Papers

We prove that the spectrum of a smooth closed three-manifold gives a positive lower bound for the length of its shortest nonconstant closed geodesic. The argument uses three curvature heat coefficients and the first positive singularity of the retarded wave trace. The heat coefficients yield a curvature–length dichotomy: sufficiently large maximum curvature forces a closed geodesic shorter than the curvature scale. We then show that a shortest closed geodesic lying below the conjugate radius produces a singularity of the sine trace. Combining these estimates gives a lower bound depending only on the common spectrum. The compactness theorem of Anderson then implies smooth compactness of the full isospectral family modulo diffeomorphisms.

In 2006, Katzman showed that the first six Betti numbers of the Stanley–Reisner ring of a flag complex are field independent. He also found flag complexes on eleven vertices whose eighth Betti number depends on the field, and asked whether the seventh is always field independent. We answer this affirmatively by proving a stronger, purely topological result. Let $\tau(d)$ be the least number of vertices of a flag complex whose $d$-th reduced integral homology has torsion. We prove that $\tau(d)\geq d+10$ for every $d\ge0$. This bound yields the field independence of the seventh Betti number. Equivalently, combining our result with Katzman's, for every finite simple graph $G$, the first seven Betti numbers of the edge ideal $I(G)$ are field independent.

A sequence of integers $1=a_0<a_1<\cdots<a_k=n$ is called an addition chain of length $k$ if $a_j=a_s+a_t$ with $0\le s,t<j$ for all integers $j\in \{1,2,\ldots,k\}$. We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$. Here we investigate the distribution of the function $\ell$ through the counting function $$ F(m,r):=\#\{n\in [2^m,2^{m+1}):\ell(n)\le m+r\} $$ and show that, for every fixed $0<c<\log 2$, there exist positive constants $K_1$ and $K_2$ such that $$ K_1^r m^r\le F(m,r)\le K_2^r m^r $$ for all sufficiently large $m$ and all integers $m^{0.9}<r\le cm/\log m$. The upper bound also holds for every integer $r>m^{0.9}$. Moreover, denoting by $G(m,r)$ the number of distinct addition chains of length $m+r$ leading to an integer $n\in [2^m, 2^{m+1})$, we show that there exist positive constants $K_3$ and $K_4$ such that $$ K_3^r \left(\frac{m^2}{r}\right)^r\le G\left(m,r\right)\le K_4^r \left(\frac{m^2}{r} \right)^r $$ provided $m^{0.9}<r<m$. This improves and generalizes previous results on the minimal length of addition chains and addresses a question raised by Paul Erdős.

We prove a multidimensional Birkhoff theorem for a new class of "$C^0$ Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.

Generalized Langevin equations describe non-Markovian dynamics in which the evolution of resolved variables depends on their past. We propose a memory-conditioned diffusion method for learning stochastic flow maps of these dynamics from observed trajectories, without identifying a memory kernel or reconstructing unresolved variables. A compact, recursively updated bank of exponential filters enables the flow map to retain predictive history over multiple time scales without conditioning on long observation windows. The next-step distribution is conditioned on the current observation and this memory state, whose storage and update costs are independent of the history length for a fixed bank size. Predictive criteria guide the memory budget, with reference-assisted selection in the vector benchmark, and an optional linear projection further reduces the conditioning dimension. A kernel-based score estimator generates conditional samples without training a score network, and these samples are used to train a neural flow map for autoregressive simulation. Three numerical examples assess long-memory retention at small conditioning dimension, predictive compression in coupled vector dynamics, and non-Gaussian conditional distributions and intermittent events. The non-Gaussian example reproduces conditional asymmetry and burst statistics in a stochastic model of the plasma scrape-off layer.

While the Jacobian of a planar harmonic mapping keeps a constant sign, the oriented and the multiplicity-weighted surface elements agree up to that sign, and the two natural moment hierarchies coincide. The nearest classification results in the literature are stated under exactly this hypothesis; this paper works past it. Once the Jacobian changes sign, the critical circle acquires a caustic as its image, part of the plane is covered several times with opposite orientations, and the hierarchies part company: integrating a radial weight against the oriented surface element $n_z$ lets overlapping sheets cancel, giving the degree-weighted moments $J_n$, while integrating against $|n_z|$ lets every sheet count positively, giving the multiplicity-weighted moments $J_n^{\pm}$. These are the classical degree and area formulas for the same map, the second weighted by the multiplicity function (Banach indicatrix). It is the second family, the one the caustic threatens, for which a closed form is no longer automatic. Our main result is that it has one anyway: $J_n^{\pm}$ is the total variation of the very same explicit primitive whose endpoint difference gives $J_n$. Crossing the caustic therefore costs only finitely many extra evaluation points, and they are the same points for every order $n$; for a monotone profile there is at most one interior breakpoint, at the critical radius itself. The family carrying this is the two-monomial planar harmonic mapping $F(w)=w+iC\bar w^k$, $w=p(u)e^{iv}$, arising as the planar projection of a Fourier-perturbed surface. On the degree-weighted side $J_n$ has a boundary-only closed form for every $n\ge 0$, the zeroth being the oriented $z$-flux. Neither weighted integral is a new object, and the caustic has been located before in the literature on harmonic trinomials; what is new is the closed evaluation of the multiplicity-weighted hierarchy across it.

We introduce partition functors: algebraic structures indexed by partitions of finite sets and equipped with restriction and transfer maps along refinements. We construct a monad PD on the category of partition functors and on several categories of partition functors with additional multiplicative structure. For a partition ring, exponential elements in its associated completed ring of symmetric functions acquire canonical logarithms after applying PD. This gives rise to a universal exponential relation between multiplicative and additive power operations. For representation rings this can be used to recover the classical relation between symmetric powers and Adams operations, while for Morava E-theory it can be used to recover Ganter's exponential relation between symmetric powers and Hecke operators. We show that the representation rings of products of symmetric groups form the initial partition ring and that, for symmetric monoidal partition functors, the monad PD is closely related to symmetric invariant tensors and the divided power envelope. We also construct a symmetric monoidal partition ring carrying the universal exponential element, so that its image under PD carries the universal exponential relation.

Given an abelian scheme $A$ over a $p$-adic ring $R$, Borger and Saha constructed a filtered module $\{\mathbf{H}_\delta(A) \supset \mathbf{X}_{\mathrm{prim}}(A)\supset \{0\}\}$ with a semilinear operator $\mathfrak{f}^*$ on $\mathbf{H}_\delta(A)$ using the theory of arithmetic jet spaces. The above object admits a canonical map $\Phi$ to the Hodge sequence $\{\mathbf{H}^1_{\mathrm{dR}}(A) \supset H^0(A,\Omega_A)\supset \{0\}\}$ of $A$ in the category of filtered modules. As a result, by restricting $\Phi$, we obtain a natural $R$-linear map $\Upsilon: \mathbf{X}_{\mathrm{prim}}(A) \rightarrow H^0(A,\Omega_A)$. In this paper, we show that the map $\Upsilon$ is an isomorphism of vector spaces over $K$, the field of fractions of $R$. As a consequence, we will show that for all abelian schemes $A$, the operator $\mathfrak{f}^*$ on $\mathbf{H}_\delta(A)_K$ is a bijection, and our object $\{\mathbf{H}_\delta(A)_K \supset \mathbf{X}_{\mathrm{prim}}(A)_K\supset \{0\}\}$ becomes a filtered isocrystal. In fact, the above results admit a generalization to the setting of semi-abelian schemes. The elements of $\mathbf{X}_{\mathrm{prim}}(A)$ are represented by primitive additive characters of the arithmetic jet spaces attached to $A$. Hence, our isomorphism given by $\Upsilon$ provides an interesting character-theoretic interpretation of $H^0(A,\Omega_A)$ in terms of primitive delta characters. As a result, to any $1$-form $\omega$, the above isomorphism associates a canonical numerical invariant that depends on deformation theoretic data of $A$. Furthermore, we also extend a comparison theorem between $\mathbf{H}_\delta(A)_K$ and the first crystalline cohomology $\mathbf{H}_{\mathrm{cris}}^1(A)_K$ in the general case when the elliptic curve $A$ is defined over the ring of integers of a $p$-adic field $K$ that is a finite extension of $\mathbb{Q}_p$.

Control barrier functions (CBF) are a popular safety filter to ensure safety for nonlinear dynamical systems. However, when the system is subject to uncertainties and disturbances, this requires the use of robust variants of CBFs, which can be difficult to construct and can be overly conservative, especially for high-dimensional systems under input constraints. In this work, we propose a new approach to solve these challenges by introducing Least-Effort Adversarial Potentials (LEAP), a certificate that quantifies the robustness of a given state against disturbances in terms of the effort required by the disturbance to cause failure. We show that LEAP is a CBF for the undisturbed system, but can also be used to construct a safety filter that is robust to disturbances whose cumulative effort is bounded. We propose a method for constructing LEAPs with on-policy deep reinforcement learning. Next, we demonstrate LEAPs in simulation on a variety of multi-agent systems with disturbances and uncertainties. Finally, hardware experiments on a quadruped and quadrotors validate that LEAPs are well suited to tackle the disturbances and uncertainties from real-world robotic systems.

In this paper, we extend some Bohr-Rogosinski type inequalities of analytic functions in $\mathbb{C}$ to the cases of holomorphic mappings with values in higher-dimensional complex space. First, we obtain the general Bohr-Rogosinski radius for holomorphic mappings with values in the closure of the unit polydisc in $\mathbb{C} ^n$. Furthermore, we consider the corresponding problems for holomorphic mappings with value in the closure of the unit ball of a JB$^*$-triple. All the radius are optimal.

We introduce and study the class of almost subtractive submonoids of additive monoids. This property relaxes the strict condition of subtractivity while still allowing for a simulated subtraction operation sufficient for many algebraic applications. We explore the structural properties of these monoids and demonstrate their significance in the theory of modules over semirings, specifically regarding quadratic forms and their companions.

In this paper, we investigate the values of zeta functions of root systems at non-positive integer points. In particular, we focus on the zeta functions of type $\mathrm{A}_r$. Since non-positive integer points are included in the set of singularities of zeta functions of root systems, we introduce the ordered limit values to define those values. We study four particular ordered limit values and give explicit formulas for them, or recurrence formulas that allow us to calculate them inductively. We then prove that three ordered limit values of zeta functions of type $\mathrm{A}_r$ vanish under some conditions that can be regarded as a multivariable analogue of the trivial zeros of the Riemann zeta function. Finally, we discuss the other zeros of zeta functions of type $\mathrm{A}_r$ at non-positive integer points and a possible connection of such zeros and non-trivial identities among classical Eisenstein series.

It is well known that an Eisenstein series $E_k$ cannot be written as a product of two lower-weight Eisenstein series, except for $E_{14}=E_4 E_{10}= E_4^2 E_6= E_6 E_8$. A similar result is also known for the Hecke eigenforms. It is also known that equalities among the monomials of the Eisenstein series can be reduced to the above-mentioned identities. Recently, the present author, along with E. Saha studied the possible monomial relations of the Poincaré cusp forms of index $1$. So far, the problem of studying the monomial relations has been restricted to the setup of holomorphic modular forms. For an even integer $k\ge 4$ and a negative integer $m$, the Poincaré series $G_{k}(z,m)$ of weight $k$ and index $m$ is a weakly holomorphic modular form having a pole of order $-m$ at $\iota\infty$. It is immediate that the Poincaré series $G_k(z,m)$ for $m<0$ can not be written as a product of two lower-weight Poincaré series of the same index $m$. In this article, we investigate the possible equalities among the monomials of the Poincaré series $G_k(z,m)$ for arbitrary $m<0$. In particular, we show that for any $m<0$ such that $0.06\le\{-2m\pi\}\le0.99$, two monomials composed of $G_k(z,m)$ of the weights $k\ge 50$ are never equal, where $\{x\}$ denotes the fractional part of a real number $x$. In view of Weyl's equidistribution criterion, at least $93\%$ of $m<0$ satisfies the above inequality.

It is well known that the choice of basis functions in hp-FEM heavily influences the stability and the computational cost in order to obtain an approximate solution. For simplicial elements in two and three space dimensions, tensor-product-like basis functions built from Jacobi polynomials with different weights yield optimal properties with respect to condition number and sparsity. In this paper we construct such a high order basis for the Nédélec spaces and for the Raviart-Thomas and Brezzi-Douglas-Marini spaces, modifying existing definitions of Zaglmayr so that the functions belong to the Nédélec spaces of first kind and the Raviart-Thomas spaces. The bases are designed so that the Nédélec space of second kind and the BDM space are extensions of the respective other spaces. In the second part of the paper we introduce biorthogonal basis functions for $H(\text{div}, {\Omega})$ continuing previous research for $H^1({\Omega})$ and $H(\text{curl}, {\Omega})$. These functions can be expressed in closed form as sums of tensor products of Jacobi polynomials, which allows for fast computation of the $L^2$-projection.

We prove that for every window $w$ of the form $w(x) = e^{-\pi |x|^2} h(x)$, where $h$ is a polynomial, there exists a uniformly discrete set of points $\mathcal{S} \subset \mathbb R^{2d}$ such that the magnitude of the short-time Fourier transform $V_w f$ on $\mathcal{S}$ determines every $f \in L^2(\mathbb R^d)$ up to a constant phase factor. The separation distance can be chosen independent of the degree of $h$ and proportional to the square root of the dimension. The proof combines a discrete norming inequality, based on a multidimensional Remez inequality and VC-dimension bounds, with tail estimates for the reproducing kernel. A formalization of our main result in Lean 4 is also provided.

Local geometric mixing localizes geometric mixing by requiring geometric convergence to equilibrium in total variation only over finitely many transitions. It accommodates local convergence rates and captures rapid local equilibration, even when global mixing is much slower. We establish and discuss local geometric mixing bounds through Dobrushin contraction. We then apply this approach to Diffusion Path Monte Carlo, a recently proposed Markov chain Monte Carlo method, aimed at leveraging advances in score-based modeling, whose ideal transitions coincide with those of the Proximal Sampler. Our analysis covers both the ideal method and its implementable Metropolis-adjusted counterpart, providing mixing guarantees under minimal assumptions. For the ideal method, these guarantees complement recent spectral gap estimates, which we develop into mixing time bounds.

In this work we study the long term behavior of the curve shortening flow of asymptotically conical curves with convex projections in $\mathbb{R}^3$.

We consider the construction of confidence intervals for population means when observations are subject to missingness. To accommodate more general missingness mechanisms than missing completely at random (MCAR), we adopt a reparametrised version of the realisable contamination model of Ma et al. (2026), which is a mixture of an MCAR version and a missing not at random version of the same base distribution $P$. We characterise the minimax length of confidence intervals that can adapt to potentially unknown parameters of the model, including the contamination fraction, for Gaussian base distributions and for nonparametric classes satisfying certain tail or symmetry assumptions. In all of these settings, we provide explicit constructions of simple, practical and finite-sample valid adaptive confidence intervals that attain the corresponding minimax rates. Finally, we provide implications of our results for causal inference.

A classical theorem of Beckner, Carbery, Semmes, and Soria states that the Fourier restriction operator for the sphere does not satisfy the restricted weak-type estimate at the conjectured endpoint. Their proof uses the Besicovitch compression phenomenon and proceeds by contradiction, as in Fefferman's disproof of the ball multiplier conjecture. As a result, it reveals little about how badly the estimate fails. We develop a general framework that converts Besicovitch compression into a quantitative failure at an endpoint. Given a family of sets that compresses, our framework produces an explicit set that violates the bound, with a rate governed by the amount of compression. The rate is sharp within the admissible family of sets. We apply this to the sphere and the paraboloid and construct a deterministic sequence of sets along which the restricted weak-type inequality degrades at an explicitly computed rate. We also show that our construction does not provide blowup for the moment curve in dimensions three and higher, where the (much stronger) restricted strong-type bound is known to hold at the endpoint.

We complete, in the Riemannian setting, the classification of isolated singularities of positive solutions to the Yamabe equation for \(3\le n\le 29\), initiated in the flat case by Caffarelli, Gidas, and Spruck in 1989. Precisely, every isolated singularity is either removable or asymptotic to a Fowler solution. The sharpness of this range is established by counterexamples for \(n\ge30\) in our subsequent paper.

We propose a class of extended generalized Pareto models for right-censored survival data, with particular emphasis on tail inference and long-term extrapolation. Our framework integrates extreme value theory and survival analysis, combining a generalized Pareto distribution with a flexible perturbation distribution on the unit interval. We consider three perturbation specifications: a parametric Beta model, a Bernstein polynomial estimator, and a histogram-based estimator. To facilitate direct comparison, all three models are fitted using a unified iterative procedure adapted to right censoring through Kaplan-Meier-based pseudo-observations. A Monte Carlo study evaluates finite-sample performance across different tail indices, censoring levels, sample sizes, and model complexities. The results reveal a trade-off between flexibility and stability: the Beta specification generally performs best for tail-index estimation, whereas the histogram estimator performs particularly well for scale estimation under low censoring. The Bernstein estimator shows intermediate performance and greater sensitivity to sample size and censoring. Applications to bladder cancer recurrence and heart-failure survival data show that models with very similar in-sample fits can nevertheless produce markedly different tail-index estimates and long-term extrapolations. These findings emphasize the importance of perturbation specification when extended generalized Pareto models are used for survival extrapolation under censoring.

We study how Difference of Convex (DC) decomposition affects cutting-plane approximation in global optimization. For a smooth decomposition $f = g-h$, let $d$ be the dimension of the span of all Hessian ranges of $g$ over the convex hull of a compact feasible set. The smallest attainable dimension equals the minimum rank of a constant positive semidefinite matrix majorizing the Hessians of $f$ on that convex hull, and is attained by a quadratic first component. Retaining earlier gradient cuts and accepting new cuts only when the model error exceeds $\epsilon$ gives an $O(\epsilon^{-d/2})$ bound on accepted cuts. For an adaptive global DC method, this also bounds the number of subproblems solved exactly and globally. Under nondegeneracy, the exponent is sharp for uniform cutting-plane approximation of quadratic first components. A structured example reduces the certified exponent from $n/2$ to $1/2$ and the global subproblem to one variable.

We establish a Petersson-Rankin-Selberg identity for Hilbert Rankin-Cohen brackets over totally real fields of degree greater than one and arbitrary narrow class number. This extends results of Zhang-Zhang and Zhang-Zhou. For even $k,\ell\ge 2$, $r\in\mathbb{Z}_{\ge 0}$, $K=k+\ell+2r$, and normalized full-level cuspidal Hecke eigentuples $f\in S_K(\omega_f)$ and $h\in S_\ell(\omega_h)$, we show $\langle f,[E_k,h]_r\rangle=C_{F,k,\ell,r}L^S(k/2,\Pi(f)\times\Pi(h)^\vee)/L_F(k,\omega_f\omega_h^{-1})\ne 0$, where $C_{F,k,\ell,r}>0$. Specializing to $k=2$, we remove the remaining GRH assumption in the product classification of Hao-Qin-Zhou. Hence, over real quadratic fields of narrow class number one, the only full-level product identities among Hecke eigenforms of even parallel weights at least two, up to interchanging the factors, are $E_4^{\mathrm{eig}}=60(E_2^{\mathrm{eig}})^2$ and $h_8=120E_2^{\mathrm{eig}}h_6$ over $\mathbb{Q}(\sqrt{5})$.

We extend the phi-FD finite-difference scheme, which handles domains described by a level-set function on a Cartesian grid, to Neumann boundary conditions. The condition is relaxed near the boundary through the level-set and an auxiliary field, which is eliminated by writing the relaxed condition at a boundary node and at its nearest interior node. The scheme uses only interior nodes and the level-set: no ghost values, flux reconstruction or cut-cell volumes. We prove optimal consistency and show second-order convergence in the L2, H1 and Linf norms on a non-convex domain. Two elementary precautions, an individual normalization of the boundary equations and a rule for the choice of the coupled interior node, make the condition number O(h^-2) and independent of the smallest cut cell, without any stabilization term.

We confirm a conjecture posed by T. Iwaniec in 1982: the $L^p$ norm of the Beurling–Ahlfors transform is $p-1$ if $p\geq 2$, and $(p-1)^{-1}$ if $1<p\leq 2$.

We show that a Riemannian metric on $RP^3$ is a surface Zoll metric if and only if its four projective area widths are equal. We also establish a weighted Crofton formula on $RP^3$ under surface Zoll metrics, from which we derive an inequality relating the systole, the volume, and the common value of the projective area widths.

For a subset $S$ of a finite group $G$, let $I_G(S)$ denote the set of commutators $[g,x]=g^{-1}g^x$, where $g\in G$ and $x\in S$. Suppose that a finite group $G$ has a Carter subgroup $C$, that is, a nilpotent subgroup containing its normalizer. Suppose that any subgroup generated by a subset of $I_G(C)$ is $r$-generated. We prove that if $G$ is soluble, then the derived subgroup $G'$ has $r$-bounded rank. We produce examples showing that the solubility condition cannot be dropped. For any finite group, we prove that the rank of $G'$ is $(r,l)$-bounded, where $l$ is the maximum rank of composition factors of $G$ isomorphic to $PSL_2(q)$ for $q\equiv 7\,(\operatorname{mod}8)$. We also prove in the general case that $G'$ has $r$-bounded rank under the additional condition that for any $x\in I_G(C)$, any subgroup generated by a subset of $I_G(x)$ is $r$-generated. The proofs rely on the classification of finite simple groups, using which we prove that if a finite simple group $G$ has an element $x$ of prime order $p$ such that any subgroup generated by a subset of $I_G(x)$ is $r$-generated, then $G$ has $r$-bounded (Prüfer) rank or is isomorphic to $PSL_2(q)$ with $q \equiv 3\,(\operatorname{mod}4)$ when $p=2$, or to $PSL_2(q)$ with $q \equiv -1\,(\operatorname{mod} p)$ when $p\ne 2$.

We establish whole-sequence convergence of the primal iterates of the smoothing-based full-splitting proximal subgradient method (S-FSPS) of Boţ, Li, and Tao (SIAM J. Optim., 35 (2025), pp.~2623–2653) with a prescribed, nonsummable sequence of vanishing smoothing parameters. The challenge is that each iteration uses a different smoothed model. We address this by constructing fixed lifted potentials, deriving compatible sufficient-decrease and relative-error estimates, and applying a scaled Kurdyka–Łojasiewicz (KL) finite-length argument that controls the residuals caused by changes in the smoothing parameter. Under a KL assumption on the corresponding fixed lifted potential and a weighted summability condition on these residuals, the primal trajectory has finite length. For power schedules $\gamma_k=(k+k_0)^{-\beta}$, with $k_0\geq1$ and $1/2<\beta\leq1$, the summability condition holds when the lift exponent is sufficiently large. Consequently, the primal sequence converges to an exact limiting lifted stationary point without imposing full-row-rank assumptions on the linear operators. As a corollary, we establish whole-sequence convergence for a variable-smoothing full-splitting projected-gradient method in nonconvex nonsmooth composite optimization. Examples distinguish primal convergence from convergence of the auxiliary dual variables and illustrate the role of nonsummability in guaranteeing exact stationarity.

We study braidings and infinitesimal braidings on categories of bi(co)modules and tetramodules, and their compatibility with duoidal structures. For an arbitrary coalgebra, we classify braidings on its category of bicomodules with the cotensor product in terms of canonical $R$-forms, dualizing the classification for bimodules obtained by Agore, Caenepeel, and Militaru. Every such braiding is a symmetry and admits only the zero infinitesimal braiding. We construct duoidal structures on categories of bi(co)modules and tetramodules over bimonoids in braided monoidal categories under suitable assumptions on equalizers and coequalizers. We then introduce pre-Cartier duoidal categories and their one-sided variants. We establish obstructions to compatible braidings for tetramodules, construct nonzero one-sided pre-Cartier structures on bi(co)modules, and give a pre-Cartier duoidal example with distinct monoidal products and a nonzero infinitesimal braiding.

The Dyson series expands the propagator of a time-dependent Hamiltonian in powers of the Hamiltonian, but its truncations are in general not unitary. The Fer expansion writes the same propagator as an infinite product of matrix exponentials, each of them unitary, and its remainder decays doubly exponentially with the number of factors. Convergence, however, is guaranteed only within a finite radius: the time integral of the norm of the Hamiltonian must be smaller than $2$. This is the best value known to date. Here we improve it by about $30\%$, raising it to about $2.6058$. The result also holds for non-Hermitian Hamiltonians that are Hermitian with respect to a fixed metric.