Papers

In this article, we characterize the holomorphic mappings from $B_{\ell_p^n}\times\mathbb{D}^{m}$ into $\mathbb{D}^{m}$ for $p\in \{2,\infty\}$. In addition, we give a simple proof for the boundary Schwarz lemma for vector valued holomorphic functions, which also extends the existing result. Also, we obtain the boundary Schwarz lemma for pluriharmonic self-mappings of the unit ball $B_{\ell_p^n}$, $p \in [2,\infty]$. Furthermore, we establish the boundary rigidity theorem for holomorphic self-mappings of $B_{\ell_p^n}$, $p \in (1,\infty)$.

In this note, we show that the space of all $\varepsilon$-nets in the real line $\mathbb{R}$ with a natural metric, equipped with either Hausdorff or Gromov–Hausdorff distance, is contractible.

We prove that a connected mean convex region in $\mathbb{R}^{n+1}$ with at least two boundary components cannot have strictly positive mean curvature. This answers a question of Gromov. We also obtain estimates for how quickly the mean curvature must decay at infinity, and generalize this result to hyperbolic space.

In this paper, we study a consensus-based optimization method for nonconvex bi-level optimization, where the objective is to minimize an upper-level function over the set of global minimizers of a lower-level problem. The proposed approach is derivative-free, and constructs its consensus point via smooth quantile selection combined with a Gibbs-type Laplace approximation. We establish convergence guarantees for both the associated mean-field dynamics and its finite-particle approximation. In particular, under suitable assumptions on smooth quantile localization, error bounds, and stability, we show that the mean-field law reaches any arbitrary prescribed Wasserstein neighborhood of the target bi-level solution with an explicit exponential rate up to the hitting time. Numerical experiments on a two-dimensional constrained problem and neural network training further support the theoretical results.

We study the problem of estimating a monotone function $f:\{0,1\}^d\to[0,1]$ from noisy observations at uniformly random vertices of the Boolean hypercube. As a measure of complexity for the target~$f$, we use the total $L^1$-influence $I(f)=\sum\_{i=1}^d(\E[f(X)\mid X\_i=1]-\E[f(X)\mid X\_i=0])$, a classical quantity in Boolean analysis that is nonnegative for monotone functions and controls the effective dimensionality of the estimation problem: through a spectral concentration result in the spirit of Friedgut's junta theorem, the Fourier spectrum of any $f$ with $I(f)\leqslant K$ concentrates on low-degree subsets of the influential coordinates. We establish minimax bounds over the class $\cF\_K=\{f:\{0,1\}^d\to[0,1],\;f\text{ monotone},\; I(f)\leqslant K\}$: \[ c\,\frac{K^2}{(\log n)^{3/2}} \;\leqslant\; \inf\_{\hat f}\;\sup\_{f\in\cF\_K}\; \E\bigl[\|\hat f - f\|\_2^2\bigr] \;\leqslant\; C\,\frac{K}{\sqrt{\log n}}, \] where $n$ is the sample size. The upper bound holds for all $K\geqslant 1$ and is uniform in the ambient dimension~$d$ (under the mild condition $\log d\leqslant n^{1-\varepsilon}$). It is achieved by a Fourier thresholding estimator that adapts to the unknown~$K$. The lower bound relies on a Varshamov–Gilbert packing on the middle layer of the hypercube combined with Fano's inequality.

We establish rapid mixing for Davies Markov semigroups associated with 2D Abelian quantum double models at any positive temperature. A condition of Dobrushin-Shlosman (DS) type holds at any temperature, and we show that the latter implies a modified logarithmic Sobolev inequality for the Davies Lindbladian. A key step in the argument is to verify a strong martingale condition for the local conditional expectations of the Davies semigroup in the regime of validity of the DS condition.

Recently the matrix $A_2$ conjecture was disproved. Indeed, the growth of the vector Hilbert transform in the matrix weighted $L^2(W)$ space was shown to be at best a constant multiple of $[W]_{\mathbf{A}_2}^{3/2}$. This bound had previously been established and it was thus proved that it is sharp and the conjectured linear growth cannot be obtained. It is a natural question to see if the $3/2$ power persists if we replace the classical matrix $A_2$ characteristic by the "fattened", larger, so-called matrix Poisson $A_2$ characteristic. We show that the 3/2 power, even in this case, cannot be improved.

We study the effect of quantum Frobenius twist on Ext-groups in the category of quantum polynomial, and prove that the existence of type of complexes, called quantum Troesch complexes, enables the construction of a spectral sequence computing the Ext groups of twisted functors from the knowledge of Ext-groups of the original functors. We then construct quantum Troesch complexes in the special case where the parameters of the quantum deformation is a root of unity of order 3.

In this paper, we establish Berry-Esseen-type bounds for federated linear stochastic approximation (LSA). Our results provide the first federated Gaussian approximations for LSA that explicitly capture communication-computation trade-offs and heterogeneity-aware error terms, quantifying the effects of local step size, number of local updates, and heterogeneity on convergence rates. We present results for both (i) constant step size regime and (ii) decreasing step size with an increasing number of local iterations, recovering the recent rates of Bonnerjee et al. [2025] as a special case. As a primary application of our results, we develop an online multiplier bootstrap procedure for inference on the last iterate, which avoids explicit estimation of the asymptotic covariance matrix, and obtain non-asymptotic validity guarantees for this procedure.

Deep generative models have emerged as state-of-the-art for solving inverse problems, but applying them to inverse problems for PDEs, like electrical impedance tomography (EIT) remains challenging. Because physical domains are naturally discretized as unstructured meshes rather than regular grids, standard convolutional architectures are often inadequate. In this paper, we propose a novel framework that extends diffusion posterior sampling (DPS) to graph-structured data. We develop an unconditional score-based diffusion model directly on a 2D triangular mesh to learn an accurate prior over the physical solution space. Furthermore, we introduce a regularized variant, RDPS, which incorporates explicit regularization terms, such as total variation and generalized Tikhonov, to complement the implicit diffusion prior and mitigate severe ill-posedness. Extensive experiments on synthetic and real 2D EIT datasets demonstrate that RDPS produces stable, physically plausible reconstructions. Our approach generalizes well to out-of-distribution inclusion geometries, is highly robust to measurement noise, and outperforms current state-of-the-art solvers (e.g., GPnP-BM3D, DP-SGS) in reconstruction accuracy and artifact reduction.

Matrix-structured parameters frequently appear in many artificial intelligence models such as large language models. More recently, an efficient Muon optimizer is designed for matrix parameters of large-scale models, and shows markedly faster convergence than the vector-wise algorithms. Although some works have begun to study convergence properties (i.e., optimization error) of the Muon optimizer, its generalization properties (i.e., generalization error) is still not established. Thus, in this paper, we study generalization error of the Muon optimizer based on algorithmic stability and mathematical induction, and prove that the Muon has a generalization error of $O\big(\frac{1}{N\kappa^{T}}\big)$, where $N$ is training sample size, and $T$ denotes iteration number, and $\kappa>0$ denotes minimum difference between singular values of gradient estimate. To enhance generalization of the Muon, we propose an effective mixed Muon (MiMuon) optimizer by cautiously using orthogonalization of gradient, which is a hybrid of Muon and momentum-based SGD optimizers. Then we prove that our MiMuon optimizer has a lower generalization error of $O\big(\frac{1}{N}\big)$ than $O\big(\frac{1}{N\kappa^{T}}\big)$ of Muon optimizer, since $\kappa$ generally is very small. Meanwhile, we also studied the convergence properties of our MiMuon algorithm, and prove that our MiMuon algorithm has the same convergence rate of $O(\frac{1}{T^{1/4}})$ as the Muon algorithm. Some numerical experimental results on training large models including Qwen3-0.6B and YOLO26m demonstrate efficiency of the MiMuon optimizer.

We give both a sufficient condition for and an obstruction to the derived category of a commutative ring being generated by its residue fields. As an illustration, we exhibit a ring for which Foxby's small support classifies localizing subcategories despite the failure of the local-to-global principle. We also conclude that the residue fields do not generate for a polynomial ring in infinitely many variables. Finally, we give a necessary and sufficient criterion for the derived category to satisfy the local-to-global principle; it turns out to depend solely on the topology of the spectrum.

We generalise Hinich's Theorem of descent of Deligne groupoids to the case where the dgLas involved have no negative cohomology. We apply this result to study the infinitesimal deformations of a morphism $\alpha: {\mathcal F} \to {\mathcal G}$ of coherent sheaves, where both the sheaves $ {\mathcal F}$ and $ {\mathcal G}$ and the map $\alpha$ can be deformed, on a smooth variety over a field of characteristic zero. In particular, we provide an explicit dgLa that controls these deformations via the Deligne functor, applying the Thom-Whitney totalisation to a specific semicosimplicial dgLa, constructed from geometrical data.

In this paper, we investigate the inverse logarithmic coefficients associated with the class $\mathcal{C}_e$ of analytic and univalent functions satisfying the subordination condition \[ 1+\frac{z f''(z)}{f'(z)} \prec e^z, \quad z\in\mathbb{D}. \] If $F_{f^{-1}}(w) = \log\!\left(\frac{f^{-1}(w)}{w}\right) = 2\sum_{n=1}^{\infty}\Gamma_n w^n$ denotes the logarithmic expansion corresponding to the inverse function $f^{-1}$, then we establish sharp estimates for the initial inverse logarithmic coefficients and prove that \[ |\Gamma_n| \le \frac{1}{2n(n+1)}, \qquad n=1,2,3. \] We further derive the sharp coefficient-difference inequality \[ -\frac{1}{2\sqrt7} \le |\Gamma_2|-|\Gamma_1| \le \frac1{12}, \] and obtain the sharp bound for the second-order Hankel determinant associated with the inverse logarithmic coefficients: \[ \left| H_{2,1}\!\left(F_{f^{-1}}/2\right) \right| \le \frac{85}{12096}. \] Additionally, we evaluate the sharp lower and upper bounds of the generalized Fekete–Szegö functional $F_{\lambda, \mu}(f) = \big| a_3(f) - \lambda a_2(f)^2 \big| - \mu |a_2(f)|$ within this setting and establish relationships associated with the starlike class $\mathcal{S}^{\ast}_{\rho}$. The extremal functions corresponding to all obtained estimates are explicitly constructed, thereby showing the sharpness of the results.

Open-source libraries are have a catalytic role in research pipelines, where new methods must be compared against up-to-date baselines. We present the GLobal Optimization Benchmark (GLOBe) modular Python library that unifies classical and recent continuous global optimization algorithms, including decision-based and mathematically founded particle-based methods, in a single framework. A central contribution of GLOBe is its modular architecture, which factors common algorithmic patterns into reusable familyl evel components and enables plugins to be implemented once and makes them available to all algorithms in the corresponding family. This modular design leverages recent advances in mathematical formalization of global optimization, where structural commonalities across algorithms have been identified and used to develop broadly applicable, formally grounded algorithmic features. The C++ backend relies on Eigen 5 for efficient linear algebra. GLOBe presently includes 14 optimizers, 19 analytical benchmarks along with a random function generator, and an integrated toolkit for direct algorithm comparison.

Absence of local conserved quantities, or nonintegrability, is often assumed when discussing various phenomena in quantum many-body systems, such as thermalization and transport. However, no concrete proof of this property is known in electron–phonon coupled systems, a typical setting for condensed matter physics. In this paper, we show that the one-dimensional Holstein model has no nontrivial local conserved quantities other than the Hamiltonian itself and the total fermion number operator. We further show that the absence of nontrivial local conserved quantities also holds for the more general Holstein–Hubbard model. We also establish the absence of nontrivial local conserved quantities in these models on higher-dimensional hypercubic lattices. Our result has accomplished an advance in nonintegrability proofs by expanding their scope to systems in which particles with different statistical properties are mixed.

In this paper, the first Chen inequality is proved for CR-warped product submanifolds in complex space forms. This inequality involves intrinsic invariants (a leaf-wise $\delta$-invariant and the sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer to Problem [1]. We carefully distinguish the leaf-wise $\delta$-invariant of a factor (used in the bound) from the intrinsic Chen invariant of the same factor, the two being related, on the totally real factor, by the Bishop–O'Neill formula. The bound is sharp and is uniform in the sign of the holomorphic sectional curvature $c$. As a geometric application, we derive necessary conditions for the immersed CR-warped product submanifold to be minimal in a complex space form, providing a partial answer to a well-known problem proposed by S.S. Chern (Problem [2]). For further research directions, we address a couple of open problems (Problem [3]} and Problem [4]).

In this study, we firstly establish the well-posedness of a degenerate parabolic equation under Dirichlet boundary conditions. Following this, we introduce a shape design problem, which acts as a framework for approximating the degenerate parabolic equation through a series of uniformly parabolic equations. Finally, as a tangible application of this shape design approach, we deduce a boundary observability inequality associated with the degenerate parabolic equation.

We study wrinkling patterns in a thin elastic annulus subjected to radial stretching within the framework of the Föppl–von Kármán theory. Building on the analysis of the Lamé problem in Bella and Kohn, we investigate the asymptotic regime $h\to0$ and establish a $\Gamma$-convergence result for suitably rescaled energies after subtraction of the relaxed membrane energy. The limiting functional is a scalar convex measure-valued energy coupled with a constraint on the marginal of the limiting measure, describing the distribution of wrinkle frequencies. We also prove existence and qualitative properties of minimizers of the limiting functional.

Skew partial difference sets (skew PDSs) are recently-introduced combinatorial objects closely related to partial difference sets (PDSs). To date, only one construction approach for non-trivial skew PDSs is known, using bent partitions: this produces examples of Latin square type. In this paper we show that these examples are not an isolated phenomenon; we present new constructions for families of skew PDSs using cyclotomy in finite fields. We provide the first constructions for skew PDSs of Paley type, and new constructions for Latin square type (with different parameters to those from bent partitions). Moreover, we show how skew PDSs relate to disjoint and external partial difference families (DPDFs/EPDFs), and provide new cyclotomic constructions of both standard and relative DPDFs and EPDFs.

Unconditional polytopes are convex polytopes that are symmetric with respect to all coordinate hyperplanes and arise naturally from anti-blocking polytopes by reflection. This paper investigates algebraic relations between an anti-blocking lattice polytope and its associated unconditional lattice polytope. We prove that the toric ring of an anti-blocking lattice polytope is normal if and only if the toric ring of the associated unconditional lattice polytope is normal. We also show that the toric ideal of an anti-blocking lattice polytope is generated by quadratic binomials if and only if the same holds for the associated unconditional lattice polytope. As an application, we obtain a graph-theoretic characterization of quadratic generation of symmetric stable set ideals.

We prove that the inhomogeneous variant of Khintchine's Theorem holds in dimension $2$ without any monotonicity assumption. This resolves the last remaining case in the metric theory of inhomogeneous Diophantine approximation: while the monotonicity assumption is known to be unnecessary in dimensions $m\geq 3$ and necessary in dimension $m=1$, the two-dimensional case has remained open. It also settles the final outstanding case of a Khintchine–Groshev-type theorem for the approximation of systems of linear forms, confirming a conjecture of the first and third authors. Our results bring the inhomogeneous theory of metric Diophantine approximation into alignment with its homogeneous counterpart.

We introduce a flow that is designed to flow maps $u:\Sigma\to \mathbb{R}^n$ which map the boundary of a general domain surface $\Sigma$ into a given (not necessarily connected) submanifold $N\hookrightarrow \mathbb{R}^n$ towards a free boundary (branched) minimal immersion supported by $N$. In the case when $\Sigma$ is the unit disc $D$, this task can be achieved by means of the Plateau-flow introduced in the work [15] of the second author. When $\Sigma\neq D$, however, also the conformal type of the domain metric plays a role and it no longer suffices to deform the trace of the given map into a half-harmonic map as in [15]. In order to overcome this issue, here we combine ideas of the Plateau-flow from [15] with ideas of the Teichmüller harmonic flow from [12], in order to flow both an initial map $u_0$ with trace $u_0\colon\partial \Sigma\to N$ and an initial domain metric $g_0$ in a way that produces, as time tends to infinity, a half-harmonic map from $\partial \Sigma$ into $N$ whose harmonic extension is conformal and hence is a (branched) minimal immersion.

We study the soft covering phenomenon through the lens of Neyman–Pearson hypothesis testing: given a channel output sequence $y^n$, can one decide whether it was produced when the channel was driven by a random codeword, or generated independently from the output marginal? We derive exact exponential decay rates for the jointly averaged false-alarm (FA) probability $\alpha_n(\tau,R)$ and missed-detection (MD) probability $\beta_n(\tau,R)$, as functions of the decision threshold $\tau$ and the codebook rate $R$. The derived single-letter formulas of the exponents $\EFA(\tau,R)=-\lim_{n\to\infty}\frac{1}{n}\ln\alpha_n(\tau,R)$ and $\EMD(\tau,R)=-\lim_{n\to\infty}\frac{1}{n}\ln\beta_n(\tau,R)$ are tight in the random coding sense. The analysis reveals a rich phase structure. For $R < I(X;Y)$, there is a genuine exponential tradeoff between the two error types over the interval $\tau \in (0, I(X;Y)-R)$. At $R = I(X;Y)$, this tradeoff interval collapses to the single point $\tau = 0$, where both error exponents simultaneously vanish, a fact which manifests the soft covering phenomenon in the Neyman–Pearson sense. For $R > I(X;Y)$, the same instantaneous collapse persists at $\tau = 0$; moreover, for every $\tau$ at least one exponent is zero: the FA exponent is zero for $\tau \le 0$ (FA probability does not decay exponentially), and the MD exponent is zero for $\tau \ge 0$ (and finite, channel-specific for $\tau<0$; see Remark~\ref{rem:jump}). There is no interval of $\tau$ where both exponents are simultaneously positive. A sharp phase transition in the MD exponent occurs at $\tau^* = [I(X;Y)-R]_+$ for all rates.

A smallish monoid M is a monoid that has a unique 0-minimal ideal I(M) that is a 0-simple subsemigroup and such that its regular J -classes are the group of units and the two in I(M). We show constructively how to embed an arbitrary finite semigroup S into the evaluation semigroup of a smallish monoid S^{Ev} . We use the theory of flows to show that a group mapping semigroup S admits an aperiodic flow if and only if S^{Ev} admits one. This reduces the computation of Krohn-Rhodes complexity 1 to the class of smallish monoids.

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density estimate for Dirichlet $L$-functions, we may unconditionally take $U= N^{\frac{4}{49}}\exp(\log^{\frac{2}{3}+\varepsilon}N)$ for any $\varepsilon>0$. If we assume the Generalized Riemann Hypothesis instead, we may take $U= \log^{4+\varepsilon}N$.

The rise of Physics-Informed Neural Networks (PINNs) has popularized the concept of solving differential equations via residual minimization. However, neural networks are often viewed as “black boxes" requiring significant computational overhead and stochastic optimization. Moreover, PINNs typically treat boundary conditions (BCs) as “soft constraints" within the loss function and this makes the optimization process struggling to enforce the BCs properly. This paper introduces the Spline-based Physics-Informed Numerical Scheme (SPINS), a numerical framework designed to solve both initial and boundary value problems of ordinary differential equations (ODEs). By replacing the neural network architecture of traditional PINNs with a structured spline basis, SPINS achieves high accuracy and interpretability with a minimal parameter set. In addition, the BCs are automatically satisfied from the choice of the splines architecture. Therefore, SPINS provides smooth numerical solutions for ODEs allowing analytical differentiation. Moreover, SPINS benefits from the automatic differentiation where computing the gradient of the physics-informed loss function is an easy task making the optimization process very fast using gradient-based optimizers such as the L-BFGS-B algorithm. We demonstrate the efficacy of SPINS on nonlinear second order ODEs with several choices of BCs using cubic and quintic interpolating splines and present its natural extension to high order ODEs.

This paper presents a learning-augmented trajectory planning framework for cooperative unmanned aerial vehicle (UAV) and unmanned ground vehicle (UGV) handover missions. While centralized trajectory optimization ensures dynamic feasibility and task optimality, its high computational cost limits real-time applicability. We propose a neural surrogate planner utilizing decoupled encoder-decoder long short-term memory (LSTM) networks to generate coordinated handover trajectory predictions from the task specifications. These predictions serve as informed warm starts for the downstream centralized optimizer, thereby accelerating convergence to dynamically feasible solutions. Benchmark evaluations demonstrate that the learning-augmented planning framework achieves more than a threefold speedup and 100% optimization success rate compared to cold start optimization. The results indicate that combining data-driven inference with model-based refinement enables fast and reliable trajectory generation for heterogeneous multi-robot systems.

Exploiting newly available data on Gopakumar-Vafa invariants at high genus for one-parameter hypergeometric Calabi-Yau threefolds, we study numerically the growth of the 5D indices, stable pair (PT) invariants and rank one Donaldson-Thomas (DT) invariants at large charges. For the 5D index $\Omega_{5D}(d,m)$, below a critical value of the angular momentum $m$, we find perfect agreement with the Bekenstein-Hawking-Wald entropy of rotating 5D BMPV black holes, including the subleading correction from four-derivative interactions. When $m$ exceeds the critical value, the 5D index is instead dominated by black rings with the smallest possible dipole charge. The stable pair invariant $PT(d,m)$, which is determined by 5D indices, has a similar black ring/hole transition at negative $m$ (now interpreted as the D0-brane charge) but surprisingly exhibits two other phase transitions at positive $m$: first, to a plateau and then to a polynomial growth $\sim m^{2d-1}$. In each phase, we derive an approximate expression for the invariant. Finally, the rank one DT invariant $DT(d,m)$ is similar to $PT(d,m)$ at negative $m$, and then transitions to a phase dominated by D0-branes, with entropy of order $m^{2/3}$. Along the way, we determine the fixed genus, large degree behavior of GV invariants (including the overall $g$-dependent constant), extend it to an approximate formula valid also for large $g$, point out the unreasonable effectiveness of a simple PT/MSW relation, and study the growth of topological free energies at fixed degree, confirming a conjecture of Mariño.

Quantum graph homomorphisms, introduced by Mančinska and Roberson, form a natural quantum relaxation of classical graph homomorphisms. Since this relaxation may create new comparabilities, it could in principle collapse antichains and other order-theoretic configurations. We prove that this does not happen: the quantum homomorphism quasi-order of finite directed graphs is countably universal, and the quantum homomorphism quasi-order of finite planar graphs of maximum degree at most $7$ is also countably universal. Consequently, the same universality holds for finite undirected graphs and for the corresponding quotient partial orders. The result is constructive for finite patterns. Given any finite poset $P$, we explicitly construct finite planar graphs $G_p$, $p\in P$, with $\Delta(G_p)\le 7$, such that \[ p\le_P q \quad\Longleftrightarrow\quad G_p\toq G_q. \] For directed graphs, the proof uses disjoint unions of clockwise directed cycles, where quantum and classical homomorphisms coincide. For undirected graphs, the main ingredient is a finite ordered indicator whose terminals are quantum endpoint-forcing. This gives a quantum analogue of the classical ordered-indicator method: the classical endpoint-image condition is replaced by the projection-level vanishing condition \[ F_{a,p}F_{b,q}=0 \] for every illegal ordered terminal pair. The fixed indicator encodes directed-cycle constructions inside planar bounded-degree graphs without creating extra quantum homomorphisms.