Papers

We prove that a smooth projective surface over $\mathbb{C}$ admitting a full exceptional collection of line bundles is rational.

This paper presents the executable Python implementation~\cite{noroozi2026bahnstromems} of the two-layer energy-management system developed mathematically in Part~I~\cite{norooziP1I}. If needed, the software first converts hourly railway operation data into a 15-minute timetable and calculates the amount of gross motoring and available regenerative powers in a 15-minute resolution using the timetable handling layer. Given the energy prices are available either cached or by sending request to ENTSO-E Transparency Platform through the price handling layer, the software constructs a deterministic hourly day-ahead plan with either in the so-called must-run converter commitment mode or in a sub-hourly feasibility certificate mode, where the latter demands more computational complexity. Given the inputs from the timetable, day-ahead planning layers and initial measurements of the intraday-resolutions network's physical quantities, the intraday handling layer executes a risk-neutral intraday scenario model predictive controller by area-wise quadratic programming and consensus ADMM. The retained S1 forecast layer combines a one-week seasonal-naive point forecast with causal residual-path resampling, while the current measured stage is identical in all scenarios. We specify the day-ahead and intraday data contracts, absolute-UTC coupling, local-QP assembly, warm starts, residual tests, solver-status interpretation, physically validated recovery paths, state and key-performance-index updates, and the railway-owned generation and pumping increment. The Python package version, dependency lock, frozen-plan manifests, deterministic fixtures, run logs, and hash manifests make the software configuration executable and auditable. The paper describes software behavior and numerical contracts; comparative performance and stress-test claims are reserved for Part~III~\cite{norooziP1III}.

We study the minimisation of the first solenoidal Maxwell eigenvalue $\lambda_1^{\mathrm{Max}}(\Omega)$ in perfectly conducting cavities through the scale-invariant functional $J_{\mathrm{Per}}(\Omega):=\mathrm{Per}(\Omega)\lambda_1^{\mathrm{Max}}(\Omega)$ in $\mathbb R^3$. While the corresponding unconstrained problem is known to be ill-posed, we focus on bounded convex cavities and prove that $J_{\mathrm{Per}}$ has a positive infimum in this class. The proof relies on the study of the three main possible behaviours for sequences of convex domains with bounded perimeter: convergence to a bounded convex domain with non-empty interior, collapse to a planar domain or collapse to a one-dimensional domain. The case of a planar collapse is handled through projection arguments, yielding a lower bound, while that of a one-dimensional collapse is controlled by a different argument, showing that in that case $J_{\mathrm{Per}}$ must diverge to $+\infty$. Although the question of existence of an optimal shape (that is, ruling out planar collapse) remains open, we derive several negative results and obstructions to existence; typically, no minimiser with a $\mathscr C^{3,+}$ boundary exists. This relies on a fine analysis of first-order optimality conditions.

Let $p$ be an odd prime and let $S\subseteq\mathbb{F}_p^2$ have cardinality divisible by $p$. We prove that, for all sufficiently large primes $p$, no such set has exactly four special directions, and obtain a conditional extension to larger numbers of special directions under an affine-independence assumption on the corresponding projection functions. A separate second-moment argument shows more generally that, for every fixed $k\ge4$, no subset of $\mathbb{F}_p^2$ has exactly $k$ special directions once $p$ is sufficiently large. In fact, the result holds uniformly for $k$ up to a positive constant times $\sqrt p/\log p$. In contrast, for multisets every prescribed collection of at most $p$ directions can occur as the set of special directions of a $\{0,1,2\}$-valued multiset on $\mathbb{F}_p^2$.

In this paper, we prove that symmetric, doubly-periodic, three-dimensional hydroelastic waves of small amplitude, bifurcating from a non-uniform shear flow, are necessarily two-dimensional to leading order. A detailed analysis of the solvability condition at quadratic order reveals that all Fourier coefficients with transverse dependence vanish identically. The dimensional reduction is controlled entirely by the vertical structure of the shear flow and is robust under changes to the dynamic boundary condition.

We show that a simply connected compact symmetric space of dimension $n\ge 3$ admits a conformal Killing $2$-form which is not Killing if and only if it is homothetic to one of $\mathbb{S}^n$, $\mathbb{CP}^m$ or $\mathbb{HP}^q$.

Phylogenetic networks provide a flexible framework for representing reticulate evolutionary processes, such as hybridization, introgression, recombination, and horizontal gene transfer. However, their combinatorial complexity makes even basic enumeration problems difficult. Building on our previous work for networks with up to three reticulations, we derive an explicit closed-form formula for the number of unrestricted rooted binary phylogenetic networks with four reticulations on \(n\) labeled taxa. Our approach is based on tree-component graphs. We classify the 79 possible component graphs corresponding to networks with four reticulations into ten groups. We then enumerate the networks associated with each group by combining known counts of one-component networks, forests, and networks with fewer reticulations. Summing these contributions yields the desired formula. This result extends the exact enumeration of unrestricted binary phylogenetic networks to four reticulations and further demonstrates the effectiveness of component graphs for systematically organizing and counting increasingly complex network classes.

Let $G$ be a finite abelian group, and let $k$ be an infinite perfect field containing a primitive root of unity of order the exponent of $G$. We prove that the $G$-equivariant version of Gabber's presentation lemma proposed by Bachmann holds over $k$ if and only if $G$ is cyclic of $p$-power order, for some prime $p$.

We consider the stabilization of two-dimensional periodic hydroelastic waves in deep water beneath a massless nonlinear Cosserat sheet by means of a localized pres- sure feedback. For sufficiently regular solutions, under an explicit condition relating gravity, flexural rigidity and the localization profile, we prove a uniform spacetime estimate and a localized dissipation observability inequality. As a consequence, the hydroelastic Hamiltonian decays quantitatively as long as the solution remains in a small geometric regime. The proof combines an energy dissipation identity with a nonlinear multiplier method adapted to the infinite-depth geometry, together with a coercivity analysis of the Cosserat bending energy.

Let $B$ be the bluebird combinator with reduction rule $Bxyz \to_{w} x\left(yz\right)$, let $M$ be the mockingbird combinator with reduction rule $Mx \to_{w} xx$, and let $I$ be the identity bird combinator with reduction rule $Ix \to_{w} x$. A fixed-point combinator, called a sage bird by Smullyan, is a closed term $Y$ such that, for a fresh variable $x$, $Yx$ is equivalent to $x\left(Yx\right)$ under these reduction rules. For a fixed variable $x$, we construct an invariant $\mathrm{Tr}_{x}\left(u\right)$ of a $BMI$-term $u$ with respect to $\to_{w}$. This invariant traces the occurrences of $x$ in the leftmost-innermost reduction sequence of $u$. We then prove that $\mathrm{Tr}_{x}\left(Yx\right) \neq \mathrm{Tr}_{x}\left(x^{r}\left( Yx \right)\right)$ for every $x$-free $BMI$-term $Y$ and every $r\geq 1$. Consequently, there exists no fixed-point combinator in $BMI$-combinatory logic. This provides a negative answer to the problem posed by Smullyan in 1985.

In this paper, we prove that a mean system utilization smaller than one is a necessary condition for the feasibility of real-time systems. Such systems are defined as stable. Stable systems have two distinct states: a transient state, followed by a steady-state where the same distribution of response times is repeated infinitely for each task. We prove that the Liu and Layland theorem holds for stable probabilistic real-time systems with implicit deadlines, we provide an analytical approximation of response times for each of those two states and a bound of the instant when a real-time system becomes steady.

We study macroscopic thermalization in the ferromagnetic Ising model with a weak generic (highly nonlocal and many-body) random quantum perturbation. We prove that a single typical perturbation makes every initial state $|\Phi(0)\rangle$ in a specified energy shell thermalize: a measurement of a large class of macroscopic observables in the time-evolved state $|\Phi(t)\rangle=e^{-i\widetilde H_Lt}|\Phi(0)\rangle$ yields their thermal equilibrium values, within a prescribed precision and with overwhelming probability, at sufficiently large, typical times $t$. The result covers the full finite-temperature range, including the ordered phase, where we use plus boundary conditions. With periodic boundary conditions in the ordered phase, every energy eigenstate in the shell is shown to be an almost balanced macroscopic Schrödinger-cat state. Moreover, every pure initial state becomes such a cat at sufficiently large, typical times, with each branch in the corresponding macroscopic thermal equilibrium. The proof combines the theory of Roos, Sugimoto, Teufel, Tumulka, and Vogel with rigorous Ising large-deviation estimates.

We study the dynamics of weighted backward shifts on Ces`aro spaces associated with leafless locally finite rooted trees. We first characterize their boundedness in terms of adjacent level cardinalities and edge weights. We then characterize their $\mathcal{F}$-transitivity by a growth condition involving level cardinalities, products of weights along paths, and a level-dependent Ces`aro factor. As consequences, we obtain criteria for hypercyclicity, weak mixing, topological ergodicity, and topological mixing. We also characterize the existence of nonzero orbit limit points and chaotic weighted shifts, the latter in terms of normalized fixed points and unit flows satisfying an explicit summability condition. Examples show that $\mathcal{F}_{\underline{d}>0}$-transitivity need not imply frequent hypercyclicity, that a nonhypercyclic weighted shift may nevertheless have a nonzero orbit limit point, and that topological mixing need not imply chaos.

We develop single-loop stochastic projected damped extragradient methods for nonconvex–(strongly) concave minimax optimization, with complexity guarantees for both game stationarity (GS) and optimization stationarity (OS). Our approach combines a stochastic projected damped extragradient (SPDE) method with a recursive variance-reduced variant, VR-SPDE, both of which retain a single-loop structure. Under an unbiased stochastic gradient oracle with uniformly bounded variance, SPDE finds an $\varepsilon$-game-stationary point with stochastic first-order oracle (SFO) complexities of $O(\kappa\varepsilon^{-4})$ and $O(\varepsilon^{-5})$ in the nonconvex–strongly concave and nonconvex–concave settings, respectively, where $\kappa=L/\mu$. Under an additional mean-square Lipschitz condition on the stochastic gradients, VR-SPDE improves these GS complexities to $O(\kappa^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-9/2})$, respectively. For an $\varepsilon$-optimization-stationary point, SPDE achieves SFO complexities of $O(\kappa\varepsilon^{-4})$ and $O(\varepsilon^{-6})$, while VR-SPDE achieves $O(\kappa^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-6})$, in the two settings, respectively. These OS guarantees match the best-known bounds achieved by multi-loop methods while preserving a single-loop implementation. To the best of our knowledge, our results provide the best-known SFO complexity guarantees among single-loop stochastic first-order methods for the respective stationarity criteria and problem classes.

Let $\gamma(G)$ denote the number of cliques in a graph $G$ and let $\Gamma(n):= \{\gamma(G):|V(G)|=n\}$ be the set of values of $\gamma(G)$ that can be attained on $n$ vertices. We improve on a result by Erdős and Erné to show that $| \Gamma(n)| \geq 2^{n-4\ln(2)\log^3(n)}$ for sufficiently large $n$.

Let $m\geq1$, $k\geq0$, $n=m+k$, and write $x=(v,z)\in\mathbb R^m\times \mathbb R^k$, where $v$ is the active velocity variable and $z$ is passive with respect to $Y=v\cdot\nabla_y+\partial_t$. We consider \[ \operatorname{div}_x(A\nabla_xu)-Y(wu)=0, \] where $w=w(x)\in A_2(\mathbb R^n)$ and the measurable matrix $A$ has ellipticity and size controlled by $w$. For arbitrary such weights, including weights depending on the active variables, we prove a weighted hypoelliptic transfer estimate, a kinetic Sobolev inequality, local boundedness, the weak Harnack and Harnack inequalities, local Hölder continuity, and the strong minimum principle. The Harnack argument uses a full-variable weighted measure-data compactness theorem. An $\mathrm L^p$ velocity-averaging estimate gives compactness of smooth moments of $wu$, and weighted Poincaré inequalities in all diffusive variables reconstruct $u$. At a two-phase limit, testing with non-negative profiles produces transport directions given by weighted mean velocities. A $\mathrm{BV}$ argument applied to the limiting subsolution inequality yields the no-jump principle needed for expansion of positivity and a weighted kinetic ink-spots lemma. The theory includes the weight $|\lambda|^{1-2s}$ in the extension of Garofalo and Tralli for fractional powers of the Kolmogorov operator.

Conjecture~2 of Hinrichs and Vybíral asserts that every continuous, nonnegative, positive definite function $f$ on $\R^d$, normalized by $f(0)=1$, satisfies $[f(x_j-x_k)]_{j,k=1}^n\succeq \one\one^T/n$. We give an explicit trigonometric polynomial on $\R^7$ and eight points for which this inequality fails. All hypotheses are verified directly: positive definiteness follows from nonnegative Fourier coefficients, and pointwise nonnegativity follows from interpolation on a cube. For an explicit vector of signs, the quadratic form is $22/5$, whereas the proposed lower bound is $9/2$. The resulting gap is exactly $-1/10$. The example is compatible with the established bound for functions of the form $|g|^2$ with $g$ positive definite.\\ The solution was found in a single prompt try by a colleague using his private licence of ChatGPT 6. Our aim is to make the solution public, as well as to collect some ideas that emerged during the analysis of the solution.

This paper shows that there are intimate connections between the quadratic transform technique for solving fractional programming (FP) problems and the Schur-complement technique in matrix analysis. We demonstrate that the quadratic transform technique is related to two aspects of the Schur complement: (i) the linear matrix inequality (LMI) condition for positive semidefiniteness and (ii) the matrix determinant formula. Specifically, we establish that the quadratic transform and the Schur-complement LMI condition imply each other. This connection allows us to provide new interpretations of the auxiliary variable in the quadratic transform, and it allows us to rederive the Schur-complement determinant formula. Furthermore, this connection leads to generalizations of the quadratic transform in FP and the Schur-complement LMI that can accommodate generalized matrix inverse. As an application in information theory, we apply the generalized FP framework to the least-favorable-noise minimax formulation of the Gaussian vector broadcast channel sum capacity problem. When the least-favorable noise covariance is singular, matrix-inverse-based Karush-Kuhn-Tucker (KKT) analysis would require a careful analysis of the input and output spaces of the channel. We show using generalized FP that an auxiliary-variable representation of the singular matrix fraction directly yields the reciprocal multiple-access channel and recovers the uplink-downlink duality relation for sum capacity.

Reconfigurable intelligent surfaces (RISs) are poised to revolutionize wireless multicasting by enabling extended coverage and reliable operation in obstructed environments. These benefits, however, can be undermined by security vulnerabilities arising from practical deployment factors. This work addresses three such critical factors, (i) the discrete nature of RIS phase shifts, (ii) the presence of colluding eavesdroppers, and (iii) the inefficiency of static illumination beams, each threatening security if not properly accounted for in system design. To mitigate these issues, we formulate a joint resource allocation problem that minimizes the wiretap signal-to-noise ratio (SNR) across all eavesdroppers by co-optimizing the RIS configuration and the base station (BS) beam management. This yields a complex, nonconvex mixed-integer nonlinear program (MINLP), which we equivalently reformulate into a tractable mixed-integer quadratically constrained program (MIQCP) solvable to global optimality. Numerical results confirm that the proposed scheme significantly bolsters security, suppressing the wiretap SNR by up to 58% compared to existing baselines.

We give a weak formulation of spectral scalar curvature bounded from below via establishing a dihedral rigidity result. Given some special convex polyhedron, if another metric are of non-negative spectral scalar curvature in the interior, with weighted mean-convex faces, and with its dihedral angles less than or equal to their flat polyhedral model everywhere along the edges, then the metric must be flat. This is motivated by Gromov's definition of a weak notion of non-negative scalar curvature.

This paper studies the Floer-theoretic aspects of homological mirror symmetry inspired by proposals of Perutz and Siebert and the Gross–Siebert intrinsic mirror symmetry program. Given a maximally unipotent degeneration of smooth projective Calabi–Yau manifolds over the punctured disk, we construct a ring using the fixed point Floer cohomology groups of the iterates of the monodromy of the degeneration equipped with the pair of pants product. Under the assumption that this ring is commutative, we can consider a candidate mirror family defined by the relative Proj construction. Further assuming that a smooth fiber $X_t$ contains a so-called tropical Lagrangian section, we construct a fully faithful embedding from the derived category of perfect complexes on our candidate mirror family into the Fukaya category of $X_t$. We verify both of these assumptions for certain Batyrev–Borisov toric degenerations, as well as some toric degenerations of Calabi–Yau threefolds coming from the Gross–Siebert reconstruction algorithm. These two geometric hypotheses are both phrased to support the general study of mirror symmetry for maximally unipotent degenerations of Calabi–Yau manifolds, largely reducing the symplectic inputs for proving homological mirror symmetry to the problem of constructing tropical Lagrangian sections.

This paper proposes and analyzes a high-order finite volume element method on curved-edge quadrilateral meshes for elliptic equations. Unlike existing theories, which are primarily based on straight-edge meshes, this study is the first to establish an analysis of the stability and optimal convergence of the finite volume method on curved-edge meshes. By constructing a dual mesh based on Gaussian points, we overcome the accuracy degradation issues caused by geometric deformation and Jacobian non-uniformity of curved-edge meshes. We prove the coercivity of the discrete bilinear form under weak mesh regularity conditions, thereby obtaining an optimal error estimate in the energy norm. Furthermore, using orthogonality and the Aubin-Nitsche technique, we derive an optimal $L^2$ error estimate. Numerical experiments cover problems with constant and anisotropic coefficients, different dual partition strategies, complex curved boundary domains, and interfaces with large deformations. Numerical results indicate that this method consistently achieves the optimal convergence order in both the $H^1$ and $L^2$ norms on a variety of curved-edge meshes. Compared to straight-edge meshes, curved-edge meshes offer significant advantages in approximating complex curved boundaries and demonstrate better resistance to distortion in cases involving sudden changes in coefficients and large deformations at interfaces. This paper provides a unified theoretical framework for the finite volume element method on curved-edge meshes and verifies the efficiency and robustness of the proposed method.

Discovering the structure and kinetics of chemical reaction networks (CRNs) from time-series concentration data is a fundamental challenge in chemical kinetics, with existing approaches relying on prior mechanistic assumptions or suffering from high computational cost and noise sensitivity. In this work, we present integral chemical reaction neural networks (iCRNNs), a framework that combines the interpretable, physics-constrained, architecture of chemical reaction neural networks (CRNNs) with an integral collocation formulation of the governing dynamics. Rather than solving ODEs forward in time at each training step, we approximate the integral of the learned rate functions directly using numerical quadrature. This yields an entirely algebraic forward pass consisting only of matrix operations, eliminating the repeated adaptive ODE solves of standard CRNN and producing smoother, more predictable training. We provide a recovery error analysis characterising the structural sources of ill-conditioning (conservation laws, reaction reversibility, and shared reactant sets) that limit network identifiability. On two benchmark CRNs, iCRNN trains in roughly half the wall-clock time of the baseline CRNN method on a four-species mechanism and between three and four times faster on a five-species mechanism, while completing every training run reliably and attaining comparable or lower loss.

In this paper, we address the joint parameter estimation of drift and diffusion coefficients for heterogeneously interacting particle systems. Unlike the homogeneous setting, the interactions are governed by a graphon-weighted mean-field framework, which introduces significant analytical complexity: while homogeneous systems yield i.i.d. limits, our setting results in particles that are independent but non-identically distributed in the limit. Based on discrete observations of the system over a fixed time interval $[0, T]$, we propose a contrast function based on a pseudo-likelihood approach. We prove the consistency of the resulting estimators as the discretization step $\Delta_n \to 0$ and the number of particles $N \to \infty$. Furthermore, we establish asymptotic normality under the additional constraint $N \Delta_n \to 0$, demonstrating that the underlying graphon structure can be rigorously handled despite the lack of identical distribution in the limit.

The main purpose of this paper is to define and study the notion of restricted Rota-Baxter operator on restricted Lie algebras and restricted Lie triple systems. We provide the relevant properties and the usual connections with pre-Lie structures. We show that the operad of pre-Lie triple systems is the splitting of the operad of Lie triple systems. Moreover, we prove that Jacobson identities hold in pre-Lie triple systems in positive characteristic and then introduce the notion of restricted pre-Lie triple system.

This paper proposes a new small area estimation approach that integrates Partial Least Squares within the Fay-Herriot model to address the challenges posed by high-dimensional and highly correlated auxiliary variables sets. The resulting Partial Fay-Herriot (PFH) estimator constructs supervised components that maximize their association with the target variable, enhancing model stability and predictive efficiency. Monte Carlo simulations demonstrate that PFH estimator achieves lower mean squared error than the standard Fay-Herriot estimator and outperforms principal components-based alternatives while relying on fewer latent dimensions. The methodology is applied to the estimation of district-level per capita consumption in Mozambique, where the survey data source is complemented by large set of correlated census variables. The resulting estimates highlight pronounced geographic heterogeneity and uncover spatial clusters of deprivation. Overall, the findings show that the proposed supervised dimension-reduction approach represents an effective and easily interpretable tool for producing reliable indicators in high-dimensional contexts.

Let $S$ be a free profinite group on a finite ordered basis $X$, and let $S^{(n,p)}$, $n=1,2,\ldots,$ denote its lower $p$-central filtration. There is a natural duality between $S^{(n,p)}/S^{(n+1,p)}$ and $H^2(S/S^{(n,p)},\mathbb{F}_p)$. These $\mathbb{F}_p$-linear spaces admit natural bases indexed by Lyndon words of length $\leq n$ in the alphabet $X$. These bases are known to be unitriangularly dual. We prove that they are much closer to being fully dual, by showing that the pairing between two basis elements vanishes unless the corresponding Lyndon words are permutations of one another. We further show that the value of the pairing is essentially independent of $n$.

We provide kernel estimates for pseudo-differential operators in the $S(m,g)$ Weyl-Hörmander calculus classes. These new results lead to boundedness theorems like the weak (1,1) type for these classes and the boundedness of these operators from Hardy spaces to Lebesgue spaces $L^p$. We prove that under a suitable hypoelliticity condition on the weigh $m,$ one can get the weak (1,1) inequality for the Fefferman order established previously by the second author in analogy to the classical $L^p$-boundedness theorem due to Fefferman. Our approach makes an extension to $S(m,g)$ classes of previous estimates proved by Álvarez and Hounie, Nagase, Kumano-Go, etc, for the Hörmander classes including new regularity properties in this framework from the Hardy space $H^p$ to $L^p$ and also on Lorentz spaces. We have identified that the Uniform Hörmander Condition plays a fundamental role in some of our boundedness theorems.

Uniform distribution modulo 1 is a classical notion of pseudo-randomness for sequences in the unit interval, which is quantified in terms of the discrepancy. Sequences whose discrepancy is of the smallest possible asymptotic order are called low-discrepancy sequences. The Poissonian pair correlation is another notion of pseudo-randomness, which studies the distribution of the gaps between pairs of elements of the sequence on a local scale. It is known that Poissonian pair correlation implies uniform distribution mod 1, and that the opposite implication is not true in general. It has also been observed that classical examples of low-discrepancy sequences fail to have Poissonian pair correlation, and it has been speculated that the two properties might be irreconcilable due to the high degree of structural rigidity that is required for low-discrepancy behavior. As we prove in this paper, this is not the case: we construct an example of a low-discrepancy sequence with Poissonian pair correlation.

We study random signed Takagi–Landsberg bridges with independent Rademacher Faber–Schauder coefficients. At index $H=1/2$, we prove that every fixed deterministic refining sequence of partitions with vanishing mesh yields quadratic variation $t$, almost surely and uniformly in $t\in[0,1]$. For partitions that need not be refining, the mesh condition $o(1/\log n)$ is sufficient, and its order is sharp. Our proofs use an operator representation of the quadratic sums and moment bounds for Rademacher chaos. We also show that asymmetric signs can destroy this invariance. At every index $H\ne1/2$, we construct a deterministic refining sequence along which the critical $p$-power sums, with $p=1/H$, have two distinct finite accumulation points almost surely. At $H=1/4$, uniform thirds grids provide an explicit alternative to the dyadic limit. Thus $H=1/2$ is the unique index in the symmetric family at which critical variation is invariant along every fixed deterministic refining sequence. Nevertheless the variation index equals $1/H$ almost surely across a large sequence of partitions.