Papers

We study the design of a statistical privacy mechanism under a bounded $\alpha$-lift leakage constraint. An agent observes useful data $Y$, which is arbitrarily correlated with private data $X$, and releases $U$. The goal is to maximize Pearson $\chi^2$-information $\chi^2(Y;U)$ while satisfying the point-wise privacy constraint. This utility is an $f$-information that is nonnegative, obeys data processing, and measures the aggregate normalized change from the prior $P_Y$ to the posteriors $P_{Y|U=u}$. The main benefit of this choice is that $\chi^2(Y;U)$ is exactly quadratic in the posterior perturbations; hence, no local approximation of the utility is required. For the nonlinear $\alpha$-lift constraint, we derive a uniform Taylor-remainder bound that remains valid for the optimizing, $\epsilon$-dependent perturbation. When $P_{X|Y}$ is invertible, this yields a closed-form quadratic surrogate, an explicitly scaled binary mechanism that satisfies the original $\alpha$-lift constraint, a computable sufficient range for $\epsilon$, and an $O(\epsilon^{3/2})$ bound on the utility gap. For $\alpha=2$, the solution is exact over an explicit range. We transfer these results to point-wise Rényi-divergence privacy, including an exact-feasible mechanism, an explicit privacy-budget range, and matching gap bounds. We also give an exact formulation for arbitrary leakage matrices and evaluate exact-feasible mechanisms numerically.

We prove optimal dual Furstenberg estimates for Frostman and Katz–Tao configurations for which the shadings are not locally concentrated near pencils. The starting point is the sticky estimate of Orponen and Shmerkin in a companion paper. We deduce restricted projection theorems in the plane and in higher dimensions. In particular, Ahlfors-regular measures on the circle satisfy a Marstrand-type projection theorem. Applications include restricted projections to surfaces in the $\mathbb{G}(4,2)$ Grassmannian, pinned distances, discretized sum-product, and a new proof of the dimension formula for planar self-affine sets.

We consider the Helmholtz equation in a Dirichlet angular sector (which may be locally perturbed) covered on one side by a thin homogeneous layer. First, we propose an asymptotic expansion of the solution at any order and for any angle when the thickness of the layer tends to 0, using a matched asymptotic expansion. This consists in introducing different asymptotic expansions in three subdomains – the vicinity of the corner, the layer and the rest of the domain – that are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have complex singular behaviors. We overcome this difficulty using a novel system of algebraic tools, including formal series and operators on these series. Moreover, we justify the asymptotic expansion with error estimates and provide details on the form of the expansion depending on the angle. We numerically validate this expansion up to order 5. This requires computing the behavior at infinity of some corner profiles. To do this, we introduce a Galerkin-type method enriched by complements at infinity. These complements are known functions that appear in the behavior at infinity of the corner profiles with coefficients that become unknowns of the system. Finally, we propose effective models of order 3, resp. 5 where, far from the corner, the layer is replaced with an effective Robin, resp. Ventcel, boundary condition and, near the corner, an artificial boundary is introduced and a Dirichlet-to-Neumann (DtN) type condition is imposed. The DtN map relies on a modal decomposition of the truncated asymptotic expansion at order 3, resp. 5. The modes take into account the singular behavior of the expansion terms, the corner profiles and some algebraic matching coefficients. We prove the well-posedness of our effective models, as well as error estimates, and provide numerical results.

We introduce spread-out measures, a notion of local non-concentration near pencils for measures on Grassmannians. We establish an optimal $\delta$-discretized dual Furstenberg estimate for “sticky spread-out” shadings. A simple-to-state corollary is the following. Assume that $\nu$ is an Ahlfors $s$-regular measure on $\mathbb{G}(2,1)$ with $s > 0$. Let $E \subset \mathbb{R}^{2}$ be a Borel set with equal Hausdorff and packing dimension. Then, $\dim_{\mathrm{H}} \Pi_{V}(E) = \min\{\dim_{\mathrm{H}} E,1\} \quad \text{for $\nu$-a.e. } V \in S^1$. The general $\delta$-discretized estimate for “sticky spread out” configurations here is the starting point for a more general theorem in the companion paper of Cohen, Shmerkin, Wang and Zakharov, where the “stickiness” hypothesis is removed. For instance, the assumption of equal Hausdorff and packing dimension in the result stated above will be dispensed with in the companion paper.

We consider the Burgers-type stochastic PDE \[ du = \theta\,\partial_x^2 u\,dt + f(u)\,dt + g(u)\,\partial_x u\,dt + \sigma\,dW(t), \] driven by space–time white noise on the one-dimensional torus, where \(\theta>0\) is the diffusivity parameter and \(\sigma>0\) the noise intensity. The transport term \(g(u)\,\partial_x u\) admits no classical interpretation, and we work with the mild solutions defined via rough paths following~\cite{Hairer2011}, for which the solution decomposes as \(u=\psi+R\) with \(\psi\) a stationary centred Gaussian process solving a damped stochastic heat equation and \(R\) a remainder. We first show that, away from the initial time, \(R\) is Lipschitz in space and Hölder in time of order strictly greater than \(1/2\), hence strictly smoother than \(\psi\) in both variables. We then compute exactly the spatial quadratic variation and the temporal quartic variation of \(\psi\), almost surely and not merely in \(L^{2}(\Omega)\), and prove that the corresponding variations of \(u\) are unaffected by the remainder. Combined with an exact time-scaling identity, this separation of regularity scales yields strongly consistent pathwise estimators of \(\theta\). To our knowledge this is the first inference procedure for an SPDE whose solution is defined only pathwise, as a measurable function of a rough path, rather than as an Itô process.

We establish a Thouless–Anderson–Palmer (TAP) variational formula for the free energy of spherical $2$-spin models whose interaction matrices have a compactly supported limiting spectral distribution. The formula holds at every inverse temperature and, in particular, extends the TAP representation previously obtained for the spherical Sherrington–Kirkpatrick model. Our main contribution is a new microcanonical proof designed to handle marginality. We partition the sphere according to the smallest collection of top eigendirections carrying a prescribed fraction of the norm. On every element of this partition, the remaining degrees of freedom are effectively at high temperature and their contribution can be computed through spherical-integral asymptotics. The resulting auxiliary variational problem separates the macroscopic low-temperature contribution from the high-temperature fluctuation term; its maximization recovers the usual TAP functional and Plefka condition. This decomposition also shows why neighborhoods of the TAP maximizers alone do not capture the Gibbs mass: all eigendirections associated with large eigenvalues contribute on the same exponential scale, a manifestation of the marginality of spherical $2$-spin states.

We introduce a discrete spectral curve from $\mathscr{N}=2$ double-scaled SYK whose topological recursion defines quasi-polynomial volumes in integer boundary lengths. The curve has two edge theories: a local $\mathscr{N}=2$ theory at the lower edge and a bosonic theory at the upper edge. Focusing on the local $\mathscr{N}=2$ theory, we compute its finite-$q$ volumes and independently check the first few using the string equation. In the $q\rightarrow 1$ continuum limit, these recover the super Weil-Petersson volumes of Turiaci and Witten. Finally, we give an all-genus intersection-theoretic interpretation of these local volumes on the moduli space of stable marked curves, in terms of a deformed spin class with a gap-dependent $\kappa$-class dressing.

In [1, Proposition 8.1], Alexakis and Mazzeo showed that the renormalized area of minimal surfaces in the hyperbolic space $\mathbb{H}^3$ approaching the ideal boundary $\partial_\infty\mathbb{H}^3$ is the negative of the Willmore energy of their Euclidean conformal compactifications. In this paper, we investigate whether there are other geometric surface equations for which these two quantities satisfy a universal proportionality relation. We answer this question in the affirmative by extending the Alexakis–Mazzeo result to a broader class of geometric equations containing, as two distinguished cases, the equations defining generalized singular minimal surfaces in $\mathbb{R}^3$ and linear Weingarten surfaces in $\mathbb{H}^3$. Moreover, for rotational linear Weingarten surfaces, we express both energies in terms of generalized hyperelliptic integrals and show that their doubles form a family of rotational tori in $\mathbb{R}^3$ whose Willmore energies realize all values in $(2\pi^2,\infty)$, with a particular subfamily converging to the Clifford torus in $\mathcal{W}^{2,p}$ for every $1\leq p<\infty$, as the optimal value $2\pi^2$ is approached.

We study sampling of a diffusion on $[0,T]$ conditioned on its terminal position lying on a prescribed ray, i.e. on an exactly observed bearing. For a linear drift we derive closed-form expressions for the Doob $h$-function and its gradient. For a broad class of tracking models admitting a Kronecker block structure, the guiding term is obtained from a single row of a matrix exponential, and the conditioned process decomposes into independent radial and angular components. For a nonlinear drift, we construct a guided process using the $h$-function of the linear model and prove absolute continuity of the law of the true conditioned process with respect to the law of the guided process, with an explicit likelihood ratio. This result is used to devise an auxiliary particle filter for bearings observed sequentially in time without noise. Numerical experiments show that the proposed filter approximates the filtering distribution and estimates the log-likelihood considerably more accurately than the shifted Rayleigh filter, at the cost of a higher computing time.

We apply the generalized monoid of words and logarithmic-prefix functionals of \cite{C1} to the regularization of the divergent product \[ P_{\omega}=\prod_{n=1}^{\infty} n^{(-1)^{\omega(n)}}, \] where \(\omega(n)\) is the number of distinct prime divisors of \(n\). The associated Dirichlet series \(D_{\omega}(s)=\sum_{n\ge1}(-1)^{\omega(n)}n^{-s}\) has a natural boundary at \(\Re(s)=0\), as established by Meretov (1987) and recovered here as a special case of the Estermann–Dahlquist theory of Euler products, which makes zeta regularization impossible: the derivative \(D_{\omega}'(0)\) does not exist. This product therefore lies completely beyond the reach of classical regularization methods. We construct the canonical net encoding the sign sequence, define the logarithmic-prefix functional with exponent \(\alpha=2\), and extract a finite value through the weighted evaluation functional of the monoid framework. The evaluation functional is shown to converge absolutely, and this convergence is unconditional: it follows from the classical zero-free region of \(\zeta(s)\) (de la Vallée Poussin), which yields the effective bound \(T_{\omega}(x)=O(x\,e^{-c\sqrt{\log x}})\) for some \(c>0\). Numerical computation up to \(N=10^7\) gives the regularized value \[ P_{\omega,\mathrm{ren}}\approx 0.7731, \] a new constant of arithmetic origin that cannot be obtained by any classical summation method. We also compute the second derivative of the Dirichlet series at \(s=1\), \[ D_{\omega}''(1)=8\,C_2\ln 2, \] where \(C_2\) is the Hardy–Littlewood twin prime constant, which provides an unconditional analytical link to the arithmetic structure of the sequence \((-1)^{\omega(n)}\). A self-contained proof of the natural boundary, based on the Estermann–Dahlquist theorem, is given in Appendix~A.

We prove that on a compact locally conformally Kähler manifold, the existence of a non-trivial parallel vector field with respect to a Gauduchon connection $\nabla^t$, for $t\ne \frac{1}{3}$, forces the metric to be either Vaisman (for $t=-1$), globally conformally Kähler (for $t=1$), or Kähler for all other values of $t$. As a consequence, we characterize compact Vaisman manifolds in terms of their Bismut holonomy group. Moreover, we show that in complex dimension two, the existence of a non-trivial $\nabla^\frac{1}{3}$-parallel vector field forces the metric to be globally conformally Kähler.

We establish a general metastability principle showing that the local stability of a McKean-Vlasov equilibrium implies exponentially long confinement of the associated particle system. Consider interacting diffusions on $\mathbb{R}^d$ with a general smooth and bounded interaction kernel and a globally Lipschitz drift that is dissipative at large distances. Assume the McKean-Vlasov equation has at least one equilibrium which is locally exponentially stable. We prove that the probability that the empirical measure leaves a sufficiently small fixed Wasserstein-1 neighborhood of the selected equilibrium before time $T$ is bounded by $C(1+T)e^{-cN}$, where $N$ is the number of particles and $C,c>0$ are independent of $N$ and $T$. Consequently, confinement holds up to times $e^{\gamma N}$, for every $0<\gamma<c$, with exponentially small exit probability. The proof combines an explicit Volterra construction of the linearized Fokker-Planck semigroup, a mild representation of empirical fluctuations, and exponential maximal estimates for stochastic convolutions. Our approach does not require a gradient flow structure or reversibility of the equations.

In the context of Fell bundles over discrete groups, we introduce and analyze suitable notions of direct sum and tensor product of actions of Fell bundles on Hilbert bundles. For $C^*$-dynamical systems, where actions arise from equivariant representations, we discuss how the tensor product of two such actions is related to the action associated with the tensor product of the underlying equivariant representations. Likewise, since any action gives rise to a $C^*$-correspondence between the full cross-sectional $C^*$-algebras of the involved Fell bundles, we examine the connection between the $C^*$-correspondence associated with the tensor product of two given actions and the tensor product of the associated $C^*$-correspondences. We also discuss how the tensor product of actions naturally appears in Rieffel-like induction processes. Finally we present related results on the Fourier-Stieltjes and Fourier algebras of a Fell bundle.

By a celebrated result of Birman and Series, the set of simple complete geodesics form a nowhere dense subset of a hyperbolic surface. We show that a segment of finite length is in a (small) neighborhood of this set if it can be "extended simply" in both directions for long enough. We deduce that, for large enough genus $g$, there exist surfaces with a Birman-Series set that is roughly $\frac{1}{g^{\sfrac{2}{3}}}$-dense.

Fault diagnosis in asset-intensive industries requires multiclass models trained on noisy, sensitive data collected across geographically distributed client sites. Federated learning enables collaborative training without centralizing data, but most approaches do not provide formal privacy guarantees for individual records stored at client sites and can degrade under label noise. To address these limitations, we develop a differentially private, distributionally robust framework for federated multiclass classification that provides sample-level privacy for records stored at each client. We build on a federated Wasserstein distributionally robust multiclass objective with the Crammer-Singer loss, solved using an Alternating Direction Method of Multipliers (ADMM)-based algorithm. To protect the released client updates, our method injects calibrated Gaussian noise into the proximal linearized client updates of both the local model and the local distributionally robust variable before aggregation at the central server, and we quantify the cumulative privacy loss using the moments accountant. We provide per-round sample-level $(\epsilon,\delta)$-differential privacy guarantees and an overall composed privacy guarantee across all training rounds, together with convergence results and an explicit privacy-utility tradeoff. Experiments on simulated industrial fault data and UCI benchmark datasets show that our method outperforms differentially private federated empirical risk minimization and stochastic gradient descent baselines in most tested settings, with the largest gains under strong privacy-induced noise.

An inverse spectral problem for a $(1+1)$-dimensional wave equation with wave speed $c(x)$ is considered. The spectral data consist of the eigenvalues of the Neumann–Dirichlet problem for the operator $-c^2(x)\partial_x^2$ together with their derivatives with respect to a Robin parameter at the boundary. We propose a two-step recovery procedure. In the first step, the Neumann-to-Dirichlet map $\Lambda$ is reconstructed from the spectral data using the results of arXiv/0406582. In the second step, the reconstructed $\Lambda$ is used within the regularization strategy for the boundary control method presented in arXiv:1509.04478. As an improvement, the regularization of the control problems employs a weighted combination of $L^2$ and $H^1$ penalty terms, and we prove a uniform convergence result for the family of optimal controls, thereby allowing the regularization parameters to be tuned simultaneously. We also prove that the forward spectral map, which assigns to a wave speed its spectral data, is Lipschitz continuous, and that the inverse spectral map underlying the reconstruction procedure is Hölder continuous on a subclass of wave speeds. The method is implemented numerically and validated on both smooth and piecewise smooth wave speed profiles. Numerical experiments demonstrate that the recovery of the wave speed exhibits high accuracy and robustness.

We present a fully discrete energy conserving discretisation of the rotating shallow water equations with arbitrary order in space and time. A space time finite element discretisation is used, built from compatible finite element methods in space and continuous Petrov-Galerkin schemes in time. Projected auxiliary variables for Hamiltonian derivatives are necessary for exact energy conservation. We provide an iterative solver approach based on monolithic patch smoothers, which allows us to efficiently converge a Newton method sufficiently that energy errors due to the solver are very small. We extend this construction to the energy conserving Stochastic Forcing by Lie Transport formulation. Then we demonstrate the convergence, conservation and robustness in numerical tests.

We study the two-dimensional compressible Navier-Stokes-Poisson system in a general simply connected bounded domain with Navier-slip boundary conditions and a positive, possibly non-flat, doping profile. We establish the global existence of a unique strong solution and of a global weak solution for arbitrarily large initial data allowing vacuum when the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than 4/3. The main difficulties arise from the interaction between the Poisson force and the general boundary geometry. To overcome them, we introduce a modified material derivative and derive quantitative $L^p$ estimates for the density by means of a Neumann duality argument together with a harmonic correction generated by the Navier-slip boundary condition. We further obtain a pointwise representation of the effective viscous flux by pulling back the Neumann Green function on the unit disk through a conformal mapping. The resulting commutator estimates, combined with elliptic estimates for the Poisson equation, yield a uniform-in-time upper bound for the density and allow us to close the higher-order a priori estimates. Finally, the global weak solution is constructed through a charge-preserving approximation and compactness argument.

We establish endpoint partial regularity for suitable weak solutions to the three-dimensional chemotaxis–Navier–Stokes system with constant sensitivity and linear consumption in a half-space under no-slip and no-flux boundary conditions. The full space–time singular set, including its boundary part, has vanishing one-dimensional parabolic Hausdorff measure. This follows from a Caffarelli–Kohn–Nirenberg type criterion at the natural $r^{-1}$ dissipation scaling, valid at interior and boundary points without additional pressure smallness. The key estimate combines weighted mass control with the localized entropy–energy inequality to close the iteration without a separate $n\log n$ quantity and accounts for the mass term generated by rescaling. Boundary-compatible pressure and caloric decompositions yield local Hölder regularity. We also prove a lower-order criterion requiring smallness only of the velocity, signal gradient, and pressure oscillation, with a threshold depending on an arbitrary finite density mass bound. We construct global suitable weak solutions under natural entropy and energy assumptions without smallness restrictions on the initial data. The potential is time-independent with bounded gradient; boundedness of the potential itself is assumed only for the large-time results. If its gradient is also Hölder continuous, a solution can be selected to become bounded and classical throughout the closed half-space after a finite waiting time. For the same solution, we obtain decay and Neumann Gaussian asymptotics for both scalar components without initial spatial moment assumptions. The profile coefficients are the conserved bacterial mass and the limiting signal mass, respectively, and the latter is strictly positive whenever the initial signal is nonzero.

In an earlier study of the Laplacian eigenvalues with strongly attractive Robin boundary conditions in polygons it was found that if each corner is either concave or non-resonant, then the asymptotic behavior of the individual eigenvalues is determined by an effective one-dimensional operator on the boundary with Dirichlet boundary conditions at the corners. The non-resonance condition was formulated in terms of an eigenvalue bound on truncated sectors, and it was only shown to be satisfied by the right and obtuse opening angles. In the present work, we determine the full range of non-resonant angles: it turns out that the non-resonance condition fails only for the opening angles $\frac{\pi}{2n+1}$ with $n=1,2,3,\dots$, and these critical values are equivalently characterized in terms of the jumps of the eigenvalue counting function and the existence of bounded generalized eigenfunctions. This resolves the numerically supported conjecture stated by Dauge and Lafranche.

We analyze distributional drift in nonstationary weighted risk minimization by decomposing the excess risk into estimation, drift, and approximation components. The drift error is commonly bounded through discrepancy-type quantities that control changes of the risk over classes of predictors. We instead relate the drift error directly to the movement of the best predictor within the model class and analyze its behavior in parametrized regression models. Our analysis shows that discrepancy-type quantities can be overly conservative: unlike our oracle drift, they may scale at a larger order than the resulting excess-risk contribution and can respond to changes in the risk surface that do not affect the optimal predictor. For linear regression, we characterize this gap sharply and combine the resulting drift bounds with estimation guarantees. We further extend the analysis to nonparametric regression.

We establish the following novel heterovariant Gaussian Brunn-Minkowski inequality (involving general covariance matrices $\Sigma_1,\Sigma_2,\Gamma$) for origin-symmetric convex sets $K,L \subset \mathbb{R}^n$ and $p,\lambda \in (0,1)$: \[ \gamma_{\Sigma_1}(K)^{1-p} \gamma_{\Sigma_2}(L)^p \le \left( \left( \frac{1-p}{1-\lambda} \right)^{1-p} \left( \frac{p}{\lambda} \right)^p \right)^{n} \gamma_{\Gamma}((1-\lambda) K + \lambda L) , \] whenever $\Gamma \leq \Sigma_1 \sharp_p \Sigma_2$, the $p$-geometric mean of $\Sigma_1,\Sigma_2$. This immediately implies both the dimensional Gaussian Brunn-Minkowski inequality of Eskenazis-Moschidis (by using $\Sigma_1 = \Sigma_2 = \Gamma$ and optimizing over $p$) and the strong Gaussian (B)-theorem of Cordero-Erausquin-Fradelizi-Maurey (by using $K=L$, $p=\lambda$ and diagonal covariances), fusing them into a single unified statement. In fact, the inequality applies to convex sets whose appropriate Gaussian barycenter is at the origin. Furthermore, it reveals a new *bilinear* version of the strong Gaussian (B)-theorem (involving *two* convex sets); in particular, for all $a,b > 0$ and $\lambda \in [0,1]$: \[ \gamma(a K)^{1-\lambda} \gamma(b L)^{\lambda} \leq \gamma \left ( a^{1-\lambda} b^\lambda \left ((1-\lambda) K + \lambda L \right) \right ) . \] A more general heterovariant version of the (bilinear) (B)-theorem replaces dilation by general linear transformations. A different version yields a heterovariant form of the recently established Gaussian conjugate Rogers-Shephard inequality. By averaging these Gaussian inequalities with respect to the covariance matrices, we obtain dimensional Brunn-Minkowski and conjugate Rogers-Shephard inequalities for Gaussian mixtures. The proofs are based on the Gaussian Forward-Reverse Brascamp-Lieb inequality established in our previous work.

We prove that the local Dirichlet-to-Neumann map of $\Delta_g+q$ determines a compact connected real-analytic Riemannian manifold of dimension at least three and a real-analytic real-valued potential, up to a diffeomorphism fixing the observed boundary and the conformal gauge of the equation. The result is new even when \(q\) is constant, i.e. for the fixed-frequency (Helmholtz) equation \(\Delta_g u+\lambda u=0\). The measurements are restricted to an arbitrary nonempty open boundary set $\Gamma$. The proof develops conformal Poisson embeddings method and uses coordinates given by quotients of solutions.

The proximal Galerkin algorithm is a structure-preserving solver for variational inequalities that results in discretization-independent nonlinear iteration counts. However, in large three-dimensional simulations, the arising Newton saddle point systems cannot be feasibly solved by direct methods. To leverage an iterative linear solver, we propose a robust and scalable block preconditioner. Instead of using the usual dual Schur complement that acts on the latent variable, we design a preconditioner for the primal Schur complement. This leads to a block preconditioner that requires two approximate inverses of the latent block and only one of the primal Schur complement. The latent block represents an algebraic equation which, after a small perturbation, can be cheaply inverted. We then introduce two choices of approximations for the primal Schur complement: a PDE-operator-based and an algebraic approach. Either approximation can be efficiently inverted by geometric multigrid. We investigate the effectiveness of these strategies when proximal Galerkin is applied to obstacle, Signorini, and strain-constrained elasticity problems and also compare with a monolithic multigrid approach. Choices of standard relaxation methods, in the block preconditioner route, lead to robust iteration counts with respect to the mesh size and discretization degree. Moreover, in the context of the Signorini, specialized relaxation and prolongation also result in effective solvers for nearly-incompressible elastic materials. The linear solves are parameter-robust in 2D, with only mild growth in 3D. This allows for the solution of a three-dimensional contact problem, with a Poisson's ratio of 0.499, discretized with over ten million degrees of freedom.

Recent results [arXiv:2602.04378, arXiv:2602.22608] have established worst-case objective-error lower bounds of order $1/T^2$ after $T$ linear minimization oracle (LMO) calls, matching the upper bound [arXiv:1406.1305] in its dependence on $T$ and thereby resolving a long-standing open question. Upon close inspection, however, both lower bounds are somewhat brittle. The first [arXiv:2602.04378] applies only to the standard Frank–Wolfe algorithm, and, as we show, its prescribed slow exact-line-search trajectories can amplify sufficiently small initialization errors exponentially, corresponding to roughly one additional bit of initialization accuracy per iteration. The second [arXiv:2602.22608] obtains lower bounds for any deterministic LMO-based first-order method via a resisting oracle that adversarially adapts to the queries of the algorithm. In both cases it is conceivable that randomization may circumvent these lower bounds; in particular, neither establishes a lower bound for unrestricted randomized first-order LMO methods. We close this gap by proving an $\Omega(L\operatorname{diam}(S)^2/T^2)$ distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets. While our construction reuses the family of [arXiv:2602.22608], our proof characterizes the conditional distribution of the hidden permutation after adaptive queries and bounds the expected decrease of a combinatorial statistic controlling optimizer variance. We further show that the same error threshold holds with probability at least $3/4$ and yields a stopping-time tail bound.

We'll call a map from basic to clopen sets of binary sequences an encoding if it preserves uniform Lebesgue measure and the disjointedness relation. For some notion of randomness $R$ we can ask if there is a computable encoding $E$ such that for every non-Martin-Löf-random (non-MLR) sequence $\alpha$, the sequences in $E(\alpha)$ are non-$R$. Thus we can use $E$ and randomness test for $R$ to construct a randomness test for MLR. We show that there is no such $E$ for partial computable randomness (PCR), but there is one for Kolmogorov-Loveland randomness. Indeed, there is a total computable $E$ and a pair of total computable non-monotonic betting strategies such that on every sequence in $E(\alpha)$ at least one of the strategies wins whenever $\alpha$ is non-MLR.

We study the de Rham cohomology and Deligne splitting for a class of Eisenstein-Kronecker forms by introducing a modular weight grading. As an application, we compute a particular cohomology class that gives rise to generating series of Gromov-Witten invariants of elliptic curves.

Chromatic villainy measures the largest number of vertices that must change color to repair a rearrangement of an optimal coloring while preserving every color multiplicity. Weak villainy allows different multiplicities in the repair. On the odd cycle $C_{2r+1}$, we determine the largest repair distance from arbitrary color assignments with a prescribed color-count vector (profile) $\mathbf m$ to proper colorings with any realizable target profile $\mathbf p$: it equals $2r+1-\sum_X\min\{p_X,\lceil m_X/2\rceil\}$. In particular, every feasible profile $(a,b,c)$ has ordinary and weak repair radius $\lfloor a/2\rfloor+\lfloor b/2\rfloor+\lfloor c/2\rfloor$, and $B(C_{2r+1})=r$ for $r\ge2$, answering a question of Clark et al. For arbitrary complete multipartite graphs, we give an arithmetic formula for ordinary villainy with at most $k^2$ candidates and an exact finite formula for weak villainy, where $k$ is the number of parts. Both values are fixed-parameter tractable in $k$ for binary-encoded part sizes. We also determine the maximum of each parameter among all $n$-vertex $k$-chromatic graphs. For $k\ge3$, the ordinary maximum is $\min\{n,4(n-k)\}$, except when $n=4(n-k)-1$ or $(n,k)\in\{(5,3),(7,3)\}$, when it is $n-1$. For $k\ge2$, the weak maximum is $n-\max\{\lceil n/k\rceil,2k-n\}-\varepsilon$, where $\varepsilon\in\{0,1\}$ is determined explicitly by $n$ and $k$. Both extrema are attained by complete multipartite graphs, and a weak extremizer needs at most three distinct part sizes.

We study the long-time behaviour of multi-dimensional first-order symmetrizable quasilinear hyperbolic systems with nonlinear feedback boundary conditions. Assuming strict interior and boundary Lyapunov inequalities for the linearized system and non-characteristic boundaries, we prove global existence for small initial data and exponential H^s decay, with s>d/2+1 an integer. We consider both bounded smooth domains and flat periodic cylinders, and illustrate the results with physical examples and numerical simulations.

We place the Appell shift theorem for asymptotic expansions inside a Borel–Laplace framework and apply it to two classical functions of analytic number theory: the Hurwitz zeta function \(\zeta(m,q)\) and the Lerch transcendent \(\Phi(z,m,a)\). The resulting shifted expansions have coefficients given by Appell polynomials evaluated at the shift, with remainders controlled uniformly on compact sets. The generating sequences are identified explicitly in terms of the Taylor coefficients of the Borel kernels. In a second direction, we extend the Appell–Mellin framework of Navas, Ruiz and Varona to Dirichlet \(L\)-functions. For a primitive non-principal character \(\chi\) modulo \(q\), we construct an explicit Appell–Mellin generating function \(A_\chi(t)\) whose associated Appell sequence satisfies \(R_n^\chi(x) = L(-n,\chi,x)\), and we prove the functional equation \(A_\chi(t) = \chi(-1)A_\chi(-t)\). We further establish five structural properties of the polynomials \(R_n^\chi\): identification with generalized Bernoulli polynomials, a difference equation, an explicit formula, a Fourier expansion, and the trivial zeros of \(L(s,\chi)\). To the best of our knowledge, these results do not appear in the literature.