Papers

We study gradient descent with predetermined nonnegative stepsizes on smooth strongly convex functions. Let $p_{\mathrm{sil}}=\log_2(1+\sqrt2)$ and $\kappa$ be the condition number. We prove the iteration lower bound $\Omega\left(\kappa^{\frac{1}{p_{\mathrm{sil}}}-o(1)}\log\frac1\delta\right)$ for both relative squared distance and relative function error, uniformly over $0<\delta<1$ and sufficiently large $\kappa$. This matches the polynomial exponent of $\kappa$ for the Silver stepsize schedule established in [Altschuler and Parrilo, 2025].

We construct a pencil of complex quintic surfaces with every smooth member containing exactly $85$ lines. We show that one smooth member has Picard number $43$ and by introducing the notion of a $d$-saturated lattice, we prove that its Néron–Severi group is generated by the lines and four conics. We also construct smooth quintic surfaces with $79$ lines, including examples with $24$ pairwise skew lines, and a smooth surface of degree $10$ with $356$ lines, including $112$ pairwise skew lines.

Let $M^n$, $n\ge3$, be a closed oriented minimal hypersurface in the unit sphere with constant scalar curvature and constant $f_3=\tr(h^3)$, where $h$ is the second fundamental form. We prove that $S=\norm h^2>n$ implies $S\ge2n$. Equality holds if and only if each connected component is locally congruent to a Cartan minimal isoparametric hypersurface with three distinct principal curvatures.

We study representations for relation algebras corresponding to certain edge colourings of complete graphs. Previously suitable colourings were obtained for the number of colours $n$ up to $2000$, with two exceptions: $n = 8$ and $n = 13$. Using a method that we have called the FBF (Fusion Beautiful Fusions) method, we find colourings for $8$ and $13$ colours. Our method is a new guess-and-check approach that we describe as an adaptation of Comer's finite-field method. Using FBF we construct novel colourings that are non-isomorphic to existing published colourings for 5, 6, 7, 9, 10, 11 and 12 colours.

Let $f$ be an analytic function in the unit disc whose modulus is bounded by $1$. We present a new proof of Bohr's theorem, which asserts that its majorant function $Mf(r)$ is bounded by $1$ for $0\leq r \leq 1/3$ and that $1/3$ is optimal. The argument also yields a sharp upper bound for $Mf(r)$ in terms of $\lvert f(0) \rvert$, complementing a result due to Bombieri.

We study the reconstruction of a directional albedo field from confocal non-line-of-sight measurements. For mirror-symmetric vector fields in Sobolev spaces, radial preprocessing reduces the data to spherical means of the divergence with centers on the relay wall. Full data determine this divergence uniquely, with divergence-free fields forming the entire ambiguity. An explicit Fourier–sine formula recovers the irrotational Helmholtz component and reproduces the data. The Fourier–sine transforms of the model data form a weighted Hilbert space that we characterize exactly, with norm equal to the Sobolev norm of the reconstructed field. This weighted range remains well defined even when the preprocessed data fail to belong to standard Sobolev spaces. For Schwartz fields in the model class, such Sobolev regularity holds exactly when the divergence has zero depth integral. Measurements on an open subset of the relay wall, for all radii $0<r<R_{\max}$, uniquely determine the divergence in the union of the corresponding balls. An FFT-based algorithm with Stolt interpolation implements the reconstruction in $O(N^3\log N)$ operations on an $N^3$ grid. Tests on explicitly defined fields assess reconstruction accuracy and invariance under divergence-free perturbations. We present imaging examples of potential and vector-field reconstruction from model-generated, rendered, and measured transients.

Residual-based adaptive GMsFEM for high-contrast elliptic problems repeatedly evaluates local weighted $H^{-1}$ indicators on every coarse neighborhood, making indicator evaluation a recurring cost in repeated-query settings. We introduce a non-intrusive Gaussian-process (GP) surrogate for the indicator scores used in Dörfler marking. The operational predictor is the GP posterior mean, algebraically equivalent to a kernel ridge regression (KRR) estimator under the stated convention; it uses compressed local solution and spectral features without changing the multiscale solve, local spectral construction, or basis enrichment. A nonuniform perturbed-marking result quantifies how pointwise score errors affect the exact indicator mass captured by surrogate-selected neighborhoods, while a conditional bounded-discrepancy KRR pathway identifies sufficient assumptions for such score bounds. In controlled held-out in-distribution tests, the surrogate-guided method gives error-versus-DoF trends comparable with classical $H^{-1}$-residual offline adaptivity and evaluates the online indicator component 2.0-2.1 times faster, excluding offline data generation and GP training.

We prove Miller's conjectured bound for solvable groups: if $G$ is a finite solvable group and $\chi\in\Irr(G)$, then $\chi(g)$ is zero or a root of unity for at least half of the elements $g\in G$. We also prove the same bound for monomial irreducible characters of arbitrary finite groups.

We establish a fully discrete weak-to-strong (W2S) multi-entropy stability theory for arbitrarily high-order finite-volume and discontinuous Galerkin (DG) approximations of the ideal compressible magnetohydrodynamics (MHD) equations on general polytopal meshes, whereby a single numerical update simultaneously satisfies discrete entropy inequalities for any prescribed finite family of convex Harten entropy pairs. Because physical entropies are defined only for positive density and pressure, the central analytical difficulty lies in reconciling discrete entropy stability with positivity preservation, the magnetic divergence constraint, and the nonconservative Godunov–Powell coupling. Building upon the provably positivity-preserving MHD framework of Wu and Shu, we develop a weak multi-entropy analysis of the common cell-average evolution of finite-volume and DG methods. Through convex decomposition and relative entropy, we establish two sufficient criteria for this weak stability general polytopal meshes. Compatible magnetic corrections and locally divergence-free approximations cooperatively offset the nonconservative Godunov–Powell coupling and provide the discrete cancellations essential to both positivity-preservation and entropy stability. A W2S lifting then unifies positivity and entropy limiting into a single cellwise scaling limiter, achieving strong multi-entropy stability while strictly preserving cell averages and the locally divergence-free magnetic field. High-order temporal accuracy follows from variable-step strong-stability-preserving multistep methods. Numerical experiments confirm the theoretical findings and demonstrate computational robustness. This framework provides a rigorous, fully discrete foundation for unifying physical admissibility, magnetic divergence control, and multi-entropy stability in high-order MHD approximations.

In a recent preprint (arXiv:2511.09556), we exhibited an inverse energy cascade for the 3D Navier-Stokes equations which results in "instantaneous" Type I blow-up and failure of uniqueness in certain sharp regularity classes. Our purpose here is to elucidate this phenomenon further in the setting of the Obukhov dyadic model, and to compare it to the better-known phenomenon of finite-time blow-up. When intermittency is low ($\alpha\leq 2$), we recover our previous result from the Navier-Stokes setting; in the energy-supercritical case where intermittency is high ($\alpha>2$), we show that the same can occur in the class of finite energy solutions. We present two different proofs: a soft approach using a Lyapunov function and an explicit multiscale construction. For the inviscid system we illustrate that a similar phenomenon occurs at lower regularity. Finally, we state a finite-time blow-up theorem for a mixed Desnyansky-Novikov-Obukhov model, in which a forward cascade from finitely supported data and no force produces a singularity that is of Type II only by a small margin, with features comparable to the recent forced Navier-Stokes blow-up of OpenAI.

Robin's inequality reformulates the Riemann hypothesis as a bound on the sum-of-divisors function $\sigma$. We introduce a finite-truncation version of Robin's inequality determined intrinsically by the arithmetic structure of $n$, \[ \frac{\sigma(n)}{n} < e^\gamma \sum_{j=0}^{\omega(n)} \frac{(\ln\ln\ln n)^j}{j!}, \qquad n>5040, \] where $\omega(n)$ denotes the number of distinct prime factors of $n$, and $\gamma$ is the Euler-Mascheroni constant. The resulting bound is pointwise stronger than Robin's inequality and we prove it unconditionally for integers with $\omega(n)\le 6$, primorials, odd integers and square-free integers. We also prove that this inequality is equivalent to Robin's inequality, and hence to the Riemann hypothesis. Consequently, the Riemann hypothesis is equivalent to the truncated inequality holding for all colossally abundant numbers. We further show that if the inequality fails, its minimal counterexample must be a superabundant number, placing any obstruction within a rigid extremal class of integers.

Using Soundararajan's resonance method, we obtain extreme values of the central derivatives \[ L^{(r)}(1/2,f\otimes\chi_d), \] where $f$ is a normalized newform and $\chi_d$ ranges over primitive real characters associated with negative fundamental discriminants. The twisted first moment estimate used in the proof may be of independent interest. Applying Gross–Zagier-type formulas, we obtain correspondingly large Néron–Tate heights of Heegner points on elliptic curves and large absolute values of Beilinson–Bloch heights of Heegner cycles attached to modular forms.

We study decentralized partially observable team decision problems with low-rank latent dynamics and unknown system models. The proposed framework combines team-theoretic equivalence with low-rank model representations to address cooperative decision-making in partially observable Markov decision processes without prior knowledge of the transition model. Each team member makes decisions based on local private information and delayed common information shared across the team. Using only this available information, each member learns an approximate low-rank Markov decision process and applies least-squares value iteration to compute its policy. This yields a fully decentralized learning and planning algorithm that requires neither a centralized coordinator nor centralized training. We show that the resulting member-side solutions approximate the centralized team solution: despite partial observability, unknown dynamics, and delayed common information, each member recovers the corresponding component of an approximate team-optimal policy. We further establish finite-sample performance guarantees and derive a corresponding sample-complexity bound for the proposed algorithm.

Birnbaum's theorem states that the sufficiency and conditionality principles jointly imply, and are implied by, the likelihood principle. To enable formal certification of the results in Lean, we make the conditionality relation explicit through a finite-sample formulation based on \cite{Evans2013}. We reformulate the theorem as a statement about classes of statistical procedures that preserve statistical relations, and show that the proof reduces to the propagation of evidential equality along chains of conditionality-related inference bases (experiment–observation pairs). The main results are illustrated by Venn diagrams, and a worked finite example exhibits two pairs of inference bases: one related by conditionality but not by sufficiency, and the other related by sufficiency but not by conditionality. For finite parameter spaces with at least two points, the class of likelihood-invariant procedures with codomain $[0,1]$ is strictly contained in the class of sufficiency-invariant procedures.

The Odlyzko-Poonen conjecture states that a monic polynomial of constant coefficient $1$ and with remaining coefficients chosen independently and uniformly from $\{0,1 \}$ is irreducible over the rational numbers with probability tending to one as the degree tends to infinity. We prove the Odlyzko-Poonen conjecture unconditionally.

This paper investigates the ambiguity of noisy short-arc angles-only orbit determination for a Molniya-orbit case study. Conventional Gauss Initial Orbit Determination followed by Gaussian Least-Squares Differential Correction is first shown to be unreliable under high-noise short-arc conditions, frequently failing to converge or converging to hyperbolic local solutions. To explore the candidate solution space, a Lambert-based initialization procedure is used over a grid of assumed ranges and angular-observation pairs, and the resulting states are refined using nonlinear least-squares optimization. The unconstrained solution set reveals multiple orbit families, including reentry elliptic, bounded elliptic, xGEO elliptic, and hyperbolic trajectories, all of which can reproduce the observed angular arc with comparable residuals. The solutions exhibit a structured range-velocity relationship, indicating that the angular measurements primarily constrain apparent line-of-sight motion rather than absolute range. Finally, physically motivated constraints are introduced to isolate Molniya-like bounded elliptic solutions. The results demonstrate that short-arc angles-only orbit determination is fundamentally non-unique without a priori information and that constraints should be interpreted as orbit-family selection tools rather than proof of uniqueness.

Decoherence free subspaces are a steady-state structure of the open quantum system which preserves quantum coherence between the states lying with in it and thus has found a variety of applications throughout quantum information science and technology. In this paper we study the computational complexity of deciding whether an open quantum system admits a decoherence free subspace or not. More specifically we study this problem with in the context of Markovian open quantum systems, governed by the time-independent Lindblad master equation. Along the way we introduce the $k$-Local Lindbladian problem, which captures the difficulty of computing purity decay rates under Lindbladian dynamics. We show that both problems are hard for the complexity class Quantum Merlin Arthur (QMA) when the locality $k \geq 5$, with the first under perfect completeness and the second being complete for QMA. Our hardness construction generalizes Kitaev's clock Hamiltonian construction to the open quantum system setting by encoding the execution of a quantum circuit into the steady subspace of a Lindbladian containing both pure and mixed history states. This subspace is then mixed depending on the output of the encoded circuit. Our results suggest that deciding whether a generic Markovian open quantum system admits a decoherence free subspace is intractable even for quantum computation.

Bogomolov–Guan manifolds form the only known series of compact simply connected non-Kähler holomorphic symplectic manifolds. We prove that all Bogomolov–Guan manifolds of complex dimension at least four are nonformal over $\mathbb{Q}$. The proof constructs a uniform degree-eight $a$-Massey product in a finite-dimensional invariant de Rham model of the symmetric quotient of the Bogomolov–Guan precursor. Its indeterminacy vanishes, and its nontriviality is detected by an explicit top-degree pairing arising from an $S_{n+1}$-invariant tensor contraction. A nonzero-degree map then transfers nonformality to the smooth Bogomolov–Guan manifold. We also compare the Bogomolov–Guan precursor with the nilmanifold appearing in Guan's original construction, showing that they are related by a natural $S_{n+1}$-equivariant finite cover of degree $(n+1)^2$ and have the same invariant de Rham model. Using Guan's resolution, which we prove to be projective and semismall, together with Saito's decomposition theorem, we further show that the third Betti number of every Bogomolov–Guan manifold of complex dimension at least four is equal to $3$.

Let $\mathcal{C}$ be a differential $\mathbb{Z}/2$-graded category over $\mathbb{C}$. Its periodic cyclic homology $HH^{per}_*(\mathcal{C})$, when viewed as a vector bundle over the formal punctured disk, is equipped with a canonical connection $\nabla^{\mathcal{C}}_{\partial_t}$ called the Getzler-Gauss-Manin connection in the $t$-direction (or the categorical $t$-connection). Our main result is that when $\mathcal{C}$ is smooth and proper, this connection has a regular singularity at $t=0$ and quasi-unipotent monodromy, affirming a conjecture of Katzarkov-Kontsevich-Pantev \cite{KKP}. Our proof follows a reduction mod $p$ argument using a spreading out technique of Toën \cite{To} and a regularity criterion of Katz \cite{Ka1}. The main novelty is the proof of a multiplicative property of the $p$-curvature of $\nabla^{\mathcal{C}}_{\partial_t}$ through an interpretation in terms of the two-colored Kontsevich-Soibelman operad. We then explore two applications of the main result. First, we give an explicit description (under additional assumptions) of the non-commutative Hodge filtration on the periodic cyclic homology of a smooth proper d$(\mathbb{Z}/2)$g category, following a construction of Shklyarov \cite{Shk}. The second application, which is special to our particular method or proof, is an upper bound on the sizes of Jordan blocks of the monodromy of $\nabla^{\mathcal{C}}_{\partial_t}$, which simultaneously generalizes Scherk's local monodromy theorem for isolated hypersurface singularities \cite{Sche} and (partially) a recent result of Pomerleano-Seidel on the quantum connection of a closed monotone symplectic manifold \cite{PS2}. As a specialization, we show that the sizes of these Jordan blocks are bounded above by the diagonal dimension of $\mathcal{C}$ plus one.

We consider simultaneously testing hypotheses about thousands of units, e.g., genes or proteins, where each unit yields a handful of replicate measurements and we test whether its mean is zero. A widely used approach in genomics, implemented in the limma software, borrows strength across units to learn the distribution of the unit-specific variances, then computes moderated t-statistics. Here we develop the first procedures using moderated t-statistics that (i) control the false discovery rate (FDR) in finite samples under independence across units and null group invariance, without assuming limma's hierarchical model, and (ii) match the power of an oracle local false discovery rate procedure in a sparse asymptotic regime under limma's working model. Benjamini-Hochberg (BH) with standard t-test p-values has asymptotically zero power in the same regime. Our approach learns the variance distribution from statistics that depend on each unit's data only through its orbit under a compact group of transformations that preserves the null distributions, such as sign flips or orthogonal rotations. We then calibrate the resulting moderated t-statistics against group-transformed statistics pooled across units to obtain compound p-values, which we use with BH and a close variant. For small finite groups, we also construct a Selective SeqStep+ procedure using the same learned scores. Our approach extends to two-sample tests and tests of linear model coefficients.

A $\textit{magic-lined hypercube}$ (or, simply, $\textit{magic hypercube}$) of order $k$ and dimension $n$ is an arrangement of the numbers $1,\dots,k^n$ in a $k\times\cdots\times k$ ($n$-fold) grid such that every line of $k$ numbers parallel to a coordinate axis has the same magic sum. While such hypercubes exist for every order $k\ge 3$ and every dimension $n$, no magic-lined hypercube of order $2$ exists in any dimension $n\ge 2$. For hypercubes of order $2$, we thus relax the magic condition from lines to two-dimensional faces or planes. We call an arrangement of the numbers $1,\dots,2^n$ in a $2\times\cdots\times2$ ($n$-fold) grid magic-faced if every $2\times2$ face has the same magic sum. We prove that a magic-faced hypercube of order $2$ exists in every dimension $n\ge0$, and that it is unique up to a certain natural set of transformations of size $2^n(n+1)!$ when $n$ is even and $2^n n\cdot n!$ when $n$ is odd. As an application, we recover and generalize the classical $4\times4$ Khajuraho magic square, answer a question of Coxeter on the group acting on the $384$ “most-perfect" $4\times4$ magic squares, and extend the picture to higher dimensions. In particular, we prove that, in dimension $n$, these most-perfect objects form a single orbit under a certain natural action of the Weyl group $W(B_{2n})$.

We study the closed spans of finitely many joint orbits of holomorphic multipliers on the Hardy space $H^2(\mathbb D^d)$ of the polydisk. If fewer than $d$ symbols are used, every such span has infinite codimension. The symbols may be unbounded, provided that all orbit vectors belong to the Hardy space. We give two elementary proofs. The first uses common level sets and independent point evaluations; for bounded symbols, these evaluations yield joint adjoint eigenvectors. The second uses finite Taylor sections and the nilpotent structure of truncated convolution, also known as the Jury product. For $r$ generators and $q<d$ symbols, the orbit dimension on a cube of side $n$ is $O(n^q)$, whereas the ambient dimension is $n^d$. In the bidisk we obtain the explicit codimension bound $MN-r(M+N-1)$, which is sharp for a single orbit. The coordinate multipliers attain the parameter threshold. We also explain how these obstructions relate to kernel interpolation and the established representation of dynamical frames by compressed shifts on model spaces.

We determine explicit formulae for the Betti numbers of the defining toric ideals of two families of numerical semigroups of Sally type, resolving and extending beyond several conjectures of Goel–Şahin–Singh–Srinivasan. Our approach uses Apéry resolutions and their specialisations. For the family \[ \Gamma_m(n)=\langle m,m+1,\ldots,\widehat{m+n},\ldots,2m-1\rangle, \qquad 2\leq n<m, \] where the hat denotes omission, we also compute the Poincaré series of the residue field and characterise Golodness. For fixed multiplicity $m$ and $3\leq n<m$, we show that these rings share the same rational residue field Poincaré series, but exhibit arbitrarily late departures from Serre's upper bound.

We study the transmission of two independent confidential classical messages over a quantum broadcast channel, one for each receiver. Each message must remain secret from the other receiver, including when that receiver knows its own message. For classical inputs and quantum outputs, we establish the classical Marton-type inner bound under average reliability and conditional strong secrecy. The encoder selects pairs of codewords from independently generated codebooks using normalized likelihood weights. We prove reliability through a change-of-distribution argument. Our main technical result is a bipartite classical-quantum resolvability theorem that accounts for the dependence created by pair selection and establishes secrecy for the same encoder. We also obtain a multi-letter capacity characterization and extend it to arbitrary quantum inputs under secrecy against the other receiver, together with the Stinespring environment. We recover confidential capacity regions for deterministic classical and degraded classical-quantum channels, with an explicit evaluation for the classical Blackwell channel. For coherent isometric extensions of injective deterministic classical broadcast channels, we show that the confidential classical capacity region equals that of the corresponding classical channel. We then compare confidential classical communication with quantum transmission. For the coherent isometric extension of the Blackwell channel, we determine the unassisted quantum-capacity region and show that some achievable confidential classical rate pairs lie outside it. The Platypus channel provides another example of this separation.

We analyze an inexact, proximal trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function. Our algorithm leverages the proximal gradient as a benchmark to guarantee global convergence. However, evaluating the proximal gradient involves solving a nonsmooth convex optimization problem that in many applications cannot be performed exactly. As such, we develop a framework that permits inexact evaluations of the proximity operator and we show the proposed inexactness conditions yield descent properties analogous to the exact proximal gradient. We further discuss practical procedures for computing approximate proximity operators for two common applications: (i) the nonsmooth term is the sum of nonsmooth convex functions where one is composed with an affine map and (ii) using alternative inner products to facilitate the evaluation of the proximity operator. We demonstrate our algorithm and confirm our analysis with two numerical experiments arising in PDE-constrained optimization.

We show that among Möbius bands in $\mathbb{S}^3$ bounded by a great circle, the minimal Willmore energy is realized by an embedded minimal Möbius band with Morse index two. To prove this, we introduce a $2$-parameter “canonical family" associated to any non-orientable surface in $\mathbb{S}^3$ with boundary a great circle and apply a min-max argument. The canonical family detects the Euler number of the surface and is inspired by the $5$-parameter family discovered by F.C. Marques and A. Neves detecting the genus of an orientable surface in $\mathbb{S}^3$. For $\mathbb{Z}_2$-invariant Klein bottles, this reduces R. Kusner's 1989 conjecture that $\tau_{1,2}$ minimizes the Willmore energy for a Klein bottle immersed in $\mathbb{S}^3$ to any of several conjectural characterizations of the Lawson Möbius band.

Let \(\mathbf{E}\to M\) be a finite-rank Riemannian vector bundle over a compact Riemannian manifold with nonempty smooth boundary, equipped with a compatible connection. We develop an intrinsic trace theory for Sobolev sections defined through weak covariant derivatives. For every integer \(m\ge1\) and every \(1\le p<\infty\), we construct continuous trace operators for the covariant derivatives up to order \(m-1\) and characterize the Sobolev space with homogeneous Dirichlet boundary conditions as the kernel of the corresponding trace operator. As an application, we establish the existence of ground state solutions for a class of quasilinear elliptic equations on Riemannian vector bundles with generalized \((p,q)\)-growth. The proposed framework unifies several important geometric operators, including the Bochner \(p\)-Laplacian, the \((p,q)\)-Bochner Laplacian, and prescribed mean curvature-type operators.

A seminal result of Hastings showed that entangled inputs can increase the rate of classical communication through independent uses of a quantum channel. While Hastings' result proved the existence of this phenomenon, it has remained a major open problem to find an explicit channel with this property. In this work, we resolve this question by constructing an explicit family of finite-dimensional quantum channels for which entangling inputs between pairs of channel uses yields an unbounded separation from the achievable communication rate using unentangled inputs. We describe a circuit implementation of our channel that uses only deterministic qudit Clifford unitaries and a single binary measurement with classical feedforward. The key ingredient is a careful design of the measurement and feedforward process that yields the required one-copy and two-copy output entropy bounds from moment estimates alone, without requiring strong convergence of the underlying unitary family.

Families of explicit analytic solutions of the magnetohydrodynamic equilibrium equations are presented, equivalent to steady incompressible Euler flow. The solutions are non-axisymmetric and possess exact nested toroidal flux surfaces. No expansion is made in inverse aspect ratio or in the deviation from axisymmetry. The magnetic field and flux surfaces are given explicitly in Cartesian coordinates using elementary functions. The field, current density, and scalar pressure are smooth over the toroidal domain. The pressure gradient vanishes only on the magnetic axis. One family of solutions has uniform rotational transform $\iota=2$, while another family has a sheared $\iota$ profile. These counterexamples to Grad's conjecture are valuable for understanding the existence and regularity of 3D equilibria and for testing numerical codes.

We prove that the real $K$-theory functor is lax symmetric monoidal. Moreover, it is initial among those real functors $\mathsf{Wald}_\infty^\mathrm{gd}\to\mathsf{Sp}^{C_2}$ that are additive and lax symmetric monoidal.