, irreversible damage model for anisotropic media in large deformations to address the damage process of the myocardium following a cardiac infarction. We model damage to the collagen in the cleavage planes resulting from an increased load to the passive tissue. Starting from variational principles, we derive a quasi-static differential model governing the irreversible evolution of the damage. We apply our model to two test cases. First, a rectangular passive slab geometry to which an indenter-like load is applied. Then, we couple our model with a model for the numerical simulation of left ventricular electromechanics. Our results show that a redistribution of the load within a left ventricle occurring following a cardiac infarction can produce the damage to the collagen scaffold in the infarcted area consistent with experimental results. This is indicative of the fact that passive load-dependent dissipative phenomena are implicated in post-infarction remodelling.

Exceptional points provide a mechanism for producing strong spectral sensitivity in non-Hermitian wave systems. For subwavelength resonators with highly dispersive material parameters, however, the resonance problem is not a linear eigenvalue problem in the frequency. In this paper, we use the Lippmann-Schwinger formulation and the associated volume-potential reduction to study exceptional points in dispersive resonator arrays. The reduction leads to a finite-dimensional nonlinear matrix pencil whose entries encode both the local material response and the volume-mediated interaction between the resonators. We characterize exceptional points as defective characteristic values of this pencil and derive explicit conditions for dimers and higher-order systems. We show that passive damping and detuning can generate exceptional points without imposed gain, and we formulate an inverse-design condition for placing such degeneracies at prescribed frequencies. We also prove the corresponding Puiseux splitting laws and identify how material dispersion enters the leading sensitivity prefactor. Numerical experiments, including halide-perovskite-inspired dimers and trimers, illustrate the theory and show that the exceptional-point mechanism persists for explicit highly dispersive passive material laws.

We propose a hybridized staggered discontinuous Galerkin–mixed finite element method for the strain gradient elasticity model, a fourth-order singularly perturbed problem governed by a material length scale parameter $\iota$ and the Lamé constants $\lambda$ and $\mu$. Introducing the total stress together with the scaled Cauchy and hyper stresses as auxiliary unknowns, we recast the model as a first-order system and discretize the displacement and the total stress by staggered discontinuous Galerkin spaces, the two scaled stresses by Raviart–Thomas pairs, with symmetry imposed strongly on all three stresses. The higher-order Dirichlet boundary condition enters the variational formulation naturally, so that the scheme avoids the numerical boundary layer that limits displacement-based methods. By establishing an inf-sup condition on the symmetric subspace and a discrete Korn inequality, we prove algebraic stability and optimal convergence, both uniform in $\iota$ and $\lambda$. We further develop a hybridized scheme in which local static condensation leaves only two multipliers on the mesh skeleton as globally coupled unknowns. Numerical experiments confirm the predicted convergence rates and the parameter robustness.

For the $k$-self-similar naked singularity solutions of Einstein–scalar field equations constructed by Christodoulou, we construct a family of interior gravitational perturbations leading to trapped surface formation that remain large but finite at the threshold Hölder regularity, while converging to zero in all regularities below the threshold. This extends the interior spherically symmetric result below the threshold in \cite{Li25} to non-spherically symmetric setting. We further construct a genuinely localized family of perturbations whose angular support shrinks to a single point, with smooth convergence to zero away from that point. This exploits the additional angular freedom available beyond spherical symmetry. Moreover, when measured instead in Sobolev regularity, such genuinely localized family of perturbations still converges to zero in all regularities below the same threshold.

We develop a biorthogonal time-dependent variational principle for real-time dynamics of non-Hermitian quantum many-body systems. Independent left and right matrix-product states obey coupled bivariational tangent-space equations whose cross-Gram matrix defines an oblique projection. A matrix-free scaled Taylor action propagates the resulting non-normal local generators without assembling dense matrices or storing a Krylov basis. We distinguish the fully coupled algorithm, which solves the cross-pairing problem and truncates the two bond bases jointly, from an efficient independently propagated approximation used for large systems. Independent truncation can make the retained left-right pairing nearly singular; overlap drift and the smallest singular value of the bond cross matrix expose this failure, while coupled truncation substantially delays it. Exact benchmarks and convergence tests validate the method. Applied to an interacting long-range non-Hermitian Ising chain, it resolves a biorthogonal dynamical quantum phase transition and shows that a weak imaginary field shifts the leading critical time from $t^{\ast}|J|=1.84$ to $1.04$.

In this work, we construct analytical kink-antikink $(K\bar{K})$ configurations in the non-integrable $\varphi^4$ model by mapping exact solutions of the integrable sine-Gordon system via a field deformation. This procedure yields two distinct classes of configurations, parametrized by the initial half-separation and velocity. We compare these profiles with the standard additive superposition Ansatz in terms of vacuum structure and equation-of-motion residuals, deriving explicit closed-form expressions for the corresponding integrated squared residuals. For small separations, the mapped soliton-antisoliton configurations exhibit appreciably smaller residuals than the naive superposition Ansatz, which in turn performs better than the mapped two-soliton configurations. Although the mapped fields are not exact solutions of the $\varphi^4$ equation of motion, they provide mathematically consistent, topologically sound, and physically motivated initial data for numerical studies of kink-antikink scattering and resonance phenomena.

We establish two results on the asymptotic equipartition property (AEP) for quantum channels. First, the AEP formulated using smoothing over completely positive trace-preserving maps fails in general. Second, the AEP with smoothing over subchannels (completely positive trace-nonincreasing maps) is equivalent to the strong converse for parallel quantum channel Stein's lemma, to a sharp threshold for the channel hockey-stick divergence, and to an AEP for the channel Lorenz divergence. The counterexample reveals a surprising separation between channel and subchannel smoothing. Although states and classical channels admit a sharp, dimension-independent completion bound, which we establish, the corresponding single-shot gap for quantum channels is unbounded already for qubits. This separation persists asymptotically: we construct qutrit channel pairs with finite max-relative entropy whose CPTP-smoothed rate strictly exceeds the regularized channel relative entropy. The asymptotic gap between channel and subchannel smoothing is unbounded across the family, even when the subchannel approximation error decays exponentially. To establish the equivalences, we develop a uniform filter that converts Lorenz smoothing into a single nearby subchannel. The four conjectured formulations thus share a common asymptotic threshold, whose validity for general channel pairs remains open. Together, these results separate this threshold question from the obstruction imposed by exact trace preservation.

Connection formulas for the asymptotic solutions of a system of n> 1 coupled Painleve'-II equations with symmetry-breaking parameters are written explicitly. An asymptotically exact WKB approach to these formulas relies on the quantum-mechanical independent crossing approximation for an explicitly time-dependent Schroedinger equation.

We introduce discrete $C^*$-Frobenius algebras as categorical data for constructing infinite-index extensions of Möbius covariant nets. The underlying representation is a countable direct sum of simple dualized modules, while the direct sum itself is not necessarily dualizable. Accordingly, rather than requiring a single bounded multiplication morphism, we work with compatible families of bounded left and right multiplication morphisms. Using Bin Gui's theory of categorical extensions, we associate to every discrete $C^*$-Frobenius algebra two (generally non-local) Möbius covariant extensions generated by left and right charged fields, respectively. If the discrete $C^*$-Frobenius algebra is commutative, the two extensions coincide and yield a local Möbius covariant extension. As applications, for every $C^*$-tensor category generated by an invertible object with trivial self-braiding such that its fusion product rule is given by $\mathbb{Z}$, we construct two classes of examples: the simple current extension by $\mathbb{Z}$ and the discrete Longo-Rehren construction.

We study boundary transmission and excursion geometry in the planar Lorentz mirror model through finite-volume numerical experiments. Cylinder penetration quantiles grow approximately linearly with circumference over the simulated range, with a systematic upward drift in the normalized quantiles when vacancies are present. Estimated slab-transmission probabilities yield exponential confinement bounds with quantified statistical confidence at the tested widths. Planar excursions conditioned to reach a distant boundary exhibit transverse ranges proportional to the exit radius, while their repeated-visit fractions vary strongly with mirror density. Endpoint-label collision probabilities substantially exceed a general gluing lower bound, although they account for only part of the observed same-column crossing probability. Finally, rerouting at repeated vertices preserves the exit while substantially shortening the sampled paths. These findings characterize finite-volume transport and excursion geometry without determining an asymptotic scaling exponent or resolving infinite-plane localization.

We develop Geometric Invariant Theory for affine superschemes under the action of reductive algebraic supergroups, formulating the theory in terms of coordinate Hopf superalgebras in order to accommodate anticommuting and nilpotent variables. Within this algebraic setting, we construct affine superquotients and establish their basic structural properties under suitable hypotheses. We then prove a supergeometric analogue of the Hilbert-Mumford criterion, giving a numerical test for the semistability and stability of points under natural constraints on the supergroup coaction. We show that this numerical criterion can be used to construct the GIT superquotient of these superschemes, and we give examples illustrating how the supergeometric GIT superquotient differs from its classical counterpart.

Hyperbolic band theory (HBT) has recently unveiled a wealth of exotic physical phenomena in negatively curved spaces. Concurrently, the quantum metric has emerged as a fundamental tensor governing quantum geometry and topological phases. In this Letter, we bridge these two frontiers by investigating the quantum metric in HBT. Because conventional Euclidean metric extraction protocols fundamentally fail in these non-commutative geometries, we introduce holonomy shaking, a novel, experimentally viable technique utilizing periodic lattice driving to dynamically extract the metric. Using the {8,3} regular hyperbolic and kagome-like lattices as concrete examples, we demonstrate the high-fidelity dynamical extraction of the quantum metric. Our results establish a robust blueprint for probing hyperbolic quantum geometries, paving the way for the discovery of exotic topological phenomena in hyperbolic quantum matter.

We classify the generators of stability-preserving semigroups on polynomial spaces with bounded coordinate degrees. The generators have differential order at most two, with principal coefficients characterized by low-degree nonnegativity conditions. In disk coordinates, positive degree damping gives strict zero-freeness and a sharp spectral gap, extending the Hermitian gap of Bravyi, Gosset, Liu, and Wong to complex generators. The resulting analytic domain yields deterministic classical algorithms for ground energies and stable product-state queries for bounded-degree Suzuki-Fisher Hamiltonians with bounded local strength and fixed positive fields. Field perturbation gives zero-field energy-value approximation schemes for bounded-degree unweighted EPR and bipartite Quantum MaxCut. We also prove that, up to scalars, the Hermitian qubit Hamiltonians whose full Gibbs tensors are Lee-Yang at every temperature in a fixed basis are precisely the edgewise phase-rotated Suzuki-Fisher family. With strictly positive longitudinal fields, these tensors are zero-free on a polydisk of radius greater than one at each positive inverse temperature. These two statements answer questions of Wong, Bravyi, Gosset, and Liu. Degree-preserving, coefficient-positive stability semigroups have concave sector growth rates, yielding token-graph concavity. We also give a counterexample to a proposed explicit ground-state radius.

Hamiltonian and Lagrangian mechanics are powerful frameworks for analyzing physical systems. Previous work has extended these formalisms to ecological systems, such as the predator-prey Lotka-Volterra (LV) system. In this Article, we derive a canonical Lagrangian for the two-dimensional LV model directly from its Hamiltonian representation. We find that the two-dimensional LV system admits a standard canonical Lagrangian formulation with one degree of freedom and a non-quadratic kinetic structure. This formulation admits a mechanical interpretation in terms of a particle moving in a potential well, where the non-standard kinetic structure produces a position-dependent damping term that can instead act as "revving." The derivation provides a direct connection between predator-prey dynamics and a canonical formulation of mechanical dynamics. As a verification of the construction, we apply Noether's procedure to the explicitly time-independent derived Lagrangian and reveal that the well-known Hamiltonian of the LV system is the corresponding conserved quantity. We also uncover a subtle redundancy associated with the choice of canonical momentum and its identification with the original population variables.