Papers

We introduce the notion of factorizable Lie conformal bialgebras and establish their correspondence with quadratic Rota-Baxter Lie conformal algebras of nonzero weight. Consequently, on the one hand, a Lie conformal bialgebra with a Rota-Baxter operator of nonzero weight gives a factorizable Lie conformal bialgebra. And on the other hand, as the conformal analogue of the fact that a quadratic Rota-Baxter Lie algebra induces a generalized pseudo-Hessian post-Lie algebra and a special partial-pre-post-Lie algebra, a quadratic Rota-Baxter Lie conformal algebra as well as a factorizable Lie conformal bialgebra induces a generalized pseudo-Hessian post-Lie conformal algebra and a special partial-pre-post-Lie conformal algebra. Furthermore, we generalize the correspondence between Gel'fand-Dorfman bialgebras and a class of Lie conformal bialgebras to the factorizable cases. There is not only a correspondence between factorizable Gel'fand-Dorfman bialgebras and a class of factorizable Lie conformal bialgebras, but also a correspondence for their corresponding quadratic Rota-Baxter counterparts as well as some induced structures. In particular, there is a construction of factorizable Lie conformal bialgebras from Gel'fand-Dorfman bialgebras with a Rota-Baxter operator of nonzero weight.

We investigate the computation of optimal bias potentials for accelerating transitions between metastable states and for computation of equilibrium properties in molecular dynamics simulations. We formulate optimal biasing as an ergodic optimal control problem (OCP), which can be recast as a linear eigenvalue problem for the infinitesimal generator of the unbiased dynamics. We demonstrate that data-driven learning methods for the generator enable reliable solution of the OCP, computation of biasing potentials, extraction of equilibrium properties, and acceleration of state transitions. We also explore the relation of the control problem to coarse grained representations and learning of coarse grained dynamics.

We prove local well-posedness for the initial-boundary-value problem for the Korteweg–de Vries equation on the right half-line at the critical regularity $s=-3/4$, with initial and boundary data in $H^{-3/4}(\mathbb R_+)$ and $H^{1/12}(\mathbb R_+)$, respectively. This reaches the endpoint left open in the previous half-line results of Holmer (2006) and Bona, Sun, and Zhang (2006), who established local well-posedness for $s>-3/4$. The proof is based on a contraction argument in the Besov-type Bourgain space introduced by Kishimoto. The main difficulty is to construct a boundary forcing operator compatible with the endpoint $b=1/2$ and the required $\ell^1$-summability in the modulation variable. We construct such an operator using a Laplace-transform representation of the linear half-line problem and a carefully designed extension to $x<0$. The extension combines the required regularity across the boundary with prescribed vanishing spatial moments, yielding favorable low-frequency behavior of the associated kernel. A pointwise kernel estimate then provides the crucial dyadic summability needed for the $X^{-3/4,1/2,1}$ estimate. Together with a time-trace estimate for the Duhamel term at $b=1/2$, this yields the critical local well-posedness result. The ideas used here have prospects to be used in other nonlinear dispersive equations on the half-lines.

Givental-Kim and Ciocan-Fontanine gave an explicit presentation of the quantum cohomology ring of the flag variety. The author and Shirato introduced an algebraic generalization of their presentation in the context of the coordinate rings of regular nilpotent Hessenberg varieties. In particular, they connect the coordinate ring of the lower unipotent group in a general linear group and the polynomial ring with quantum parameters. In this paper we consider the dual of the connection and we see that the Plücker coordinates correspond to the quantizations of Schur polynomials. As an application of the connection, we construct an action of the symmetric group on the polynomial ring with quantum parameters. Using the symmetric group action, one can define the divided difference operators on the polynomial ring with quantum parameters. We study the quantizations of Schubert polynomials in relation to the divided difference operators.

A connected skew brace of finite Morley rank is left nilpotent (resp. weakly soluble) whenever the additive and multiplicative groups are nilpotent (resp. are soluble and the left chief length is at most 3).

We establish a sufficient condition under which overlapping PEPS boundary states satisfy the approximate factorization criterion for a spectral gap. The condition is formulated in terms of the boundary response induced by cutting bonds through the center of a rectangular region. We decompose this response into a contribution common to all four associated regions and a remainder measured in the instantaneous boundary metric. A single linear transport absorbs the common contribution and yields compatible factorization operators, with an error controlled solely by the remainder. We apply this framework to Ising PEPS and to the spin-3/2 AKLT model on the hexagonal lattice. For Ising PEPS, the required boundary-response estimate reduces to a Dobrushin-Shlosman-type condition. For the hexagonal AKLT model, a rooted expansion in paths and loops isolates the common response, while a local comparison of boundary states together with a scalar Kotecký-Preiss estimate controls the remaining terms. In both cases, the factorization error decays exponentially with the overlap width, up to a prefactor proportional to the cut length. For AKLT, we establish the corresponding physical projector estimate and obtain a uniform spectral gap. For Ising PEPS, the gap implication additionally requires compatible injective regional contractions, as specified in the paper. More generally, the method provides a systematic route from locality of PEPS boundary response to spectral gaps of two-dimensional parent Hamiltonians.

Which quantum measurement extracts the most classical information from an ensemble? We introduce the posterior algebra, a new canonical operator algebra selected by mutual information. For faithful ensembles, an affine information bound is exact precisely when this algebra is commutative; its joint spectral measurement is then optimal, and every optimal finite POVM refines it. Binary ensembles have one generator; compactness covers singular states, giving a proof of Shor's finite-dimensional binary orthogonal-measurement conjecture. The framework also gives rigidity bounds and a certified posterior-spectral receiver, validated on 408 mixed-state instances.

We settle three conjectures concerning a doubly symmetric binary source $(X,Y)$ with crossover $p$. Consider Markov chains $U - X - Y - V$ with $U,V$ binary, and let $\mathcal{A}$ be the set of rate triples $(I(U;V),I(U;X),I(Y;V))$ attainable with arbitrary binary channels $X\to U$, $Y\to V$, and $\mathcal{B}$ the subset attainable with binary symmetric channels. The averaged BSC conjecture, Conjecture 5.2 of Pichler, Piantanida and Matz (2022), asserts $\operatorname{conv}\mathcal{A}=\operatorname{conv}\mathcal{B}$. We prove this for every $p\in[0,1]$. Two conjectures of Dikshtein, Ordentlich and Shamai (2022) concern the double-sided information bottleneck at $p=0$, where $Y=X$ and the two channels see the same source: their Conjecture 1 identifies the exact maximum of $I(U;V)$ at prescribed rates $I(U;X)$ and $I(Y;V)$, and their Conjecture 2 the exact minimum. We prove both for binary $U,V$: the two extrema are attained by the same pair of Z/S-channels, in opposite orientation for the maximum and in the same orientation for the minimum. The proofs were found with substantial AI assistance, and all three theorems are formalised in Lean 4 with Mathlib, depending only on the standard axioms. The development is available at https://github.com/g-pichler/bsc-averaging . The proof of Conjecture 1 of Dikshtein, Ordentlich and Shamai (2022) contains three certified computations, a polynomial bound, an interval sweep and a polynomial positivity certificate, all of which are checked in Lean.

Rapid growth of data centers and artificial-intelligence services is producing computational loads at scales once associated mainly with largest power plants. Recent grid events show that a routine transmission disturbance can cause several gigawatts of data-center demand to disconnect or transfer to backup nearly at once. This article revisits the classical synchronization and transient-stability theory needed to reason about such events. We organize four lines of work—graph-based synchronization conditions, winding-number descriptions of nonlinear power flow, separable convex network optimization, and direct energy methods—into a single cycle-space certificate framework for the lossless fixed-voltage model. The static layer gives an exact strict-cohesion test within a prescribed winding cell and reveals the widely used Dörfler–Chertkov–Bullo test as a quadratic surrogate of the same convex problem. The dynamic layer converts the critical-energy calculation into a finite family of convex boundary problems. Standard MATPOWER benchmarks illustrate both what the stronger static test gains and where it gains nothing: the 118-bus case admits $16.2\%$ more loading than the sufficient screen, while the 39-bus case is bridge-limited and the thresholds coincide. A stylized $2.7$-GW 39-bus event further shows that transient margin can change by about a factor of two depending on where balancing power is supplied, even when every final balanced operating point remains statically feasible. The result is a tutorial synthesis and an extensible deterministic certificate for emerging gigawatt-scale computational-load contingencies.

In this paper, we are concerned with matrices formed by block-diagonal factors interleaved with fixed permutations – a flexible family of structured matrices. This class has recently drawn interest in deep learning architectures for its balanced expressivity-efficiency trade-off, yet efficient computational strategies for working with it remain to be found. We approach this problem through Riemannian geometry and examine under what conditions this class admits a smooth manifold structure. For the practically important case of orthogonal two-factor matrices, we derive the essential Riemannian tools and propose efficient algorithms for their implementation. The algorithms leverage automatic differentiation, support parameter sharing within each factor, and avoid explicit dense matrix construction. We test them within the Riemannian optimization framework on the best matrix approximation problem and for parameter-efficient fine-tuning of large language models. Beyond the two-factor setting, we study the geometric and matrix-theoretic properties of factorizations with a larger number of block-diagonal factors.

We study a one-parameter family of SO(4)-invariant pluriclosed Hermitian metrics $g_c$ with vanishing Bismut Ricci form on the affine quadric $Q_3\cong TS^3$, previously described in the physics literature. For every real parameter $c$, we prove local existence and uniqueness of a real-analytic solution to the defining singular initial-value problem, together with positivity of the associated metric near the singular orbit $S^3$. We then establish global existence and completeness for $|c|\leq1$ and finite-time degeneration for $|c|>1$. The complete family exhausts all SO(4)-invariant Bismut Hermitian-Einstein metrics on the affine quadric $Q_3$, and joins Stenzel's Kähler Ricci-flat to a metric that was first considered by Chamseddine-Volkov/Maldacena-Nuñez. We also determine the leading asymptotics and derive an explicit scalar curvature formula, proving strict positivity for the non-Kähler members and identifying the change in asymptotic scalar curvature at the endpoint $c=1$. Finally, we show that all these metrics are not Bismut flat and have full Bismut holonomy SU(3), providing new examples of such manifolds.

We prove that $S^1$, ${\rm SU}(2)$, and ${\rm SO}(3)$ are compact amalgamation bases: for each of these groups, any two compact groups with specified embedded copies of it embed in a compact group in which those copies agree. For compact Lie groups, the amalgam can be chosen to be a finite-dimensional unitary group. This research was conducted with substantial AI assistance, including in the development of mathematical constructions and proof arguments.

In an $n$-dimensional cropped hypercube each of the $2^n$ cropping inequalities chops off a single corner of the $0$–$1$ hypercube by an $\ell_1$-distance $\rho$. The case $\rho = 1/2$ has been extensively studied in the literature. This paper shows that the Lasserre rank of the $n$-dimensional cropped hypercube where $\rho = 1/2$, $n \geq 2$, is the smallest integer $0\leq t \leq n$ such that $\Delta_t < 0$ in the recurrence $\Delta_{-1} = 1$, $\Delta_{0} = n-1$, $\Delta_t = (n-1)\Delta_{t-1} - t(n-t+1)\Delta_{t-2}$. It follows that the Lasserre rank can be computed in time $O(n^2 \log^2 n)$. Asymptotically, the rank is $\frac{n}{2} + c_{1/2}\sqrt{n} + o(\sqrt{n})$, where $c_{1/2}$ is the unique zero of a given function. Numerically, $c_{1/2} \approx 0.3825$. In fact, we prove such results for any fixed $0 < \rho < 1$.

Seasonal hydropower operation requires release decisions that account for uncertain future inflows and the opportunity cost of stored water. We assess the Modelling and Optimisation of Stochastic Seasonal Hydropower Storage (MOSSHOOS) framework for the Manantali reservoir using 660 monthly energy-inflow observations from 1961–2015. The framework combines multiscale inflow representation, blocked resampling and scenario reduction with stochastic dual dynamic programming, and uses a seasonal square-root diffusion and Hamilton–Jacobi–Bellman formulation as a continuous-time interpretation of marginal water value. Scenario reduction improves the energy score by 47.6% relative to independent monthly sampling, but the marginal distribution is rejected by a Kolmogorov–Smirnov test and the lower-tail p10 error remains 103.9%, indicating inadequate dry-tail calibration. Consistent with that limitation, 50 paired Monte Carlo evaluations show no detectable cost or deficit advantage of the stochastic policy over its deterministic-equivalent counterpart at the archived operating configuration. As an independent benchmark, the repaired open-source plan4res seasonal storage valuation chain yields a dominant marginal value of 81.66~USD/MWh, within 3% of the MOSSHOOS mean of 79.25~USD/MWh. These results support a coherent stochastic inflow-to-water-value methodology while showing that hydrological tail representation, rather than economic scaling, is the principal limitation to address before operational application.

In reinforcement learning, more accurate policy evaluation has been frequently shown to improve downstream policy optimization performance. For tractability, most value estimators rely on $n$-step truncation and thereby omit the long-horizon behavior of the transition dynamics. Methods that instead capture this behavior through the global structure of the dynamics can converge faster, but their computational and memory requirements limit their scalability to large or continuous state spaces. To reconcile these limitations, we introduce Limiting-Kernel Q($\lambda$) (LKQL), an off-policy value estimator that uses the limiting kernel to approximate the long-horizon terms omitted by truncation. LKQL has the same order of complexity as $n$-step estimators and integrates directly into both on- and off-policy actor-critic algorithms. We prove that, under aperiodicity and in the near-on-policy regime, the operator underlying LKQL improves the policy evaluation convergence rate over its truncated counterpart for sufficiently large $n$, and that LKQL itself converges almost surely to the optimal values in finite Markov decision processes (MDPs) under a fixed behavior policy. On the MuJoCo continuous-control benchmark, we show that LKQL improves over $n$-step baselines in most settings, particularly on long-horizon tasks.

We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at $q=\varepsilon$, where $\varepsilon$ is a primitive $N$th root of unity with $N$ odd. For the commuting layer transfer matrices $\mathbb{T}(y)$ with mixed boundary conditions, we establish the quantum Frobenius property \[ \mathbb{T}(y)\mathbb{T}(\varepsilon y)\cdots \mathbb{T}(\varepsilon^{N-1}y) = \mathscr{P}_N(y^N)\mathbb{I}, \] where $\mathscr{P}_N$ is a scalar Laurent polynomial obtained directly from the monomial expansion of $\mathbb{T}(y)$. Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting $\tau_{\; 2}$ model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.

We study lattice-counting functions on compact integral-integral affine manifolds with corners, which generalize integral unimodular polytopes. We prove analogues of Ehrhart's theorem and Ehrhart-Macdonald reciprocity in this setting.

Let $G$ be a finite group admitting a rational fixed-point-free (FPF) representation $V$, with a $G$-invariant full-rank lattice $L\subset V$. We study the induced $G$-action on the complex torus $V_{\mathbb{C}}/L$ and the quotient orbifold $V_{\mathbb{C}}/(G\ltimes L)$. We prove that every FPF group admits a unique irreducible rational FPF representation. We derive closed-form formulas for three core geometric invariants: the number of non-free points of the torus action, the number of singular points of the quotient orbifold, and its orbifold Euler characteristic.

A subset $S \subseteq \mathbb{F}_2^n$ is called a Sidon set if no four distinct points of $S$ have zero sum. It is shown that if $n \geq 7$ and $n \equiv 3 \mod 4$, then $|S|\leq 2^{\frac{n+1}{2}}-3$. As a consequence, for any even $t \geq 4$, there does not exist a binary linear $[2^t-3,2^t-2t-2,5]$-code, strengthening a nonexistence result of Brouwer and Tolhuizen from 1993. As a second consequence, we also show that for an even integer $n \geq 4$, every almost perfect nonlinear (APN) function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ has nonlinearity at least $3$.

Let $G$ be the group $\mathbb{R}_+\ltimes \mathbb{R}^n$ endowed with Riemannian symmetric space metric $d$ and the right Haar measure $\mathrm{d} \rho$ which is of $ax+b$ type, and $L$ be the positive definite distinguished left-invariant Laplacian on $G$. Let $u=u(t,\cdot)$ be the solution to $u_{tt}+Lu=0$ with initial conditions $u|_{t=0}=f$ and $u_t|_{t=0}=g$. In this article we show that for a fixed $t \in\mathbb{R}\setminus\{0\}$, \begin{align*} \|u(t,\cdot)\|_{L^{1,\infty}(G,\mathrm{d}\rho)}\leq C\ (1+|t|)\big(\|(\operatorname{Id}+L)^{\alpha_0/2}f\|_{L^1(G,\mathrm{d}\rho)} +\|(\operatorname{Id}+L)^{\alpha_1/2}g\|_{L^1(G,\mathrm{d}\rho)}\big) \end{align*} if and only if $\alpha_0\geq {n/2}$ and $\alpha_1\geq n/2-1$. Moreover, the polynomial-in-time growth $1+|t|$ in the above estimate is best possible.

We prove zero density estimates for $L$-functions of cuspidal automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$. We show that $N_\pi(\sigma, T) \ll T^{\frac{5}{2}(1 - \sigma) + o(1)}$, where $N_\pi(\sigma, T)$ denotes the number of zeros $\rho = \beta + i\gamma$ of $L(s,\pi)$ with $\beta \geq \sigma$ and $|\gamma| \leq T$. The key input is an extension of the Guth$\unicode{x2013}$Maynard argument to Dirichlet polynomials whose coefficients satisfy a weaker $\ell^q$ bound.

We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every $\delta>1/2$ there is a constant $C_M(\delta)$, independent of the group $G$, such that $\|\mu\|_{M(G)}\le1$ and $\inf_{\gamma\in\widehat G}|\widehat\mu(\gamma)|\ge\delta$ imply that $\mu$ is invertible in $M(G)$ and $\|\mu^{-1}\|_{M(G)}\le C_M(\delta)$. Together with Nikolski's negative results, this shows that $1/2$ is the sharp universal threshold. The main step is a uniform inversion theorem for Fourier algebras $A(K)$ of compact Abelian groups; its proof is based on a profile decomposition and a resolvent argument. We also obtain norm-controlled inversion for Toeplitz operators with non-vanishing Wiener symbols of winding number zero and give direct endpoint constructions showing failure of norm control at $\delta=1/2$ in $A(K)$ for every infinite compact Abelian group $K$, and in the unitization of $L^1(G)$ for every nondiscrete locally compact Abelian group $G$.

In this paper, we introduce the class of weakly localized operators on the weighted Bergman spaces induced by two-sided doubling weights. We first establish the characterization of the compactness of the operator belonging to the norm closure of weakly localized operators in terms of the boundary vanishing of the Berezin transform. Our approach here is different from the methods in the literature since the “translation" operators are not served us a direct applicable tool in our setting. We further characterize the compactness of Toeplitz operators with bounded symbols in the two-weight setting, a boundedness criterion of a two-weight Bergman projection is also presented.

We establish a Donsker-Prokhorov invariance principle in some Besov spaces. Specifically, we show that polygonal line processes associated with partial sum processes of a triangular array of row-wise independent random variables converge in distribution to Brownian motion, extending earlier results in the literature.

We introduce an explicit deterministic collection of $N$ spherical points, $\cP_N\subset\mathbb S^2$, that we call the deterministic Diamond points. For every $0<\alpha<2$ we prove that there exists a constant $C_\alpha>0$ such that \[ 0\leq \frac{2^{\alpha+1}}{\alpha+2}N^2-\sum_{x,y\in\cP_N}|x-y|^\alpha \le C_\alpha N^{1-\alpha/2}. \] This result shows that $\cP_N\subset\mathbb S^2$ has a Riesz energy deficit of optimal order in this range of the parameter. In particular, for $\alpha=1$, Stolarsky's invariance principle implies that $\cP_N\subset\mathbb S^2$ has $L^2$ spherical cap discrepancy of asymptotically optimal order \[ D_{L^2}^C(\mathcal P_N)\asymp N^{-3/4}, \] making it, to our knowledge, the first explicit configuration proved to have this property. More generally, our result implies that the point set $\cP_N$ has Sobolev $H^{s}(\Sph)$ worst-case error of optimal order for all $1<s<2$.

We study a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions whose coefficients are allowed to depend on time, and we construct conservative discretizations in time for the associated initial–boundary value problem. We consider two symmetric three-layer schemes of Crank–Nicolson type, a locally linear one and a genuinely nonlinear one, and we show that each of them preserves a discrete analogue of the total mechanical energy of the homogeneous problem with constant coefficients. For the nonlinear scheme we establish uniform apriori bounds on the discrete solution and on its discrete velocity by working directly with the discrete energies, without invoking a nonlinear discrete Grönwall inequality, the constants still grow exponentially in the final time, and we prove local second-order convergence in time, both for the solution and for the central-difference approximation of its first time derivative. The nonlinear system arising at each time level is solved by a fixed-point iteration: given iterates at the two preceding levels that satisfy the apriori bounds, that equation has exactly one solution and the iteration converges to it at a geometric rate once the time step is small enough. Since the apriori bounds are index-local, alternating them with that one-step solver constructs the trajectory stepwise, for time-dependent coefficients as well, on the local interval on which those bounds hold. Numerical experiments, carried out in a setting in which the spatial discretization contributes no error, exhibit the conservation of the discrete invariants, confirm the second order in time and verify the geometric convergence of the fixed-point iteration.

Using the idea of Venkataramana's construction of infinite families of arithmetic orthogonal hypergeometric groups, we construct some of the very first examples of infinite families of arithmetic symplectic hypergeometric groups that do NOT satisfy the arithmeticity criterion of Singh and Venkataramana. For example, we show that the hypergeometric groups associated to the pairs of polynomials $(x-1)^4P_m(x^{9})$ and $(x^4+x^3+2x^2+x+1)Q_m(x^{9})$, where $P_m$ and $Q_m$ are integral polynomials of degree $2m$ such that the corresponding pair determines a Zariski dense symplectic hypergeometric group, are arithmetic in $\operatorname{Sp}(18m+4)$ for any integer $m\in\mathbb{N}$.

In this paper, we prove that every connected sesqui-regular graph with parameters $(n,k,c)$, where $c\geq3$, and with smallest eigenvalue in $[-3,-2)$ is $1$-integrable if its valency is sufficiently large. The bound on $c$ is optimal: for every integer $t\geq2$, the Cartesian product of the Shrikhande graph and $K_t$ is a connected sesqui-regular graph with parameters $(16t,t+5,2)$ and smallest eigenvalue $-3$, but it is not $1$-integrable.

In this paper, we prove that on a smooth complex projective curve of positive genus, the Deligne–Hitchin twistor spaces for the structure groups $\mathrm{GL}(n,\mathbb{C})\ (n\geq1)$ and $\mathrm{SL}(n,\mathbb{C})\ (n\geq2)$ admit no algebraization by a complex scheme locally of finite type. When the genus is at least two, the same holds for their stable loci and hence for the corresponding Hitchin twistor spaces. Nevertheless, the Deligne–Hitchin twistor spaces admit $\mathbb{R}_{\mathrm{alg}}$-definable complex analytic structures when the rank or the genus is one.

Let $0 \xrightarrow{}I \xrightarrow{ } A \xrightarrow{ }A/I \xrightarrow{} 0$ be a short exact sequence of C*-algebras. In this paper, we prove that if $\dim({\rm Cu}(I))\leq n$ and $\dim({\rm Cu}(A/I))\leq m$, then $\dim({\rm Cu}(A))\leq n+m+1$, this result gives an affirmative answer to a problem concerning the Cu-semigroup of C*-algebras raised by Thiel and Vilalta. We further show that certain comparison and divisibility properties satisfied by $I$ and $A/I$ are inherited by the extension algebra $A$. As an application, we provide an alternative proof of equivalence that $A$ is pure if and only if $I$ and $A/I$, a result originally proven by Perera, Thiel, and Vilalta.