Papers

False discovery rate control does not by itself guarantee optimal recovery of sparse signals: a procedure may make few false discoveries while missing an unnecessarily large fraction of the alternatives. We investigate whether the classical Gavrilov–Benjamini–Sarkar (GBS) step-down procedure also achieves the sharp frequentist minimax limits for sparse recovery in independent Gaussian sequence models. Under a beta-min separation at the critical sparse detection boundary, GBS attains the exact asymptotic risk constant $\bar\Phi(b)$ for normalized Hamming loss and combined $\mathrm{FDP}+\mathrm{FNP}$ loss. The Hamming conclusion remains valid when the number of signals is only bounded above by the nominal sparsity $s_n$. For heterogeneous signal magnitudes, the corresponding risk under either loss equals the sharp benchmark $\Lambda_n(a)+o(1)$, without assuming that $\Lambda_n(a)$ converges; this is additive, rather than necessarily relative, sharpness. We assume $s_n\to\infty$, $n/s_n\to\infty$, and a vanishing nominal level $\alpha_n$ satisfying $\log(1/\alpha_n)=o(\sqrt{\log(n/s_n)})$. Unlike step-up rules, a step-down procedure must satisfy every preceding ordered $p$-value constraint before reaching a given rank. Our analysis handles this sequential obstruction directly: a finite-sample shifted order-statistic bound makes false positives negligible, while new uniform crossing arguments show that the rejection path reaches the optimal fraction of signals. The results extend the range of testing architectures known to attain the sharp minimax benchmarks, without requiring localization of a single rejection threshold.

In this paper, we consider the following critical nonlinear elliptic equation: \[ - \Delta u = a(x) |u|^{2^*-2}u \quad \text{in } \mathbb{R}^N, \quad u \in \mathcal{D}^{1,2}(\mathbb{R}^N) \] where $N \ge 3$, $2^* = \frac{2N}{N - 2}$, $a(x) \in C(\mathbb{R}^N, \mathbb{R})$ is a positive function that is invariant under the map $x \to -\frac{x}{|x|^2}$. Under some assumptions on $a(x)$, we show the existence of a positive solution to the equation that is invariant under the Kelvin transform. The symmetry condition imposed here is substantially weaker than the invariance under a noncompact symmetry group that is typically assumed in the literature. The key to the proof is a classification of the Palais–Smale sequences of the associated energy functional. To this end, we establish a new abstract profile decomposition theorem incorporating symmetries such as the Kelvin transform.

We study topological relaxation in ordered viscoelastic conformation flows at finite epsilon. In an ordered region, a positive spectral gap selects an oriented principal axis and hence an S^2-valued director with a Hopf class. We show that a change of this class must be accompanied, before the sharp-interface limit, by one of a finite list of costs: exterior gap concentration, ordered-core mass, boundary flux, FENE/collar loss, or a topology exit. The result is proved for a concrete Landau-de Gennes ordered-core closure coupled to an Oldroyd/FENE-type transport law. The structural hypotheses used in the argument are verified up to the first typed exit time: the Morse-Bott ordered well, tubular soft coordinates, massive-mode coercivity, a projected transported Ginzburg-Landau equation, exterior gap control, and tame FENE/collar coefficients. The projected Ginzburg-Landau equation separates translation modes from the remaining residuals. The translation modes give the normal line force, while the orthogonal soft, massive, geometric, and collar terms are absorbed by coercivity or charged to the corresponding exit. A modulated-energy argument propagates a nonempty class of vortex-tube data on regular intervals. On each such interval, the normalized core measures converge to an integral one-varifold satisfying a transported forced Brakke inequality with the computed force. The theorem therefore derives the force projection, open-basin propagation, and Brakke compactness estimates before invoking any limiting Brakke flow, and it records the finite-epsilon cost when the regular ordered-core description breaks down.

Existence proofs for many matrix decompositions share a recursive routine: a local transformation prepares the matrix, a slice is selected, a recursive solution is obtained, and the result is lifted and transported back. Formalizing this routine uniformly in dependent type theory is difficult because recursive subproblems may change index types, and reconstruction must preserve structural predicates across block embeddings and reindexings. We develop a Lean~4 framework that separates decomposition schemas, transformations, reduction strategies, measures, lifting, transport, and subtype induction. The framework uses general index types, packages square and rectangular matrices in universe types, and provides a decomposition driver that assembles strategy data into subtype-induction instances. It has been instantiated across PLU, LU, LDL/Cholesky, QR variants, Gauss rank normal form, Hessenberg reductions, Schur variants, normal spectral decomposition, SVD, bidiagonalization, tridiagonalization, UTV, Smith normal form, rational canonical form, and Jordan-type forms at varying levels of statement strength. Across these instances, repeated decomposition proofs are best treated not as separate tasks but as instances of a more general inductive statement whose interface records a certified proof path compatible with the chosen decomposition statement.

Nonlinear differential equations play a crucial role in modeling a wide range of phenomena, yet their solutions remain notoriously difficult to obtain. With the rapid development of quantum computing, quantum algorithms for efficiently solving such equations are actively being explored. One promising approach is based on Carleman linearization, which transforms nonlinear differential equations into linear systems. However, this method suffers from exponential divergence beyond a certain time scale. By reformulating the solutions in terms of eigenvalues and eigenvectors, we identify that this divergence originates from the Laurent expansion outside its neighborhood of convergence. To address this issue, we insert a regularized function to the divergent solution hinted by analytical continuation. We validate this divergence-correction method on both the logistic equation and some other partial differential equations like KPP-Fisher equations and Phase-Field models under periodic conditions. We implement our method for the logistic equation using the Linear Combination of Unitaries (LCU) quantum algorithm, providing a detailed complexity and error analysis.

Memory-efficient optimizers such as GaLore train large language models by projecting gradients onto a rank-r subspace recomputed every T steps, assuming this subspace is a slowly drifting object that can be tracked. We show that beyond a small reproducible core, there is no such object. Two estimates of the top-r subspace computed at the same step from disjoint minibatches disagree as much as estimates computed T steps apart (0.73 vs 0.74 of the maximal chordal distance sqrt(2r), at Pythia-160M with r=128): the apparent rotation at each refresh is dominated by estimator noise. This holds across four model families in three architecture classes from 70M to 6.9B parameters, strengthening with scale, and more weakly in a vision transformer. Only ~39 of 128 directions are reproducible across minibatches, and averaging cannot recover the rest: under N-fold averaging the gradient's spectral tail shrinks as N^(-1/4) rather than the N^(-1/2) of pure noise, so no averaging budget makes the subspace well defined. What helps instead follows from treating each refresh as a change of coordinates for Adam's state. Carrying the second moment blindly is provably about (r-k*)/2 worse than the best rotation-blind estimator, while the first moment transports exactly through the rotation, the optimal linear map under isotropic gradients and the rule LDAdam uses. At 1B over 40k steps (3 seeds), full LDAdam reaches 18.7 perplexity at beta2=0.999, beating untransported GaLore after its best beta2 fix (19.3); shortening the second-moment memory to beta2=0.99 helps the refreshing optimizers, though for canonical GaLore the effect is small and a full-rank control reverses it. One measurable fact, subspace non-identifiability, clarifies why GaLore works, which patches work, and what to check before trusting a low-rank assumption: the reproducible rank k*.

We study the abstract damped wave equation on a Banach space, allowing the damping coefficient to be unbounded. By recasting the equation as a first-order system and identifying conditions under which the associated block operator generates a $C_0$-group, we construct generalized cosine and sine families that represent mild and classical solutions, extending the classical undamped theory. We establish existence, uniqueness, regularity, invariant subspaces, growth rate, and trigonometric type identities for these families. Our setup applies to a broad class of damped wave, Klein–Gordon, and higher-order PDE examples, including cases where damping restores well-posedness that fails in the undamped equation.

Pencils of the form $\mathcal{A}({\lambda}) = {\lambda}E-A$ are studied, where $A$ and $E$ are bounded linear operators on a Hilbert space. Of interest are the spectral properties of $\mathcal{A}({\lambda})$. This is done via a corresponding linear relation in a Krein space, which is given in range representation using the two operators $A$ and $E$. Under some assumptions on $E$ and $A$, the linear relation in range representation is nonnegative or has finitely many negative squares. Then one uses spectral properties of linear relations and deduces spectral properties of the operator pencil $\mathcal{A}({\lambda}) = {\lambda}E-A$.

We prove a mesoscopic central limit theorem for linear eigenvalue statistics of correlated Hermitian random matrices. The class considered here includes Wigner and Wigner-type matrices, as well as models whose entry correlations decay polynomially in the distance between index pairs. The proof combines a multivariate cumulant expansion with multi-resolvent local laws and a detailed analysis of the resulting variance kernel on the operator-level.

We study singular fibers of elliptic fibrations on normal \(K3\) surfaces. For a normal \(K3\) surface \(Y\) with minimal resolution \(\nu \colon X \to Y\), we describe singular fibers on \(Y\) in terms of contractions of suitable ADE configurations of \((-2)\)-curves in singular fibers on \(X\). We determine ADE configurations occurring in singular fibers and describe fibers obtained after contraction. As a consequence, we obtain a description of singular fibers of elliptic fibrations on normal \(K3\) surfaces.

Frege's theorem proves that Hume's principle, in second-order logic, yields all of arithmetic. Yet the Pirahã people show one-to-one correspondence (equinumerosity) only where pairing can be enacted, with its range extended under local training, and still have no counting or arithmetic. We argue this is not a paradox but a matter of precise localization. Hume's principle includes a cardinality operator # that names cardinals as objects (often modeled as equivalence classes of equinumerous concepts), and what the Pirahã lack is not the relation but this operator. We identify the number-word practice as the cognitive realization of #, which recasts the "number-as-cognitive-technology" thesis in formal terms and locates the cognitive boundary at symbolization, not recursion. The identification is generative, not decorative: the reach of # tracks the reach of the token practice that carries it, so across languages and cultures we see a gradient, not a sharp cliff. And number words are not special as words; what # needs is any stable, reusable marker that can preserve exact cardinal identity across absence, rearrangement, delay, or modality shift: a spoken numeral, a scratch on a stick, or a knot in a cord. So the thesis is about having some symbolic token-practice, not about language specifically. It is supported by converging evidence from Nicaraguan homesigners, numerate adults under verbal interference, and cross-linguistic numeral gradients. We make no causal, acquisition, or neural claim; the identification is constitutive.

The limited-memory BFGS (L-BFGS) algorithm is a cornerstone of large-scale optimization due to its linear memory and computational costs. However, in ill-conditioned or non-convex landscapes, the implicit inverse Hessian approximation can suffer from an exploding condition number, leading to numerical instability and degraded convergence. To address this, we propose Two-Sided L-BFGS, a safeguarded variant that dynamically constrains the condition number of the inverse Hessian operator via a two-sided geometric envelope. Moreover, we show that Two-Sided L-BFGS preserves accumulated curvature information and maintains standard $O(mn)$ memory and per-iteration time complexities. We prove that this geometric envelope yields a uniform bound on the condition number of every inverse Hessian approximation generated by the algorithm. By tracking the algebraic evolution of the extreme eigenvalues through $m$ consecutive quasi-Newton updates starting from a scaled identity matrix, the resulting bound is expressed explicitly as a function of the memory depth, problem dimension, and envelope hyperparameters. Moreover, we show that Two-Sided L-BFGS preserves asymptotic global convergence in non-convex regimes under standard smoothness and strong Wolfe line-search assumptions, matching the theoretical guarantees of L-BFGS variants utilizing the Li-Fukushima cautious update rule. Numerical experiments on high-dimensional optimization problems demonstrate that the proposed method maintains well-conditioned inverse Hessian approximations and improves robustness and convergence behavior on ill-conditioned benchmarks.

For every loopless matroid $M$ and every Feichtner–Yuzvinsky building set $\mathcal{G}$ containing the top flat, we construct an integral tangent class $T_{M,\mathcal{G}}^{\mathbb{Z}}\in K_{\mathbb{Z}}(M,\mathcal{G})$; in the realizable case it specializes to the class of the tangent bundle of the corresponding wonderful compactification, it recovers the Hilbert series of the Chow ring through Hirzebruch–Riemann–Roch, and it satisfies the expected Chern-alpha lower bounds. This reproduces the tangent class and its key properties studied by the first author in arXiv:2606.22650. The main body of this paper was produced autonomously, without human mathematical guidance, by Danus, an AI mathematical reasoning agent. Danus solved the problem before arXiv:2606.22650 was publicly available, demonstrating the potential of AI agents in mathematical research. We reproduce its output faithfully, adding only editorial comments; the experiment is documented in Appendix B.

A coherent domain, in the sense of Dana Scott's domain theory, is a domain in which the intersection of every two compact saturated subsets is again compact, when the domain is equipped with the Scott topology. Coherence plays key roles in classifying Cartesian closed subcategories of domains and in characterizing Lawson compactness of domains. In this paper, we find two typical domains that fail to be coherent, and prove that a bounded algebraic domain fails to be coherent if and only if it has one of the two typical domains as its Scott-continuous retract. Similar results also generalize to bounded continuous domains, provided that the domains in consideration are hereditarily Lindelöf and weakly Hausdorff in the Scott topology.

Averaged models are widely used to analyze periodically switched linear systems, yet the stability of the averaged flow does not automatically guarantee the stability of the true switched dynamics. The discrepancy arises from noncommutativity among the subsystem generators, so stability certificates benefit from bounds that expose this dependence in a form compatible with Lyapunov contraction metrics. We derive an explicit operator-norm bound for the one-period mismatch between the switched and averaged propagators, in which the leading-order error depends explicitly on the pairwise commutator norms of the scaled mode generators, with a closed-form prefactor depending only on the generator norms. This bound yields a computable threshold for the switching period below which the switched system inherits exponential stability from its averaged model, uniformly certified over admissible duty fractions. The analysis extends to an arbitrary number of switching modes via telescoping induction, and a semidefinite program provides sampled duty-dependent Lyapunov metrics for implementing the certificate.

In this paper, we study when the Smyth powerspace $\mathcal{Q}^*_v(X)$ of a topological space $X$ is coherent, and prove that $X$ is coherent and weakly Hausdorff if and only if $\mathcal{Q}^*_v(X)$ is coherent and weakly Hausdorff. We give examples to show that neither coherence nor weak Hausdorffness of $X$ solely implies that $\mathcal{Q}^*_v(X)$ is coherent or weakly Hausdorff. As a byproduct, our work gives an affirmative answer to a question raised by Xu.

Let $n\geq5$ and $m\geq3$. Let $\Phi\colon\mathrm{B}_n\to\mathrm{B}_m$ be a homomorphism of braid groups. We prove that if the image of $\Phi$ is irreducible and not cyclic, then $m=n$ and $\Phi$ agrees with an automorphism modulo the center $Z(\mathrm{B}_m)$. This resolves in the affirmative a conjecture of Chen, Kordek, and Margalit. It also provides a partial resolution to a problem on the K3 problem list. As a consequence, we prove that every holomorphic map $\mathrm{UConf}_n(\mathbb{C})\to\mathrm{UConf}_m(\mathbb{C})$ for $n\geq5$ and $m\geq3$ is affine equivalent to either a constant map or the identity map. This resolves a conjecture of Farb for $n\neq4$.

Let $f(N)$ denote the largest size of a set $A\subseteq [N]=\{1,\ldots,N\}$ containing no distinct $a,b,c$ such that \[ \frac2a=\frac1b+\frac1c . \] We prove \[ f(N)\gg N\exp\!\left(-(2\sqrt{\log(24/7)}+o(1))\sqrt{\log\log N}\right). \] The construction filters the odd integers up to $N$ by a random affine image of a dense three-term-progression-free set in a prime field $\mathbb{F}_q$ with $q\asymp\log N$, and then deletes a controlled family of collapsed triples.

We prove that in a non-trivial conformal circle-preserving transformation (CPT for short), $e^{\sigma}F$ on a (forward or backward) complete Finslerian manifold $(M,F)$, the conformal factor has at most two critical points. Then a diffeomorphism classification based on the number of critical points as well as some curvature rigidity properties - of Finslerian manifolds that admit nontrivial conformal CPTs - is presented.

A Killing tensor field on a Riemannian space corresponds to an integral of the geodesic flow polynomial in momenta. A (contravariant) Killing tensor field is called decomposable if it is a polynomial in Killing vector fields. While all Killing tensor fields on the spaces of constant curvature and on the complex projective space are decomposable, there is an explicitly constructed family of indecomposable quadratic Killing tensor fields on the quaternionic projective spaces $\mathbb{H}P^n, \, n \ge 3$. We prove that the algebra of Killing tensor fields on the quaternionic projective space is generated by Killing vector fields and these indecomposable quadratic Killing tensor fields. We also give another proof of the fact that the algebra of Killing tensor fields on the complex projective space is generated by Killing vector fields.

We construct a $\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra from a finite-dimensional vector space with a nondegenerate symmetric bilinear form, together with its canonically twisted module. This algebra is generated by suitable noncommutative generalizations of bosonic and fermionic fields and is a noncommutative generalization of the free boson-fermion vertex operator superalgebra. Similarly to the purely bosonic and purely fermionic cases in the early works by the second author in [H1], by Fiordalisi and the third author in [FQ], and by the third author in [Q3], the usual super-commutatitive relations between creation and annihilation operators still hold while no relations exist among creation operators. In particular, normal-ordering remains well-defined. As in [H1], [FQ], and [Q3], we prove a generalized Wick's theorem in this case, which gives a formula for a product of two normal ordered products of bosonic and fermionic generating fields. Using this generalized Wick's theorem, we construct the $\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra and its canonically twisted module in this case. The construction in this paper is the algebraic part of our construction of suitable Dirac-like operators from spin manifolds.

This paper shows that bit-level reliability can be converted into block-level reliability for binary codes over BMS channels through Forney's concatenation scheme. The construction concatenates an inner code of rate R-o(1) and vanishing bit-error probability \epsilon with an outer code of rate 1-o(1) equipped with a bounded-distance decoder. We show that, when the outer correction radius exceeds \epsilon and a suitable large-deviation condition holds, the concatenated code has rate R-o(1) and vanishing block-error probability. Hence, a family with vanishing bit-error probability for rates below capacity can be converted into a capacity-achieving concatenated-code family. We further show that, when binary BCH codes are used as outer codes, inner codes with bit-error probability \epsilon=o(1/\log\log n) can be converted into a capacity-achieving concatenated family.

In this paper, we characterize John and uniform domains in doubling metric spaces. Specifically, we show that a locally quasiconvex domain in a doubling metric space is length John if and only if it is diameter John. For uniform domains, we prove that a domain in a doubling metric space is length uniform if and only if it is diameter uniform (or distance uniform) and locally quasiconvex. Moreover, in a doubling length metric space, we refine this result by showing that a domain is length uniform (resp. John) if and only if it is diameter uniform (resp. John).

We consider the numerical solution of the wave equation in materials with rapidly varying coefficients, and time harmonic sources. For these problems, direct discretization is prohibitively costly, and instead multiscale methods are used. There are several multiscale methods that directly discretize in the frequency domain. In this work we instead start in the time-domain and combine a finite difference Heterogeneous Multiscale Method (HMM) for the wave equation with the WaveHoltz method. Each WaveHoltz iteration marches the wave equation towards the time-periodic Helmholtz solution. The advantages of the WaveHoltz method relative to traditional Helmholtz solvers carry over directly to the multiscale problems considered here. Since, in addition, the time-domain solver does not artificially impose boundary conditions on the micro-scale problems, no boundary errors from the micro-scale problems are present in the homogenized frequency domain solution.

The search for Hamiltonian cycles in vertex-transitive graphs and digraphs is a classical problem at the interface of graph theory and group theory. In the undirected setting, this goes back to the well-known conjectures of Lovász and Thomassen concerning Hamiltonian paths and cycles in connected vertex-transitive graphs. Dating back to Rankin's 1946 work, the directed analogue has an even longer history, linking the search for long cycles to classical group-rearrangement problems. Trotter and Erdős showed in 1978 that connected vertex-transitive digraphs need not be Hamiltonian. In light of this result, Alspach asked in 1981 whether there exist connected vertex-transitive digraphs whose longest directed cycle misses arbitrarily many vertices. This question was only recently resolved by Bucić, Hendrey, Mohar, Steiner and Yepremyan, who constructed connected vertex-transitive digraphs on $n$ vertices whose longest directed cycle omits $(1-o(1))\log n$ vertices. They conjectured that the number of omitted vertices can grow linearly with $n$, remarking that it would already be interesting to improve their logarithmic lower bound to a polynomial bound. In this paper, we confirm their conjecture in a strong form by constructing infinitely many connected vertex-transitive digraphs on $n$ vertices whose longest directed cycle omits at least $n/12$ vertices. In the same work, Bucić, Hendrey, Mohar, Steiner and Yepremyan also proved that every connected vertex-transitive digraph on $n$ vertices contains a directed cycle of length $\Omega(n^{1/3})$, giving the first lower bound for this problem that grows with $n$. We improve this to $\Omega(\sqrt n)$, matching the order of Babai's classical theorem from 1979 for undirected vertex-transitive graphs.

It is known that a finite-dimensional Caputo fractional differential equation, though itself need not generate a semiflow, can be represented as a Volterra integral equation which generates an infinite-dimensional semiflow on the space $\mathfrak{C}=C([0,\infty); \mathbb{R}^d)$ under the standard compact-open topology. In this paper we construct a compact absorbing set and an attractor for this semiflow on $\mathfrak{C}$, and then prove that the attractor consists of equi globally Hölder continuous functions. This strengthens the previous work of Doan & Kloeden \cite{DK21} where a bounded (with respect to a weighted norm) attractor was constructed.

We consider a derivative nonlinear Schrödinger system called the Chen-Lee-Liu type system on the torus. This system is known as a completely integrable system. We prove the flow map fails to be continuous at every point in the Sobolev space $H^s(\mathbb{T}) \times H^s(\mathbb{T})$. Moreover, we establish an additional condition required for the flow map to be continuous. For the discontinuity, we take a sequence converging to the initial data for which the corresponding solutions do not exist.

We provide a simpler proof of the fact, originally proved by Seidai Yasuda, that symmetric multiple zeta values generate the entire space of multiple zeta values. Furthermore, based on this argument, we present an algorithm for expressing multiple zeta values in terms of symmetric multiple zeta values and products of multiple zeta values. Moreover, we give some results on symmetric and finite multiple zeta values of depth three.

This paper derives a design principle for structure-preserving Galerkin formulations of energy-conserving hyperbolic systems. The aim is to reproduce the modal-energy-exchange structure of the continuous system within a resolved finite-mode space. Total energy conservation follows from this structure. We introduce a state-dependent physical-energy metric H and derive the corresponding energy-compatibility identity. In the infinite-mode exact-integration model, the volume contribution has an antisymmetric representation after H-orthogonalization, yielding pairwise modal energy exchange. Interface contributions take the same exchange form. To reproduce this structure in the practical finite-mode system, we combine two constructions: a Galerkin projection coupled with the physical-energy metric that guarantees the H-metric summation-by-parts identity, and an energy-compatibility closure that removes the component of the compatibility action contributing to the scalar energy residual. With a shared numerical energy flux at interfaces, they close the total-energy balance of the finite-mode system while preserving pairwise modal energy exchange. We also compare the practical operator construction with the finite-mode exact-integration reference and obtain an O(h^p+1) defect estimate. Finally, we derive an equivalent form of the resulting equation in the fixed Galerkin basis for direct implementation.

We construct universal cycles for affine planes in $\mathbb F_q^n$ for all prime powers $q$ and all $n\ge4$, using sliding windows of length three. The construction is local-to-global: explicit local cycles are built on frame configurations, the linear $2$-subspaces are organized by a layered frame decomposition, and the resulting cycles are assembled by gluing along shared directions. The universal cycle obtained has direction set containing all $1$-subspaces. We also extend the construction to universal cycles for $3$-subspaces of $\mathbb F_q^n$.