Papers

This paper studies an inverse problem in nonlinear optical encryption. We examine chosen-plaintext attacks (CPA) on a nonlinear optical encryption strategy that integrates double random phase encryption (DRPE) into a nonlinear optical propagation model to enhance the security of the combined system. We first demonstrate that the system's phase information can be decoded from carefully designed differential CPA data. We then demonstrate that the strength of the optical device's nonlinearity can also be recovered from CPA data, indicating that including this parameter as an additional security key does not enhance protection against CPA attacks, although numerical simulations show that strong nonlinearity still poses significant challenges for CPA attacks. Finally, we provide a stability analysis to demonstrate that small errors in decoded security keys result in only small errors in the decrypted text, even though the encryption process is nonlinear.

We study inverse boundary value problems for the fully nonlinear Monge-Ampère equation $ \det D^2u=a(x,u,\nabla u) $ in a bounded domain. Motivated by Calderón-type problems, we introduce nonlinear Cauchy data and investigate the recovery of $a$ from boundary measurements. Around a strictly convex background solution, under a sign condition ensuring invertibility of the linearized Dirichlet problem, we establish local well-posedness and differentiable dependence on boundary data. The first variation yields a linear elliptic operator with principal coefficient given by the cofactor of the background Hessian and lower-order coefficients determined by derivatives of $a$. An Alessandrini-type identity reduces equality of nonlinear measurements to an anisotropic Calderón problem. Conditional on uniqueness for this linear problem, we recover the first derivatives of $a$ along the background jet. Higher-order linearization yields integral identities whose inversion depends on a product-density property. We also establish unconditional uniqueness results in special settings. For $\det D^2u=f(x)$, local nonlinear Cauchy data determine $f$ in dimension two, and in higher dimensions under real-analytic anisotropic hypotheses. We compare this result with a planar small-data method and an all-dimensional large-data approach based on the Euclidean X-ray transform. For semilinear nonlinearities around $\Phi(x)=|x|^2/2$, Schrödinger inverse results recover the accessible $z$-jet. For equilibrium optimal-transport nonlinearities, linearization yields an isotropic conductivity equation and uniquely determines the transport density.

Our aim is to investigate the existence of local maximizers for the singular Trudinger-Moser functional $$ F_{\alpha}(u) := \int_{\Omega} \frac{e^{\alpha u^2}-1}{|x|^{a}} \mathrm{d}x,\;\;\; u \in W_0^{1,2}(\Omega), $$ restricted to the manifold $ \Sigma=:\left\{u\in W_0^{1,2}(\Omega):\|\nabla u\|_2=1\right\},$ where $a\in [0, 2)$, $\alpha\ge 0$ and $\Omega$ denotes a smooth bounded domain $\mathbb{R}^2$ containing the origin. Adimurthi and Sandeep (Nonlinear. Differ. Equ. Appl. 13, 2007) showed the following singular Trudinger-Moser type estimate \begin{equation}\nonumber \sup_{u \in W_0^{1,2}(\Omega), \, \|\nabla u\|_{L^2} \le 1} F_\alpha(u)<\infty\;\;\;\mbox{iff}\;\;\alpha \leq \alpha_a:=2\pi(2-a). \end{equation} In particular, the functional $F_{\alpha}$ is bounded on $ \Sigma$ whenever $\alpha \le \alpha_a$. In addition, Csató and Roy (Calc. Var. Partial Differ. Equ., 54, 2015) were able to ensure the existence of maximizers for $F_{\alpha}$ on $\Sigma$ when $\alpha\le \alpha_a$. In the supercritical regime $\alpha > \alpha_a$, the functional $F_{\alpha}$ becomes unbounded on $\Sigma$. Nevertheless, we prove that $F_{\alpha}$ still possesses local maximizers on $\Sigma$ beyond the critical threshold, at least for $\alpha > \alpha_a$ sufficiently close to $\alpha_a$. Our approach relies on a variational analysis near the set of maximizers associated with the critical parameter $\alpha_a$, together with a suitable local compactness argument. Our result improves and complements related findings due to Struwe (Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 5, 1988).

We say a word in a free group is “partially positive” if there is at least one generator which appears in the word, but does not appear with negative exponent. We show that one-relator groups with a partially positive relator are residually solvable. This extends a theorem of Baumslag showing one-relator groups with positive relators are residually solvable.

The invariant two-point function of a scalar field in de Sitter spacetime can be expressed in terms of Legendre functions via Lorentzian harmonic analysis. Using this structure together with the Mehler–Fock transform, we obtain an integral representation of the completed Riemann $\xi$-function in which the Legendre kernel appears naturally. Motivated by this correspondence, we introduce a retarded propagator whose spectral weight is given by the $\xi$-function and analyze it within the framework of Krein space quantization, where sign-indefinite spectral measures are admissible. This construction yields a geometric and spectral interpretation of the $\xi$-function restricted to the critical line and relates the asymptotic spacing of its zeros to a mass–time scaling in de Sitter geometry. The results provide a novel interpretive framework linking de Sitter quantum field theory, harmonic analysis, and analytic number theory.

Compact graphs are graphs for which the fractional automorphism polytope has no genuinely fractional vertices. This paper proposes a quantum analogue of this idea by evaluating the fundamental magic unitary of the quantum automorphism group on states, which we show to produce a closed convex set of doubly stochastic matrices sitting between the classical automorphism polytope and the full fractional automorphism polytope. Our main result is that the natural quantum analogue of compactness is classical, that is, a quantum compact graph is classically compact. We also relate this set to the quantum orbital algebra and obtain a hierarchy of classical and quantum compactness pseudo notions. The framework recovers familiar consequences of compactness through commutants and suggests quantum analogues of generous transitivity and distance-transitivity. We also isolate examples and open problems indicating where quantum symmetries may strictly refine the classical compactness theory.

Resolution spaces play a central role in constructing solutions for nonlinear partial differential equations. One of the main goals in the area of nonlinear dispersive PDEs has been to construct effective resolution spaces which capture the known bilinear restrictions estimates for free solutions. In this paper we propose a new structure for the Schrödinger equation which effectively replicates the classical bilinear $L^2_{t,x}$ estimate. In addition, the new structure has the property that its "dual" is an effective candidate for a space for the forcing in the linear inhomogeneous Schrödinger equation, a feature that has been elusive so far in the literature. As an application, we show how these structures can recover the known global well-posedness results for derivative NLS with null structure, with Schrödinger Maps being one such model.

Large language models can often close proof gaps in interactive theorem provers, but a verified theorem is not the same thing as a reusable library contribution. We study this distinction through a detailed case study: a semi-autonomous formalization of Grothendieck's vanishing theorem. The initial version compiles with no sorries, but an expert review found serious problems in definitions, theorem generality, file organization, and the API. We then ran a review-driven refactor and compression process and obtained a second expert review. The before-and-after comparison shows a sharp split: agents adapted well to local, mechanically checkable feedback, but remained weak at choosing definitions and designing APIs. We argue that autoformalization should be evaluated not only by closed sorries, but by whether the resulting formalization survives expert review.

We continue the study of locally finite-dimensional traces introduced in our earlier work. We give new characterizations of LFD traces in terms of finite-rank projections in irreducible representations and in the socle of the bidual. We show that, for separable nowhere scattered \(C^*\)-algebras, the set of all LFD traces is convex. We also prove that quasidiagonal traces form a face of the trace simplex and record applications to strongly self-absorbing \(C^*\)-algebras. We construct an exact tracially AF algebra not KK-equivalent to any nuclear \(C^*\)-algebra.

Given a finite group $G$ acting orientably on a surface $S$ of genus $\sigma \geq 2$, the group $G$ acts faithfully on the homology group $H_{1}(S;\mathbb{Z})$, preserving the symplectic intersection form. The action on $S$ and the homology is determined by a generating vector, a tuple of elements of $G$, generating $G$ and satisfying certain properties. In this note we show how to compute the homology representation, using the generating vector, when $S/G$ has genus 0 and the genus is suitably low. A $2\sigma \times 2\sigma$ representing matrix can be determined for any element in the group, usually for a small set of generators. The matrices are computed with respect to an auto-generated basis for the cellular homology of $S$, using a regular $CW$ structure on $S$, derived from the $G$ action. We demonstrate the application of these results by computing invariant theta characteristics of the Riemann surfaces $S$ with the algorithm implemented using Sage.

We study the dichotomy between compact satellite and ring formation in proto-planetary disks. Specifically, we examine the behavior of a model system of $N$ identical particles locked into circular, gravitationally-bound orbits around a central body. We treat interactions as dominated by inter-particle collisions, and extract an effective two-particle interaction function based on both theory and simulations. We then demonstrate that the expected dynamics are equivalent to a variant of the Kuramoto model, which undergoes a phase transition as parameters vary. This offers a novel potential explanation for the transition between formation of rings versus moons.

We prove that a sequence of constant scalar curvature (CSC) metrics which is collapsing with bounded curvature to a manifold $(X,g_{\infty})$ can be perturbed to a sequence of $\mathcal{N}$-invariant collapsing CSC metrics, under a natural assumption involving the eigenvalues of the drift Laplacian on $(X,g_{\infty})$. This answers a special case of a question of Cheeger-Fukaya-Gromov. We also give some natural conditions on the limiting metric-measure space under which the eigenvalue assumption is automatically satisfied.

We introduce a variant of the Cox ring using $\mathbb{Q}$-Cartier divisors and use this to remedy various deficiencies of nonsimplicial toric varieties. Our main applications are: Cox's ideal-variety correspondence, an explicit classification of subschemes and sheaves in terms of multigraded modules, an associated toric Deligne-Mumford stack, and a stacky extension of the GKZ/Mori theory of toric varieties.

We prove a degree-four Sum-of-Squares lower bound for the standard vector-coordinate formulations of mutually unbiased bases. For every dimension $d$ and every proposed number $m$ of bases, we construct a degree-four pseudoexpectation satisfying the orthonormality constraints and the cross-unbiasedness constraints in the quartic equality formulation. The construction is expectation over $m$ independent Haar-random orthonormal bases. We also prove that the same pseudoexpectation satisfies the degree-four localizing constraints for the natural $2\times 2$ Hermitian semidefinite formulation of the cross-coherence inequalities. Consequently, degree-four vector-coordinate SoS cannot refute the existence of $m$ mutually unbiased bases, even when $m>d+1$. In particular, under the two vector-coordinate encodings explicitly described in Randomstrasse101 Open Problem 23, degree-four SoS cannot prove that seven mutually unbiased bases do not exist in $\mathbb C^6$. We contrast this with a centered projector-coordinate Gram formulation, where degree-four SoS already recovers the elementary upper bound $m\le d+1$, giving a simple separation between vector-coordinate and projector-coordinate degree-four relaxations.

We study delay-dependent and independent passivity properties of linear port-Hamiltonian systems with delay terms. We derive passivity conditions using a particular storage function and Wirtinger-based inequalities. Furthermore, we recall several conditions for delay-dependent and independent exponential stability. Finally, we present an application of our passivity and stability results to delay equations arising in the modeling of district heating networks and oscillators with delayed damping and stiffness.

We prove a finite-scale supply–tax reduction for suitable weak solutions of the three-dimensional incompressible Navier–Stokes equations near the local regularity threshold. Along an admissible chain of parabolic windows, let B_k = A_k + C_k + D_k denote the scale-critical reservoir badness. The main PDE theorem says that if B_k stays outside a decay basin for many consecutive scales, then either cutoff/window leakage accumulates or the solution generates a positive cumulative amount of untaxed critical supply. The supply terms come from nonlinear flux, pressure transport, interpolation amplification, and pressure regeneration; the tax terms come from viscous dissipation, expected decay of old reservoir coordinates, and harmonic pressure decay. The theorem is deliberately finite-scale. It is not a proof of global regularity and it is not a construction of a singular solution. Its role is to turn persistent scale-critical badness into an explicit accounting alternative: badness can survive only by paying through untaxed supply or through non-negligible leakage. The second part of the paper interprets the ledger coordinates as a PDE-realizable finite-scale defect package. This leads to a dynamic obstruction language in which a dangerous mechanism must be profitable, reproducible, Navier–Stokes-realizable, and visible or invisible through specified pressure, flux, energy, and trace channels. The final part records finite-window quotient tests, a clean positive-cone anti-phantom theorem, and a conditional localized transfer framework. These results are finite-window or conditional tools built around the main ledger theorem, not an unconditional Navier–Stokes regularity theorem.

We show that, under constant-coefficient assumptions, the Ivancevic option-pricing nonlinear Schrödinger equation admits a Betchov-type hydrodynamic formulation analogous to the one arising in the context of the vortex filament equation. Along with this structural correspondence, known Ivancevic solutions are interpreted in terms of density–velocity variables. As an illustration of the financial interpretation, we compare the normalized one-soliton density with the Black–Scholes risk-neutral density, obtaining an explicit relation between the Ivancevic coefficient ratio and standard market quantities. We further show that the probability current can be reconstructed from the temporal evolution of the density and demonstrate this reconstruction for an interacting two-soliton solution. Together, these results illustrate how the Betchov-type hydrodynamic formulation can be used to describe the evolution and transport of the Ivancevic density, while establishing a coefficient-explicit bridge between nonlinear wave models in mathematical finance and geometric fluid mechanics.

We prove an almost everywhere divergence theorem for Cesàro means of subsequences of partial sums of Walsh–Fourier series. More precisely, we show that there exists a strictly increasing sequence of positive integers $(a_n)$ and a function $f\in L^1(G)$, where $G$ denotes the dyadic group, such that \[ \left( \frac1N\sum_{n=1}^N S_{a_n}f(x) \right)_{N=1}^{\infty} \] does not converge for almost every $x\in G$. The result is motivated by the corresponding trigonometric problem. A problem going back to Zalcwasser (1936) and remaining open for almost ninety years was recently solved in the trigonometric setting by Gát, who proved almost everywhere divergence for suitable subsequential arithmetic means of partial sums. In the Walsh–Paley setting the situation is more delicate. The dyadic structure gives positive convergence results; for instance, the partial sums $S_{2^m}f$ converge almost everywhere to $f$ for every $f\in L^1(G)$. Our theorem shows that, despite this stabilizing dyadic structure, suitably chosen arithmetic means of Walsh–Fourier partial sums may still diverge almost everywhere.

This article gives an explicit local stable resolution of cusps on GIT-stable plane quartic models. More generally, we consider a smoothing of an ordinary plane cusp over a complete discretely valued field. We show that, after a finite separable extension and a suitable choice of coordinates, a single weighted blow-up with weights ((1,2,3)) gives the stable resolution. The exceptional component is an explicit semistable Weierstrass cubic. The construction is effective and is implemented in the Sage package "StabilityFunction".

This paper establishes the AKSZ theorem for shifted contact structures and its applications. In brief, to resolve certain obstructions, we first define the quotient mapping stack as the quotient of the symplectified mapping space by the constant multiplicative group action. We then prove that if $X$ is an $n$-shifted contact derived Artin stack and $Y$ is an $\mathcal{O}$-compact, $d$-oriented derived stack, the quotient mapping stack $$[\mathrm{Map}(Y, \widetilde{X})/\mathbb{G}_m]$$ admits an $(n-d)$-shifted contact structure. In addition, by formalizing the derived analogue of the graded contact AKSZ formalism, we also introduce the notion of weak shifted contact structures in derived algebraic geometry and prove that under global trivialization of the contact line bundle, the unmodified mapping stack inherits a weak contact structure. Extending this setup to spaces with boundary, we demonstrate that derived fillings naturally induce Legendrian morphisms between quotient mapping stacks, and that topological gluing of cobordisms evaluates to derived Legendrian intersections. Furthermore, we trace the transgression of the canonical shifted $1$-form to prove that our quotient mapping stacks satisfy the derived Classical Master Equation (CME). As applications of our quotient mapping stack formalism, we define the derived analogues of specific topological field theories, including the Jacobi, Courant-Jacobi, and Loop Space Sigma Models. Next, we establish the contact analogue of the PTVV theorem for moduli spaces of perfect complexes. Finally, by composing this geometric construction with the perverse linearization in a companion paper, we elevate these moduli spaces to generate Cohomological Contact Extended Topological Field Theories.

We study the finite-sample behavior of the Generalized Fisher Transformation (GFT), the parametrization of a correlation matrix $C$ by $\gamma(C)=\operatorname{vecl}\log C$. The GFT coordinates extend Fisher's transformation to dimension $n>2$: for elliptical data their finite-sample distributions are close to Gaussian. More strikingly, the coordinates are nearly uncorrelated and their covariance is largely invariant to $C$. This approximate orthogonality and invariance make GFT-based inference far better behaved in finite samples than inference based on sample correlations or element-wise Fisher transformed correlations, yielding estimation errors that are approximately Gaussian, weakly dependent, and nearly pivotal.

We develop a method for computing stable reduction of smooth plane quartics over discretely valued fields, including residue characteristic p=2. The method uses the GIT-semistable plane models constructed in an earlier part of this project, together with an intrinsic description of hyperelliptic stable curves, to characterize when the stable model is obtained from a GIT-stable plane model by resolving its cusps. More precisely, for a smooth non-hyperelliptic curve of genus 3 with semistable reduction, we show that it admits a GIT-stable plane model if and only if its stable reduction is non-hyperelliptic. In that case, the stable model is obtained from the GIT-stable plane model by replacing each cusp by a 1-tail. Together with the companion paper on explicit local stable resolution of cusps, this gives an effective approach to computing stable reduction of smooth plane quartics. The resulting algorithms are implemented in the SageMath package "StabilityFunction".

Linear logic admits a wide range of semantic presentations reflecting its resource-sensitive notion of consequence. One well-known example is phase semantics: an algebraic semantics in which formulas are interpreted in phase models, consisting of a phase space formed by a commutative monoid and a fixed subset, with respect to which an orthogonality relation is defined, and a valuation. A rather different and much more recent approach is given by base-extension semantics, which defines validity by inductively extending a provability relation on a base – a set of inference rules over atomic propositions. We establish an equivalence between the two semantics by first defining bidirectional maps between bases and phase models, and then constructing an isomorphism between a phase model (resp. base) and its image under the composition of these maps. As a further contribution, we define the base-extension semantics clauses for the exponentials of linear logic.

We give a new proof of Vinogradov's theorem asserting that free product of biorderable groups is biorderable. Our proof relies on a simple dynamical construction that provides a bi-ordering of the free product that extends the bi-orderings on the factors. Furthermore, if the bi-orders of the factors are invariant under some automorphisms, then the bi-ordering on the product turns out to be invariant under the product automorphism.

We consider a partition of the projective space PG(m + n - 1, K), with K any (commutative) field, into one space of dimension m - 1 while all other spaces of the partition have dimension n - 1. Characterizations of particular partitions of this type are given. This research is motivated by some interesting problems on translation generalized quadrangles. In the paper we apply our results to generalized ovoids and generalized quadrangles. Finally we give some ideas for further research.

Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (\kappa^2 - \Delta)^{\alpha/2} u = \mathcal{W}, \;\; \kappa \in \mathbb{R}, \; \alpha \in \mathbb{N}. \] of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to $\alpha, \kappa$ and thus allows a universal approximation scheme for the precision and covariance matrices of the entire $(\alpha, \kappa)$-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.

Deep unfolding (DU) accelerates iterative optimizers by training learnable components through unrolled iterations, but extending DU to the large-scale semidefinite programs (SDPs) common in robotics has remained limited. Unrolling a full-update conic solver such as COSMO exposes two obstacles: backpropagating through the per-iteration linear-system solve incurs memory quadratic in the problem size when the coefficient matrix is formed explicitly, and backpropagating through the positive semidefinite (PSD) cone projection is numerically unstable under repeated eigenvalues. We address the first with a matrix-free implicit differentiation rule that reduces memory from $O(n^2)$ to $O(n + \mathrm{nnz}(A))$, and the second with a stable backward rule based on the Dalečkii–Krein representation of the Fréchet derivative. Together, these enable learning lightweight hyperparameter policies and warm starts for COSMO. On standalone SDPs and second-order cone programs, ranging from max-cut and Lovász $\vartheta$ to robust estimation and control, the learned policies outperform tuned COSMO on every problem class, reducing iterations to tolerance by up to 60$\times$, and transfer without retraining to instances up to 10$\times$ larger in the size parameter while retaining 2–5$\times$ speedups. Our matrix-free solver implementation also batches naturally on the GPU, solving 64 Lovász $\vartheta$ SDPs 14$\times$ faster than sequential COSMO.jl. When embedded in sequential convex programming for nonlinear covariance steering, our learned optimizers outperform the interior-point solver Clarabel in terms of both success rate and solve time.

Configuration spaces of graphs frequently grow factorially in complexity with the number of particles they parametrize. However, for suitable families of nested graphs $G_\bullet$ with compatible symmetric group actions, Ramos and White prove that, for fixed $k$, the rational homology of the $k$\textsuperscript{th} configuration spaces of $G_\bullet$ has multiplicity stability. In the current work, we derive the stable range and use computer algebra to determine the stable representations on homology for $k=2$ and $G_\bullet$ several families of graphs, including the complete graphs, the complete bipartite graphs on $2n$ vertices, the crown graphs on $2n$ vertices, and the complete tripartite graphs on $2n+1$ vertices. We determine the stable multiplicities for certain irreducible components in the case $k=3$ and $G_\bullet$ the complete graphs.

In this study, we present a novel time-stepping scheme for multiscale differential equations describing oscillatory systems with well-separated scales, where the scale separation is controlled by a small parameter $\epsilon$. The time-stepping method is related to a multi-modal WKB approximation and relies on a transformation of variables derived in this work. The analysis reveals that, in the transformed formulation, the leading-order oscillations are either eliminated or appear only at higher asymptotic order. The method is applied to ordinary differential equations, including the well-known van der Pol oscillator. We investigate the accuracy of the proposed method numerically for different parameter regimes, in particular for decreasing values of $\epsilon$, and study how the parameters of the numerical scheme must be adapted as $\epsilon$ is reduced. In the presented numerical tests, the computational cost remains bounded as $\epsilon$ is decreased.

We revisit the transition from inspiral to plunge for extreme mass-ratio inspirals on quasi-circular, inclined orbits in Kerr spacetime from the perspective of catastrophe theory. Our goal is to uncover the mathematical structures underlying the universality of the transition dynamics, which remains governed by the same Painlevé I differential equation as for equatorial inspirals despite the additional complexity. We first analyze the solution of the Painlevé I equation selected by the physical boundary conditions of slowly evolving quasi-circular inspiral at early times. We argue that these conditions uniquely select the tritronquée solution of Painlevé I. We then compare existing high-accuracy analytic approximations of the tritronquée solution with direct numerical integrations of the Painlevé I equation, finding comparable accuracy and improved stability under differentiation and integration for the analytic solution. In the second part of this work, we show that the equilibrium structure of the Kerr radial effective potential admits a natural interpretation in terms of catastrophe theory. Equatorial orbits are associated with the fold catastrophe, while inclined orbits are described by the cusp catastrophe. In both cases, the transition to plunge corresponds to slow evolution across fold lines of the catastrophe manifold, providing a geometric explanation for the universal appearance of the Painlevé I equation in the transition dynamics.