数学前沿

We investigate optimal control problems whose state equation takes the form $\mathcal{A}x-\mathcal{B}u=b$, where $\mathcal{A}:\mathcal{X}\to\mathcal{P}'$ is an isomorphism from a Banach state space $\mathcal{X}$ onto the dual of a reflexive Banach space $\mathcal{P}$. The adjoint equation involves the Banach space adjoint $\mathcal{A}':\mathcal{P}\to\mathcal{X}'$, so that, in general, state and multiplier are of different regularity. For particular classes of partial differential equations, the typically weaker regularity of the multiplier is improved a posteriori by bootstrapping. In this work, we provide an abstract foundation for this methodology on the operator level. More precisely, we show that whenever the state cost extends to a differentiable function on a Hilbert space $\mathcal{H}$ in which $\mathcal{X}$ is densely embedded, as for $L^2$-tracking costs on Sobolev-type state spaces, the multiplier automatically lies in a subspace of $\mathcal{P}$, the dual companion, on which $\mathcal{A}'$ restricts to an isomorphism onto $\mathcal{H}'$. This yields an optimality system with balanced regularity, where state and multiplier solve equations of the same type. In applications, the dual companion can be identified by integration by parts in time or space. We illustrate the framework by linear elliptic and parabolic equations, Cauchy problems governed by general $C_0$-semigroups, and a wave equation with accumulated-state observation.

The generalized Lyapunov equation has a natural splitting that leads to the standard fixed-point iteration (sFPI) for its numerical solution. sFPI is convergent, for an arbitrarily given initial guess, if and only if the spectral radius of the associated linear operator is smaller than $1$. This means that sFPI may diverge if the spectral radius is $1$ or bigger. In this paper, we propose a compensated splitting scheme that aims to reduce the spectral radius so that the resulting compensated fixed-point iteration (cFPI) has a better convergence property, namely, cFPI may still converge even if sFPI does not or cFPI converges faster than sFPI does when the latter is also convergent. Numerical results are presented to demonstrate the superiority of cFPI to sFPI.

We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize risk stability through the \emph{critical learning rate}, defined as the largest learning rate for stable training, and obtain $\eta_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $\eta_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-\beta)\}$, and $\eta_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^q(1-\beta)\}$, where $B$ is the batch size, $\beta$ is the momentum factor, and $q>1$ is the capacity exponent. Within this admissible region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient, through power-law decay, to a noise floor. We then minimize the final-step risk over the admissible learning rates and momentum factors under a fixed data budget, yielding a three-regime batch-size phase diagram that reveals how the role of momentum changes with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the best small-batch data-scaling exponent, thereby enabling greater parallelism without sacrificing data efficiency. In contrast, Nesterov achieves better data efficiency in the large-batch regime because its look-ahead mechanism suppresses noise accumulation. Numerical experiments validate the predicted stability boundaries, risk dynamics, and batch-size phase diagram.

This study aims to construct a stochastic process called ``diffusion house-moving,'' which is a diffusion bridge conditioned to stay between its starting point and its ending point. To construct this process, statements are prepared on the weak convergence of conditioned diffusion processes, and conditioned diffusion bridges. Moreover, the sample path properties of diffusion house-moving are studied as well.

We show that independent random points support exact quadrature rules with positive weights. For any $n$-dimensional space $V$ of integrable functions a constant multiple of $n\log n$ points sampled from a suitable density yields, with high probability, positive weights which integrate the functions in the space exactly. The weights stay close to the natural importance sampling weights according to the Lewis density with respect to $V$. They are the result of a weight adaptation procedure which requires an explicit $L_1$-norm discretization result that is proved on the way. As an application, we study numerical integration in reproducing kernel Hilbert spaces. Combining the random rules with a subsampling step, we show that positive quadrature rules with constantly oversampled nodes achieve the best possible worst-case error bound in terms of the singular numbers of the embedding into $L_2$.

We show that random tuples of matrices over a finite field are uniform dimension expanders for every expansion coefficient below the optimal one, using elementary counting arguments. The same holds for the optimal coefficient along a set of dimensions of density 1. By contrast, along Fibonacci dimensions random pairs of matrices fail to be optimal dimension expanders with probability bounded away from zero.

We investigate conditions under which a semidirect product $G=HN$, and in particular a restricted wreath product, is an equational domain. For a variety of groups $\X$, we study semidirect products of $\XT$-groups (groups where $\langle A, B\rangle \in\X$ for any two $\X$-subgroups $A$ and $B$ with nontrivial intersection) and $\CSX$-groups (groups where all maximal $\X$-subgroups are malnormal). We pay particular attention to split Frobenius groups and investigate conditions under which such groups are $\XT$- or $\CSX$-groups.

In this work, we investigate the validity of the Szeg\"o type strong asymptotic formula for the Sobolev orthogonal polynomials associated with a Sobolev-type inner product depending on a parameter $s$, which is defined as $$\begin{equation*} \langle f,g\rangle _{\frac{\lambda}{n^s}}:= \int_{\mathbb{T}} f(z)\overline{g(z)} d\mu_1(z)+ \dfrac{\lambda}{n^s} \int_{\mathbb{T}} f^{\prime}(z)\overline{g^{\prime}(z)} d\mu_2(z), \quad \text{where $s$ is fixed and } \quad s\geq 2, \end{equation*}$$ and $\lambda >0$. Here, $\mu_1$, $\mu_2$ are absolutely continuous with respect to the normalized lebesgue measure and $\int_{\mathbb{T}} \mu^{\prime}_i(z) dm=1$, where $\mu_i^{\prime}(z)= \dfrac{d\mu_i}{dm} >0$ for $i=1,2$. Further, $\mu_1, \mu_2$ belongs to the Szeg\"o class and also has non-zero bounded Radon-Nikodym derivatives on the unit circle. %satisfies $0 <m \leq \dfrac{\mu^{\prime}_1(e^{\iota \theta})}{\mu^{\prime}_2(e^{\iota \theta})}\leq M < \infty$. We observe that the measure appearing in the Szeg\"o function of the strong asymptotic limit depends on $s$. As a specific case, we consider the Bernstein-Christoffel Sobolev orthogonal polynomials and study their properties, including the asymptotic distribution and location of their zeros for different values of $s$. We further establish a spectral characterization of their zeros as the eigenvalue of an explicitly constructed modification of the associated Hessenberg matrix and provide numerical evidence supporting the obtained asymptotic and zeros results.

Stratification can reduce the estimator variance in Monte Carlo simulations by partitioning the sample space according to a low-dimensional factor and sampling separately within the resulting strata, making it well suited to the valuation of path-dependent claims in high-dimensional financial models. Its performance, however, depends critically on the choice of the stratification factor, and effective low-dimensional factors are generally difficult to identify in complex continuous-time models. Neural networks provide a flexible approximation class for this purpose and have proved useful in related variance-reduction problems in quantitative finance. In this paper, we study the construction of neural stratification factors for derivative pricing under proportional allocation in continuous-time models. Since the numerical construction of a stratification factor typically involves approximation, we establish continuity results for the associated conditional expectations and, more generally, conditional distributions, and provide several counterexamples showing the limits of this continuity theory. Combined with density results for neural networks, our findings yield a universal approximation theorem for admissible stratification factors. We conclude with numerical experiments involving a constant-maturity-swap range accrual in a two-factor Gaussian short-rate model and a worst-of autocallable in a Wishart stochastic covariance model, where the learned factors outperform all considered benchmarks.

A Berge path in a hypergraph is an alternating sequence of distinct vertices and edges $v_1,e_1, v_2, \ldots, e_{k-1}, v_{k}$ such that $v_i, v_{i+1} \in e_i$ for all $i \in [k-1]$. Gy\H{o}ri, Katona, and Lemons determined the maximum number edges in $r$-uniform, $n$-vertex hypergraphs without Berge paths on $k$ vertices. The extremal examples for their bounds are disconnected hypergraphs in which each component has either too few edges or too few vertices to contain such a Berge path. We consider the setting in which the host hypergraph is connected. Let $\operatorname{ex}^{\mathrm{conn}}_r(n,\mathcal{BP}_k)$ denote the maximum number of edges in an $n$-vertex, $r$-uniform, connected hypergraph that does not contain a Berge path on $k$ vertices. This number was determined by Zhang, Broersma, Gy\H{o}ri, Tompkins, Wang for $k \leq r+2$, and Zhao, Gerbner, Zhou for $k \geq 2r+2$. In this paper, we resolved the remaining cases $r+3 \leq k \leq 2r+1$ when $n$ is sufficiently large, confirming a conjecture of Zhang, et al.

In the framework of linearized elasticity, we study thin elastic composite plates with thickness $\delta$. The plates contain small, rigid rectangular plates distributed periodically along $\varepsilon$. Between two neighboring rigid plates is an elastic beam with thickness $\delta < \varepsilon/3 < 1$. Through a simultaneous process of homogenization and dimension reduction, we obtain the limit model. Our analysis yields Korn-type inequalities adapted to the rigid-elastic geometry of the structure and provides a precise characterization of the limit deformation and displacement fields. In the $2$D limit problem, the bending is the sum of two functions, each depending on only one variable. This is due to the fact that the mixed derivatives of the outer-plane displacement vanish. Finally, the limiting 2D problem is two decoupled plates or strips, each one with just three degrees of freedom: shear along the strip axis, the cross-contraction (-extension), and the cross-bending. The corresponding correctors are defined in the same way in the periodicity cell. In the linearized setting, all the correctors are decomposed.

We present a domain-decomposition reduced-order model for two-phase geological CO$_2$ storage with fixed and moving wells. The approach builds on a nonoverlapping Schur formulation of a conservative RT0--P0 discretization solved by a sequential pressure--transport algorithm. The distinctive feature of the formulation is that local reduced models are coupled through physically meaningful interface variables: pressure traces for the pressure solve and inflow saturation traces for the transport solve. To address moving wells, we introduce a well-aware strategy that combines source-response enrichment, HAPOD-based far-field compression, and hybrid full-order/reduced-order treatment of well and near-well subdomains. We also formulate conservation-oriented hyper-reduction rules where interface fluxes, boundary terms and source terms are protected. Numerical results are presented to demonstrate accuracy, computational efficiency, robustness of the overall methodology.

In this paper, we characterize the complex symmetry of block Toeplitz operators on the Hardy-Hilbert space H2(E) with conjugation C_{u,\lambda}, where E is a finite dimensional Hilbert space. We also extend the application of this conjugation to matrix valued truncated Toeplitz operators.

We prove that every elastic knot in any non-torus twist knot class is not planar. In particular, this disproves a numerically motivated planarity conjecture for elastic figure-eight knots.

Computed tomography (CT) and magnetic resonance (MR) imaging provide complementary information for the brain, but acquiring both modalities is not always possible. The cycle-consistent adversarial network (CycleGAN) enables unpaired CT-to-MR synthesis, but because the standard L1-norm cycle?consistency penalty grows linearly with the residual magnitude, the network tends to focus on large residuals. This paper introduces an L0-norm-based cycle?consistency term, in which the L0-norm is approximated by a smooth and bounded function. This approximation not only increases sensitivity to small residuals but also effectively suppresses the influence of large residuals. The proposed model and an otherwise identical L1-norm baseline were trained on 102 images and evaluated on 9 test images. The proposed model achieves a lower mean squared error (0.02746 versus 0.03464, a 20.7% reduction), a higher mean peak signal-to-noise ratio (15.80 versus 14.77 dB), and a higher mean structural similarity index (0.5689 versus 0.5407). It also has a lower maximum MSE and higher minimum PSNR and SSIM than the baseline. These results indicate that modifying the cycle-consistency penalty can improve reconstruction accuracy on this dataset.

Given a graph $G$, its $2$-color Tur\'{a}n number $\mathrm{ex}^{(2)}(n,G)$ is the maximum number of edges in an $n$-vertex graph, such that the edges can be colored with two colors avoiding a monochromatic copy of $G$. Let $\pi^{(2)}(G)=\lim_{n\to\infty}\mathrm{ex}^{(2)}(n,G)/\binom{n}{2}$ be the $2$-color Tur\'{a}n density of $G$. What real numbers in the interval $(0,1)$ are realized as the $2$-color Tur\'{a}n density of some graph? It is known that $\pi^{(2)}(G)=1-(R_{\chi}(G)-1)^{-1}$, where $R_{\chi}(G)$ is the chromatic Ramsey number of $G$. Burr, Erd\H{o}s, and Lov\'{a}sz showed that $(k-1)^2+1\leq{R_{\chi}(G)}\leq{R(k)}$, for any $k$-chromatic graph $G$, where $R(k)$ is the classical Ramsey number. However, it is an open problem to determine how many distinct values between $(k-1)^{2}+1$ and $R(k)$ can be realized as $R_{\chi}(G)$ of some $k$-chromatic graph $G$ for general $k$. In this paper, among others, we prove that there are $\Omega(k)$ different values of $R_{\chi}(G)$ among $k$-chromatic graphs $G$. This sheds more light onto the possible $2$-color Tur\'{a}n densities of graphs.

We characterise the existence of expectation-preserving embeddings of reduced amalgamated free product $\mathrm{C}^\ast$-algebras into norm ultrapowers of the first factor, fixing its diagonal copy pointwise, in terms of complementary compression relations on Hilbert space ultrapowers. This yields new characterisations of Robert's notion of selflessness, selfless inclusions, and relative selflessness with respect to conditional expectations. We also characterise Ozawa's notion of complete selflessness, including its extension to inclusions. We then illustrate the usefulness of these characterisations by establishing complete selflessness for suitable reduced (amalgamated) free products, free Araki--Woods $\mathrm{C}^\ast$-algebras, certain Hecke $\mathrm{C}^\ast$-algebras, and $q$-Gaussian $\mathrm{C}^\ast$-algebras. Finally, we obtain dynamical characterisations of (complete) $\mathrm{C}^\ast$-selflessness of discrete groups and establish relative selflessness for suitable reduced crossed products.

Given an integer $k\geq2$ and a graph $F$, the $k$-uniform suspension $\mathcal{S}^kF$ is obtained by adjoining a fixed set of $k-2$ new vertices to every edge of $F$. In this paper, we study two extremal problems for suspensions of even cycles. Write $K^k_t$ for the $k$-uniform clique of order $t$. Let $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ and $\mathrm{ex}(n,K^k_t,\mathcal{S}^kC_{2\ell})$ denote the maximum numbers of edges and copies of $K^k_t$, respectively, in an $\mathcal{S}^kC_{2\ell}$-free $k$-uniform hypergraph on $n$ vertices. We prove that, for every $k\geq2$ and infinitely many $n$, $$\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)=\frac{n^{k-1/2}}{(k+1)!}+O(n^{k-1}).$$ This extends a folklore result for $k=2$ and, as an immediate consequence, yields the asymptotics of $\mathrm{ex}(n,\mathcal{S}^kC_4)$ for infinitely many $n$, previously established by Mubayi (for all $n$). Furthermore, for every $k\geq2$ and $\ell\in\{3,5\}$, we determine the order of magnitude $$\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})=\Theta(n^{k-1+1/\ell}).$$ This generalizes both the classical graph case $k=2$ and a previous result of Mukherjee for $k=\ell=3$. The principal difficulty in both problems lies in constructing the lower bounds. Our construction for $\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)$ incorporates a novel block-packing structure, which yields substantially more copies of $K^k_{k+1}$ than previously known constructions. For $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ with $\ell\in\{3,5\}$, we establish a natural $k$-uniform version of Wenger graphs, addressing the subtleties involved in lifting extremal graph constructions to suspensions. We also give applications of our results to Tur\'an problems for simplicial complexes.

Game-theoretic models can resolve vehicle interactions without hand-written priority rules, but whether they describe human drivers once calibrated has rarely been tested. This study formulates a general-sum differential game of vehicle interactions at urban intersections, calibrates it to observed driving, and evaluates it on data not used for calibration. Each vehicle minimizes a cost that combines speed tracking, adherence to a map-derived lane route, control effort, gap keeping based on the intelligent driver model, a predictive conflict term, and a contact barrier, while perceiving the other vehicles with a delay. The game is solved with a feedback iterative linear-quadratic algorithm, and ten behavioral parameters per vehicle are calibrated by Bayesian optimization on trajectories recorded at four intersections in Washington, D.C. The calibrated game reproduces the trajectories of 819 vehicles with a mean position error of 1.00 m. When only each vehicle's route is known, as in a simulator, it is the most accurate of the compared models on 96 held-out vehicles (7.42 m), and ten-fold cross-validation shows that it is significantly more accurate than calibrated SUMO (by 3.61 m), an inverse reinforcement learning model (by 0.93 m), and independent and cooperative variants of the same model (by 0.33 and 0.32 m). The solutions, computed in 2.1 s for a typical interaction, are verified approximate equilibria. Finally, the conflict and steering weights, which the data do not identify, can be set from safety and comfort targets; together with a stiffer contact barrier, this removes the collisions that otherwise occur in 43% of tight closed-loop conflicts.

High-fidelity Reynolds-averaged Navier--Stokes (RANS) simulations remain a standard tool for aerodynamic analysis, yet each solve on an unstructured mesh is computationally expensive, making many-query tasks such as design exploration, optimization, and multi-condition evaluation costly. Neural operators provide mesh-flexible surrogates for mapping geometry and operating conditions to flow fields; however, in many existing architectures, global parameters such as the freestream state and design variables are concatenated with pointwise inputs, leaving their influence on intermediate features implicit. This paper proposes CoPiT, a \textbf{Co}nditional \textbf{P}osition-induced \textbf{T}ransformer that builds on the Position-induced Transformer (PiT) as its geometric backbone and injects a global condition vector through multi-stage feature-wise linear modulation (FiLM), thereby making parametric modulation explicit while preserving geometry-based attention weights. The model is evaluated on two-dimensional AirfRANS airfoil fields, including scarce-data and extrapolation regimes, and on three-dimensional BlendedNet++ blended-wing-body surfaces. On the AirfRANS full-data test set, CoPiT reduces surface and volume losses by approximately 54\% and 24\% relative to Transolver, respectively, and achieves the lowest overall normalized test MSE among the compared baselines on BlendedNet++. Matched ablations show that FiLM conditioning generally improves upon input concatenation, while the preferred placement is task-dependent. CoPiT thus provides a practical operator-learning surrogate for parametric aerodynamics on unstructured meshes.