Papers

This dissertation establishes an explicit arithmetic Picard–Lefschetz formula for isolated singularities of diagonal type. We investigate proper and flat families of varieties defined over a henselian discrete valuation ring of mixed characteristic. When the smooth generic fiber degenerates into a special fiber with isolated diagonal singularities, we construct an explicit eigenbasis for the tame inertia action on the local vanishing cycles. This construction yields a complete spectral decomposition, from which we deduce explicit formulas for the intersection pairing and the variation morphism. Furthermore, by computing the action of the geometric Frobenius operator on the subspace of inertia invariants, we establish an exact trace formula expressed in terms of Jacobi sums. We apply this theoretical framework to study the arithmetic of the symmetric powers of the Airy sheaf. By relating the compactly supported cohomology to degenerating affine hypersurfaces, we completely determine the local Galois representation for the $\ell$-adic realization of motives associated to Airy moments. This application provides a precise direct sum decomposition of the Galois module, a clear identification of the inertia invariants, and the explicit characteristic polynomial of the Frobenius action.

The three-dimensional catenoid in $\mathbb{R}^4$ is a complete embedded minimal hypersurface contained in a slab, showing that the half-space theorem does not extend directly to higher dimensions. We show that this obstruction is topological in $\mathbb{R}^4$. More precisely, we prove that a connected, complete embedded minimal hypersurface $\Sigma^3\subset\mathbb{R}^4$ contained in a half-space with bounded curvature and trivial second homology must be a hyperplane. We obtain further rigidity when a connected, complete embedded minimal hypersurface is confined to a slab. In particular, we show that if in addition the minimal hypersurface is proper and has trivial second homology then it must be a hyperplane, thus obtaining flatness without the bounded curvature assumption. If instead the minimal hypersurface has bounded curvature and finite second Betti number then it must have finite total curvature or finite index. As a corollary, we also obtain a new characterization of the three-dimensional catenoid as a minimal hypersurface confined in a slab with bounded curvature and second Betti number equal to one.

We present a theory of decaying turbulence in which the statistics of a classical flow becomes quantum mechanics in loop space. The loop functional, the average of $e^{\mathrm{i}\Gamma/\nu}$ for the circulation $\Gamma$ around a loop, is the wave function, and the Navier-Stokes equation becomes the linear loop equation. An exact operator identity represents the functional as the loop-ordered exponential of the covariant derivative $\partial+\mathrm{i}v/\nu$, with the loop parameter as time and viscosity as Planck's constant. The turbulent state is the equal-weight density matrix on a decaying trajectory, a quantum form of the ergodic hypothesis. As in quantum field theory, turbulence at infinite Reynolds number is defined by regularization and renormalization; the loop equation regularized on $N$-sided polygons has an exact manifold of decaying solutions, the Euler ensemble, labeled by rational angles and spins. As a fermionic trace, its partition function and conditional vorticity correlations become, in the local limit, Gibbs averages of two Mathieu quantum pendulums, an integrable system. The energy spectrum, given by explicit quadratures, falls as $k^{-1}\log(1/kL)$ at small $k$ and as $k^{-7/2}$ at large $k$, with $L=\sqrt{\tilde\nu(t+t_0)}$ and turbulent viscosity $\tilde\nu$. At $kL=\sqrt2$ its slope diverges as a double logarithm: the decay at fixed $k$ is singular at a finite time, and the vorticity correlation oscillates with period $\sqrt2\,\pi L$. The reduction to the pendulums, with the continuum and grand canonical limits at every nonzero separation, is formally verified in Lean 4 in forthcoming work by Y. Jiang, A. Migdal et al. The theory rests on stated physical hypotheses, as statistical mechanics rests on the Gibbs distribution; its comparison with simulations and experiment is so far qualitative.

For $n\ge1$, we let $$x_n:=\tan\bigl(\sum_{k=1}^{n}\arctan k\bigr).$$ In 2008, Amdeberhan, Medina, and Moll conjectured that $x_n\not \in \mathbb{Z}$ for every $n\ge5$. This was known for a set of positive integers of density $\tfrac{120}{817}\approx0.1469$. We prove that an integer value $x_n=m$ satisfies $|m|\ge e^{(1/2+o(1))\,n\log n}$, which we use to deduce that $$\#\{\,1\leq n\le N:x_n\in\mathbb{Z}\,\}=O(\log N). $$ In particular, the conjecture holds for a density-one set of $n$. The results in this note were formalized in Lean/Mathlib and produced autonomously by AxiomProver from natural-language statements.

Distributionally robust optimization (DRO) provides a principled framework for decision-making under distributional uncertainty. Classical data-driven DRO frameworks typically construct ambiguity sets from distributional information, such as moment constraints, divergence neighborhoods, or Wasserstein balls, specified before the downstream loss is considered. We propose a task-aware DRO framework based on targeted integral probability metrics. The ambiguity set is defined directly through the loss functions induced by feasible decisions, thereby controlling the loss discrepancy between an adversarial distribution and a data-driven reference distribution. This construction leads to an expected hinge-constrained formulation that is equivalent to an infinitely constrained loss-discrepancy formulation. It also yields finite-sample guarantees that bypass the ambient curse of dimensionality: whenever an appropriate scalar pointwise concentration inequality is available for the induced loss estimator, the ambiguity radius can be calibrated at the canonical $\widetilde{\mathcal O}(N^{-1/2})$ rate after uniformization over the decision class. As a result, the framework applies broadly to settings including heavier-tailed sub-Weibull losses, Markovian data, outlier-corrupted data, and incomplete data. We derive exact infinite-dimensional dual reformulations, establish out-of-sample and excess-risk guarantees, and develop a conservative Monte Carlo approximation scheme with convergence and suboptimality guarantees. For piecewise affine losses, the sampled problems admit tractable conic reformulations. Numerical experiments in inventory management under heavy-tailed demand and regression with outlier corruption demonstrate strong out-of-sample performance relative to existing approaches.

How fragile is the routing time window of delivery systems against spatial distributional uncertainty? We study the tardiness risk of Traveling Salesman Problem (TSP) solutions with respect to a service deadline (target) over the routing time. Using the robust satisficing model, we introduce the TSP tardiness index to quantify the target's fragility under distributional uncertainty in customer locations. Assuming there are m potential customer locations from historical samples on a service region D (of area |D|), we prove that the TSP tardiness index is {\Theta}(n * sqrt(|D|m) / {\tau}) for n realized locations with respect to the routing time target {\tau}, under non-boundary conditions. This result establishes a new scaling law that extends beyond the existing deterministic and probabilistic TSP bounds. We further extend it to a multi-vehicle case and derive simple partition rules for managing delivery systems. Our numerical experiments using synthetic and real-world routing data validate the value of the TSP tardiness index in characterizing and managing the overtime risk of routing systems.

This paper is concerned with the decay of solutions and the analysis of the local and global exact controllability to trajectories for a class of one-dimensional nonlinear parabolic systems with weakly degenerate diffusion coefficients. The system is controlled through the coefficient of the reaction term. Our approach relies on a well-known local inversion method combined with suitable a priori estimates specifically adapted to the degenerate setting.

The theory of real semigroups developed by M. Dickmann and A. Petrovich provides an algebraic framework for abstract real spectra and real algebraic geometry, yet its application to rings of continuous functions is historically hindered by the topological constraints. In this paper, we bridge this gap by introducing generalized Marshall quotients over rings of real-valued continuous and differentiable functions, yielding new explicitly calculated examples of real semigroups. Furthermore, we conclude that the group of invertible elements of these quotients constitutes a real reduced hyperfield (which is categorically equivalent to reduced special groups), addressing the open problem of characterizing when the units of a real semigroup form a reduced special group. Finally, we apply this hyperalgebraic machinery to translate topological and differential phenomena into hyperalgebraic identities, establishing generalized versions of the Łojasiewicz-type inequalities.

Degenerate diffusion problems, where the governing parabolic equation can change type to either an ordinary differential equation or an elliptic equation, model many real life applications. Due to the presence of free-boundaries, accurate numerical simulation of such problems require extremely small mesh and time step sizes locally. To remediate this issue, in this work, we consider a space-time formulation of the problem based on an efficient splitting of the nonlinearities. First, an iterative linearization scheme is proposed to resolve the nonlinearities that effectively reduces to solving a sequence of heat equations. Unconditional convergence of the scheme is proven even for double degenerate cases with linear convergence achieved if the problem is non-degenerate. Next, the dual norm of the nonlinear residual is decomposed into a linearization error component and a discretization error component corresponding to the heat equation. This leads to reliable and fully computable a posteriori estimates for the problem that are robust with respect to the nonlinearities/degeneracies. These estimates are used then in a fully adaptive (discretization + linearization) space-time solver. Numerical experiments for multiple test cases (one and two dimensions in space) demonstrate that this solver efficiently allocates the computational resources in the space-time domain, resulting in a rapid decay of error in terms of total degrees of freedom spent.

We show that finite twin-width does not imply accessibility for finitely generated groups, which answers a question of Esperet. That is, we prove that there exists a finitely generated group $\Gamma$ that has finite uniform twin-width but is not accessible. In particular, for every finite generating set $S$ of $\Gamma$, the Cayley graph $Cay(\Gamma ; S)$ has finite twin-width but is not accessible. The example is obtained by combining Wilkes construction of a finitely generated inaccessible residually $p$-finite groups with a result of Bonnet, Geniet, Tessera and Thomasse regarding the twin-width of groups acting faithfully on regular rooted trees.

Understanding how spatial patterns evolve over time is a complex task that often arises in the analysis of public health data. In this work, we investigate the use of vineyards from topological data analysis (TDA) in this setting by applying them to time series data related to the overdose epidemic in the state of Ohio. We begin by proposing statistical tests that can be used in order to evaluate whether vineyards are a reasonable technique to study a spatiotemporal dataset. We then apply these tests to the data of drug overdose deaths in Ohio and, finding the data suitable, perform a subsequent analysis using vineyards to visualize the evolution of local maxima of death rates throughout the Ohio overdose epidemic. We conclude by developing statistical methods to quantify the significance and uncertainty of vineyard features and by exploring how vineyard-derived summaries can be used for forecasting.

We discuss a cancellation property of sieve weights, one that is applicable to the distribution of primes and almost-primes in very short intervals.

For positive integers $1\le r<d<n$ consider subsets $S\subseteq \mathbb{G}(r,V)$ of the $r$-plane Grassmannian of an $n$-dimensional Hilbert space $V$ saturated in the sense that the $r$-plane $\eta''$ belongs to $S$ whenever it is the orthogonal projection of $\eta'\in S$ onto a $d$-plane through $\eta\in S$. Motivated by such closure operators' natural occurrence in projective-geometry and linear preserver problems, we classify said saturated sets as precisely the disjoint unions of Grassmannian spines, with cores standing in a relation of mutual separation that can be made precise (a spine being the set of $r$-planes containing a fixed core $k$-plane $\pi$ for some $0\le k\le r$). This generalizes the author's results describing saturated $r$-plane sets in the tame dimensional regime $2r\le d$, where the disjoint unions in question by necessity collapse to single spines.

Peer-to-peer energy trading among electric vehicles (EVs) can improve charging flexibility under limited charging infrastructure, but effective EV–EV trading requires coordinated provider–consumer matching under journey-specific conditions. This paper proposes EVTradeMatch, a prediction-guided multi-objective optimization framework for mobility-aware EV–EV energy trading. Building on the EVNextTrade study, a prior learning-to-rank model for charging-node recommendation, we define charging-node suitability as a prediction-derived score reflecting the appropriateness of assigning a provider–consumer pair to a candidate charging node based on mobility, energy, and contextual trading features. This score is used as a guidance signal and as an explicit optimization objective rather than as a hard selection rule. The EV–EV matching problem is formulated as a multi-objective mixed-integer linear program that maximizes matching coverage, transferred energy, and charging-node suitability while minimizing mobility cost under spatial, temporal, one-to-one matching, and charging-node capacity constraints. To approximate Pareto-efficient solutions in wide-area dynamic settings, we develop a tailored non-dominated sorting genetic algorithm II (NSGA-II). Experimental results show that EVTradeMatch improves transferred energy by 74.2–82.8% and charging-node suitability by 8.3–84.4% compared with proximity- and auction-based state-of-the-art methods, while improving matching coverage by 3.74–25.07 percentage points. Balanced NSGA-II solutions achieve 53.07$\pm$0.76% matching coverage and transfer 1701.94$\pm$17.03 kWh, with higher mobility cost as an explicit trade-off against travel-minimizing methods. Pareto-front analysis shows that the framework supports flexible selection among high-coverage, high-energy, low-mobility-cost, and high-suitability solutions according to operational priorities.

We establish quantitative equidistribution estimates for spherical observables sampled along Kronecker flows. For the monotone measure-preserving inverse-CDF parametrization $\Phi_d:\mathbb{T}^d\to\mathbb{S}^d$ and a Diophantine frequency of exponent $\tau$, every $\varphi\in C^s(\mathbb{S}^d)$, $0<s\le1$, has time averages converging to the spherical mean at rate $O(T^{-\vartheta})$, where $\vartheta=\frac{s}{(1+s)(\tau+d)}$, uniformly in the initial phase and averaging window. Since $\Phi_d$ is discontinuous across a polar cut, the proof localizes the pullback away from the polar degeneracies and combines de la Vallée Poussin approximation, Fourier decay, and the Diophantine lower bound. The same exponent holds for a continuous measure-preserving variant. For comparison, mollification followed by Koksma–Hlawka gives, for every $\varepsilon>0$, the rate $O(T^{-s/[d(\tau+1)]+\varepsilon})$. Its power exponent is larger exactly when $\tau(d-1-s)<ds$; at equality the nominal exponents coincide, while the localization estimate has no $T^\varepsilon$ loss. On data vanishing near the polar degeneracies, the cutoff-free argument gives exponent $s/(\tau+d)$. We apply the estimates to shrinking spiral-type trajectories, integrable angular data, and homogeneous singularities, including a contrast between power-law and exponential radial contraction. Finally, on $\mathbb{S}^2$, for every prescribed $R(T)\to0$ we construct a rationally independent frequency and a smooth mean-zero observable with $\|\varphi\|_{C^1}\le1$ for which the error is not $O(R(T))$.

Discrete global grid systems are conventionally designed over a prior coordinate reference system and inherit its compromises. Hex9 inverts the direction of design: we ask what requirements a hierarchical grid must satisfy to be geometrically coherent – intrinsic orientability, flat mode transport, vertex closure, refinement commutativity – and show that these requirements essentially determine the grid. The admissible cell primitive is the triangle; the admissible seed is the octahedral triangulation of S^2; admissible refinement is uniquely aperture 9, and uniquely up to chirality, with exactly one orientation of the hexagonal dual lattice per chirality – fixed by a machine-verified exhaustive enumeration. The surviving structure is a shifted-aperture-9 hexagonal hierarchy in which every cell carries a unique address derived from the construction alone: truncated at level L, the address is a DGGS zonal identifier in the sense of OGC Topic 21; carried to the limit, a function recovers from it a point on the reference ellipsoid to arbitrary precision. The addressing is quasi-continuous – position is recoverable everywhere except on a measure-zero set of seams – rather than continuous in the strict ISO 19111 sense; the same object serves as both zonal identifier and position-recovery coordinate, with no prior coordinate reference system as input. A separable geometric realisation – an analytical octahedral base projection composed with an optimal-transport-derived area-correcting warp – places the grid on WGS84 with quasi-uniform cell areas: at level 5, 99% of the 708,588 cells lie within 0.005% of ideal area, with residual deviation confined to the six octahedral vertices required by the topology. The combinatorial grid is projection- and ellipsoid-independent; only the warp is specific to the reference body, and is recomputable for any ellipsoid, terrestrial or planetary.

We present a novel coupling method for local and nonlocal diffusion problems in 1D. Unlike other methods, our coupling method exclusively uses local boundary conditions. This is possible because our nonlocal operators enforce them by construction. Leveraging this advantageous property, we construct a seamless coupling that is remarkably natural. The utilization of local boundary conditions allows for the transfer of well-established numerical methods from local problems to nonlocal ones. Our local-to-nonlocal coupling method is inspired by the domain decomposition method, which we would like to transfer to nonlocal problems. The main result of our study is the construction of a local-to-nonlocal coupling method with a quantifiable $O(h)$ convergence that holds for an arbitrary solution. For discretization of the local and nonlocal problems, the finite element method and the Galerkin projection are employed, respectively. We verify our convergence rate with extensive numerical experiments.

In this paper, we establish kernel-type fractional extensions of reverse Callebaut, Rogers–Hölder and Cauchy–Schwarz inequalities on time scales. The proposed results are formulated by means of a nonnegative kernel operator associated with the diamond-$\alpha$ integral, which enables the inclusion of memory effects within the framework of dynamic inequalities. By choosing particular kernels and time scales, the obtained inequalities reduce to their continuous, discrete and quantum counterparts. Moreover, when the kernel is taken as the identity kernel, the results recover the corresponding classical diamond-$\alpha$ dynamic inequalities. Several consequences are derived as special cases, including fractional reverse Cauchy–Schwarz inequalities and weighted reverse Rogers–Hölder inequalities. The results provide a unified approach for studying reverse inequalities in continuous, discrete, quantum and fractional dynamic settings and may be applied to obtain a priori estimates for Volterra-type and delay dynamic equations.

A complex semiring is a subset of the complex plane that is closed under the standard addition and multiplication of complex numbers and contains both $0$ and $1$. A complex semiring $S$ is called a bi-UFS if both its additive monoid $(S,+)$ and its multiplicative monoid $(S\setminus \{1\}, \cdot)$ are unique factorization monoids (UFM). The Bi-UF Positive Conjecture states that $\mathbb{N}_0$ is the only subsemiring of the nonnegative cone of the real line that is a bi-UFS. In this paper, we prove that no simple semiring extension of $\mathbb{N}_0$ by a quadratic algebraic number is a bi-UFS, identifying a natural class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. We also identify another class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. Then we extend the statement of the Bi-UF Positive Conjecture by motivated by a structural theorem we established for semidomains whose additive monoid are finite-rank free commutative monoids. Finally, we consider the bi-HF property, which is a relaxed version of the bi-UF property. We prove that $\mathbb{N}_0$ is the only positive rational semidomain having the bi-HF property, and we provide two methods to construct bi-HFS complex semirings that are distinct from $\mathbb{N}_0$.

Phylogenetic trees are rooted trees with branch lengths that record genetic divergence or elapsed time, and quantifying differences between them is central to a wide range of evolutionary and epidemiological analyses. Graph-polynomial encodings of rooted trees provide an accurate, interpretable, and computationally efficient way to compare tree shapes, but existing polynomial encodings must be paired with auxiliary structures to study rooted trees with branch lengths. We introduce a bivariate polynomial encoding that incorporates branch lengths directly into a recursive computation from the leaf vertices to the root vertex of a tree. We prove that, for rooted trees with branch lengths and no vertices of degree two, which include all standard phylogenetic trees, two trees have the same polynomial if and only if their underlying unlabeled trees are isomorphic and the branch lengths of corresponding edges are equal. We apply the polynomial encoding to three published HIV-1 phylogenies sampled in different epidemiological settings and show that it accurately separates the three datasets based on their tree topologies and branch lengths, outperforming previous polynomial-based approaches for analyzing rooted trees with branch lengths.

Fenley proved that any foliation almost transverse to a quasigeodesic pseudo-Anosov flow in a closed atoroidal 3-manifold has the continuous extension property, meaning the inclusions of leaves into the universal cover continuously extend to their ideal boundaries. This article gives an alternate proof of an upgraded version of this: the associated Cannon-Thurston map for the flow, constructed by Frankel and Fenley, organizes all of the leafwise continuous extensions. The proof uses the fact that the boundary of the flowspace is naturally a universal circle for the foliation.

In this note, we give some conditions of solvability of a finite group $G$ based on its number of conjugacy classes of subgroups $Con(G)$. We classify all finite groups $G$ with $Con(G)\leq 5$. A new characterization of $A_5$ is also given.

We attach to a finite group $G$ and a structured payoff probe $\phi$ an integer payoff-difference lattice $M_\phi(G)$ and its conductor $C_\phi(G)$: the primes at which $M_\phi(G)$ loses rank modulo $p$. Our main result is an exact computation: for any CA-group the commuting conductor is rad$(b-1)$, where $b$ is the number of maximal abelian subgroups. In particular, conductor primes need not divide $|G|$: the prime $3$ occurs for a $2$-group of order $64$ with $b=7$. The commuting Smith spectrum is an invariant of the isoclinism class and obeys an exact direct-product law, giving ${\rm C_{comm}}(G\times H) = {\rm C_{comm}}(G) \cup {\rm C_{comm}}(H)$ unconditionally. A Galois-orbit-trace character probe reads a complementary layer: an index-$2$ subgroup forces $2\in {\rm C_{char}}(G)$ while no odd prime is forced, and ${\rm C_{comm}}(D_{2q}) = \{q\}$, ${\rm C_{char}}(D_{2q}) = \{2\}$ for all odd primes $q$. Certified exhaustive computation ($|G|\le128$ commuting, $|G|\le64$ character) and a deformation-family analysis support the general program: classify the Smith torsion of the compressed centralizer-type incidence matrix $B_G$.

We study the stability of solutions to the discrete-time contingent-claim problem over a finite investment horizon when uncertainty is modeled by random variables with finite discrete support. Our main contribution is to use Rockafellian perturbations as a framework for this stability analysis: we construct perturbations of the underlying probability distribution, of the contingent claim, and of both jointly, and we establish epi-convergence of the corresponding approximating Rockafellians for the primal problem. The associated hypo-convergent approximations yield stable dual problems which, in turn, imply convergence of the dual variables, interpreted as shadow prices. This analysis reveals a connection between the duality gap and the value of perfect information and it provides conditions under which strong duality holds. We also construct examples in which epi-convergence fails due to critical scenarios with vanishing probabilities but unbounded impacts, illustrating the boundary between well-behaved and ill-conditioned contingent-claim problems.

Rapidly scaling autonomous science is limited not only by algorithms, compute or data volume, but by which physical records a platform exposes before action. We formulate physically accessible decision-making (PADM) and a measurement-access risk frontier: the Bayes-optimal target risk minimized over records realizable under cost, bandwidth, latency, disturbance, memory and actuation constraints. The frontier gives a no-free-autonomy limit: automation cannot collapse decision uncertainty by computation alone; an optimal controller cannot remove target components absent from its record, and closing that gap requires expanded access, auditing, tolerated disturbance, slower operation or restricted deployment. In monitored feedback, displacement-only control remains exposed to a hidden switching force, whereas a finite-bandwidth cue recovers part of the missing projection before action. A chemistry-aware candidate-ranking audit with a 1000-target stress panel, Gaussian sensing, hidden-regime decisions and cost-aware/thermodynamic channel selection provide reproducible checks. PADM identifies target-specific audit value and residual oracle gaps before deployment.

We study products of independent Haar matrices interspersed with a fixed corank-one projection. As the matrix dimension $N$ and product length $L$ grow with $L/N\to\tau$, we prove that the empirical distribution of squared singular values converges to the free multiplicative Poisson law with $S$-transform $S(z)=\exp(\tau/(1+z))$, for both the unitary and orthogonal ensembles. Lagrange inversion yields its moments explicitly, proving Beenakker's conjectured Erlang formula for every moment order.

A square matrix is said to be Hessenberg whenever each entry below the subdiagonal is zero, and each entry on the subdiagonal is nonzero. A Hessenberg matrix is called circular whenever the top-right corner entry is nonzero, and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. In 2022, Jae-ho Lee conjectured that a circular Hessenberg pair satisfies two relations called the tridiagonal relations. In the present paper, we prove Lee's conjecture. Our proof is not elementary.

We study the controllability and reachability of sphere-constrained evolution equations under degenerate (low-mode) forcing, with the harmonic map heat flow as the principal application. Exploiting the underlying geometric structure, we reformulate the problem as an infinite-dimensional control-affine system in Fourier variables and analyze the Lie algebra generated by the controlled vector fields. We prove that iterated Lie brackets generate new admissible directions, providing a mechanism through which finitely many control modes propagate their influence across infinitely many Fourier components. The results provide a Lie-algebraic framework for controlling manifold-valued evolution equations.

Battery charging of Autonomous Mobile Robots (AMRs) in warehouses is a critical operational challenge that heavily impacts both order processing times and throughput. In this study, we address the dynamic AMR charging problem under stochastic order arrivals, where robots must learn optimal charging decisions. Traditional fixed-rule heuristics often prove suboptimal in dynamic environments and fail to account for multi-AMR coordination, leading to severe resource inefficiencies. To overcome these limitations, we propose a Proximal Policy Optimization (PPO)-based Deep Reinforcement Learning (DRL) framework designed for multi-block warehouses with fixed charging stations. Our model dynamically learns two key decisions: charging station selection and optimal charging duration, explicitly accounting for anticipated queuing times at the stations. Extensive numerical experiments benchmark the proposed model against state-of-the-art DRL and traditional heuristic approaches. Results demonstrate that our PPO framework increases order-completion rates by up to 6% compared to the strongest baseline, while significantly reducing the total time dedicated to recharging operations. Furthermore, we validate the model's robustness across diverse warehouse configurations and stochastic arrival rates. Finally, we interpret the learned DRL policy, offering valuable operational insights into its superiority over standard benchmarks.

We extend recent work on the relation between classical surface theory and partial differential equations, focusing on equations of pseudo-spherical type in the sense of Chern–Tenenblat and on a non-trivial generalization motivated by the Lund–Regge system describing surfaces immersed in $S^3$. As our main application, we study a generalized Konno–Oono system with three dependent variables introduced in a previous paper by one of the authors. We construct an associated parameter-dependent overdetermined linear problem and we establish the existence of infinitely many non-trivial local conservation laws, hence, integrability. The latter is the most technically demanding part of this paper: it requires a refined analysis of a Riccati pseudo-potential expansion, the use of stereographic coordinates at the full equation manifold level, the construction of special representatives, and a direct proof of non-triviality in horizontal cohomology. We also analyse an illustrative class of travelling wave solutions and show that they can be used to generate surfaces immersed in $S^3$ whose Gaussian curvature changes sign periodically, while their mean curvature are non-vanishing periodic functions. In a limit case, we obtain surfaces that are locally congruent to generalized Clifford tori.