Papers

We study Fano manifolds of Picard number one equipped with a locally flat VMRT structure whose VMRT at a general point is a smooth linearly non-degenerate complete intersection. We prove that such a manifold is biregular to the standard hyperquadric provided that one of two additional assumptions is satisfied. The proof uses equivariant compactifications of the vector group $\mathbb C^n$, Euler-symmetric projective varieties, fundamental forms, and special birational transformations.

We investigate the automorphism group of a non-normal affine toric surface. It is known that for a normal affine toric surface the connected component of the automorphism group is generated by the acting torus and root subgroups. While this fact is also true for some classes of non-normal affine toric surfaces, it fails in general. We give an example of a non-normal affine toric surface, whose singular locus is a point and the connected component of the automorphism group is not generated by the acting torus and root subgroups.

We study dependence measures between two discrete random variables. The measures vanish at independence and satisfy the data processing inequality (DPI): stochastic post-processing of one variable cannot increase dependence. We ask which such measures admit an exactly unbiased estimator from a fixed finite number of i.i.d. samples, a property equivalent to being polynomial in the joint probabilities. We distinguish fixed-alphabet DPI, which preserves the alphabet size of the variable being processed, from dimension-changing DPI. Under one-sided fixed-alphabet DPI, every measure with a continuously differentiable extension to a neighborhood of the closed probability simplex vanishes whenever one processed state has zero probability. If the processed alphabet is larger than the other alphabet, the measure is identically zero. If it has $n$ states and the other alphabet has at least $n$, every nonzero polynomial measure has degree at least $2n$ and requires at least $2n$ samples for unbiased estimation. Both bounds are attained by the Gram determinant. Under one-sided dimension-changing DPI, every family with the same boundary regularity is identically zero. These results yield sharp task bounds for multi-task peer prediction. For two agents with $n$ signals each, $2n$ tasks are necessary and sufficient for nontrivial expected payments that vanish for independent reports and cannot increase under memoryless stochastic garbling. Under these requirements, unequal signal-space sizes force the larger-alphabet agent's expected payment to vanish identically for any finite number of tasks.

In this article we give regularity results for the geodesic X-ray transform on nonsmooth simple manifolds. As an application, we improve previous injectivity results for the geodesic X-ray transform acting on $L^p$ functions. The results are based on symbol smoothing arguments for the normal operator and mapping properties of pseudodifferential operators with low regularity symbols.

Convex-concave saddle point problems with nonlinear coupling term have wide applications in signal processing, machine learning, robust optimization, and generative models, among others. Primal-dual algorithms are widely used for convex-concave saddle point problems. However, when handling nonlinear coupling term in convex-concave saddle point problems, primal-dual algorithms usually encounter geometric mismatches between variable and mapping spaces, delayed gradient information, and strong dependence on linesearch, resulting in less stable performance and complex convergence analysis. To address these issues, we propose a new primal-dual algorithm named PDAce by combining convex combination and extrapolation strategies. More specifically, in the update of the primal variable, we construct a convex combination point to replace the current iterative point and compute the Jacobian matrix of the vector function in the nonlinear coupling term at the convex combination point. Besides, we use the latest information of convex combination points to extend extrapolation to the mapping space in the update of the dual variable. The core innovations lie in performing linearization of the vector function in the nonlinear coupling term at convex combination points and shifting extrapolation from the variable space to the nonlinear mapping space. This design completely eliminates nonlinear residual terms and allows for rigorous convergence analysis without linesearch. Under mild convex assumptions, we construct a new Lyapunov potential function to prove that PDAce is globally convergent with an ergodic convergence rate of $\mathcal{O}(1/N)$. Moreover, we develop an accelerated version of PDAce, termed aPDAce, which achieves $\mathcal{O}(1/N^2)$ rate under strong convexity of the primal function, and linear convergence when both the primal and dual functions are strongly convex.

This thesis establishes a connection between the coarse homotopy theory of Euclidean cones and the shape theory of compact metric spaces. For a compact metric space $X$, the coarse homotopy groups of the Euclidean cone $cX$ are determined by pointed shape invariants of $X$, fitting into a $\varprojlim^1$ sequence involving the Čech homotopy groups. Using inverse mapping telescopes and strong shape theory, we prove that two compact metric spaces are strong shape equivalent if and only if their Euclidean cones are coarsely homotopy equivalent. We also prove a version of the coarse Whitehead theorem for Euclidean cones. In contrast, we derive a similar $\varprojlim^1$ sequence for arbitrary proper metric spaces, and construct a counterexample showing that the coarse Whitehead theorem fails in this general setting.

Sum-rank codes provide a generalized framework for Hamming and rank-metric codes, with codewords represented as tuples of matrices and weight given by the sum of the block ranks. In this paper, we introduce and study a block-tensor-rank invariant for sum-rank metric codes. To each code, we associate its block tensor rank: the smallest number of block-simple tensors, namely rank-one matrices supported inside single blocks, whose linear span contains the code. In general, determining the block tensor rank of a sum-rank code is challenging. Our main structural result shows that the block tensor rank decomposes additively across the blocks of the code, thereby reducing its computation to a tensor-rank problem on each block projection. Consequently, we derive two complementary lower bounds on the block tensor rank, referred to as the projection-wise bound and the coordinate-code bound. Moreover, by combining the coordinate-code bound with the classical Singleton and Griesmer bounds for codes in the Hamming metric, we obtain explicit lower bounds, called the Singleton coordinate-code bound and the Griesmer coordinate-code bound, respectively. We further construct families of sum-rank codes whose block tensor ranks attain the Singleton or Griesmer coordinate-code bounds. These constructions are based on Hamming-metric codes achieving the corresponding classical bounds. Finally, we show that, in certain cases, the block tensor ranks of two known families of sum-rank codes in the literature do not attain the Singleton coordinate-code bound.

We consider the transmission of data encoded into binary messages, with the goal of minimizing the Hamming distance, i.e., the number of bit-flips, between consecutive messages. This problem is relevant for enhancing the longevity of Non-Volatile Memories and reducing transition-induced energy consumption in data buses. Known as Write-Efficient Memory coding in the literature, this challenge has traditionally been addressed using optimal but complex schemes. In low-power computer systems the same topic is known as bus encoding. In this paper, we derive closed-form expressions to evaluate the average number of bit-flips for practical, sub-optimal encoding schemes, and propose two new schemes assisted by predefined random codebooks. We demonstrate that low-complexity solutions achieve performance very close to the optimal schemes, making them attractive for implementation in energy-sensitive and memory-critical applications. For instance, by adding 8 extra bits to 64-bits data, sub-optimal schemes can achieve a bit-flip reduction (related to energy saving) of approximately 24.7%, compared to the 26.4% reduction offered by the significantly more complex optimal scheme.

Kalman filters based on the Embedded Latent Transfer Operators (ELTO) emerge as novel statistical tools for sequential state estimation. However, a critical limitation stems from their use of simplified noise models, which fail to dynamically adapt to non-stationary processes. To address this limitation, we introduce an ELTO-based Bayesian filtering approach with a new structured parameterization for the filter's noise model. This parameterization enables structured noise adaptation, which couples the data-driven learning of an optimal time-invariant noise model with dynamic parameter adaptation that responds to changes in dynamics within non-stationary processes. Empirical results show that our structured noise adaptation improves the filter's dynamic state estimation performance in noisy, time-varying environments.

We propose and analyze a generalized compartment model for hacker dynamics in cybersecurity systems. This model is an extension of a recently introduced framework, replacing the bilinear interaction term with a broad class of nonlinear incidence functions. This provides greater modeling flexibility and allows for the description of a wider range of cyber-propagation and information-spreading processes. First, we investigate the qualitative dynamics of the model. We establish the positivity and boundedness of solutions, derive the basic reproduction number, and characterize the existence of hacker-free and hacker-present equilibria. We obtain local and global asymptotic stability results, yielding a complete description of the global dynamics in terms of the basic reproduction number. Next, we develop a second-order, positivity-preserving, nonstandard finite difference (NSFD) scheme for numerical simulations. Unlike many existing NSFD approaches, which are typically first-order accurate, the proposed method achieves second-order convergence while preserving positivity for arbitrary step sizes. Furthermore, the proposed method reproduces the asymptotic stability properties of the continuous model, making it suitable for long-time simulations. Numerical experiments validate the theoretical results and demonstrate the superior accuracy and qualitative performance of the proposed scheme compared to several first-order methods.

We determine all possible intersection arrays of completely regular codes of covering radius one in the generalized Petersen graphs \(GP(n,k)\), where \(n\geq 3\) and \(1\leq k<n/2\). In the equivalent language of perfect colorings, this amounts to enumerating all quotient matrices of perfect \(2\)-colorings, up to interchanging the two colors. Since \(GP(n,k)\) is cubic, there are only six possible nontrivial quotient matrices. For each of them, we give necessary and sufficient arithmetic conditions on \(n\) and \(k\) for its existence. The feasible cases are realized by explicit periodic colorings. The nonexistence part is obtained by reducing the local coloring conditions to cyclic systems of linear equations and applying a Fourier argument on roots of unity. Together with the previously known cases \(GP(n,2)\) and \(GP(n,3)\), the results give a complete arithmetic classification of quotient matrices, and hence of covering-radius-one completely regular code parameters, in the generalized Petersen family.

We present CORD-SLS, a real-time control method for safe deformable object manipulation, with a focus on ropes and cloth. At its core is a GPU-parallel differentiable simulator with contact smoothing which enables efficient gradient-based planning through intermittent contact. To robustly satisfy constraints under model and sensing uncertainty, we develop a real-time, GPU-parallel output-feedback robust model predictive control (MPC) algorithm that plans with this simulator. We further show that the simulator accelerates model-based RL for training neural manipulation policies. To improve real-world robustness, we use conformal prediction to calibrate visual-feedback and perception-error bounds for MPC, producing reachable tubes that enable high-probability safe control. We evaluate CORD-SLS on high-dimensional, contact-rich rope and cloth manipulation tasks in simulation and hardware, including obstacle avoidance, routing, folding, and smoothing. Across settings, CORD-SLS achieves millisecond-speed planning, exceeding baselines in safety, speed, and task success.

We construct stochastic thermodynamics of overdamped Langevin systems on nonrelaticvistic curved spaces with time-dependent metrics. The time dependence of the metric contributes to the energy balance by performing work on the kinetic energy, which is instantaneously dissipated as heat in the overdamped regime. This contribution makes our framework thermodynamically consistent so that entropy production satisfies the second law of thermodynamics. As a special case, when the metric evolves according to backward Ricci flow, the entropy balance exhibits a structure similar to Perelman's entropy functional. Our framework provides a way to quantify thermodynamic costs in dynamics on time-evolving spaces such as diffusion on membranes.

Lu proved that the set $$\mathcal{T}:=\{t\in\mathbb{Z}\setminus\{0\}:(q,\mathcal{D},tL)\text{ forms a Hadamard triple}\}$$ constitutes a spectral eigenvalue set for $\mu_{q,D}$, where $(q,\mathcal{D},L)$ is a Hadamard triple. And they prove for $s \in [0,\frac{\log \#\mathcal{D}}{\log q}]$, the corresponding spectra form a family of cardinality continuum. In this paper, we study Moran measures formed by random convolutions of finite Hadamard triples. Let $$\mu=\delta_{M_1^{-1}D_1}*\delta_{M_2^{-1}D_2}*\cdots, \qquad M_k=q_1q_2\cdots q_k,$$ where the factors are produced from finitely many triples $\{(N_j,B_j,L_j):1\le j\le m\}$, $(\omega_k)_{k=1}^{\infty}\in\{1,2,\ldots,m\}^{\mathbb N}$, and $n_k\in\mathbb N^+$, by setting $$ q_k=N_{\omega_k}^{n_k},\qquad D_k=B_{\omega_k},\qquad E_k=N_{\omega_k}^{n_k-1}L_{\omega_k}. $$ Assume a non-full-digit gap $$\rho:=\min_{1\le j\le m}\frac{N_j}{\#B_j}>1.$$ For the common Hadamard triple multiplier set $$\mathcal{T}_*:=\bigcap_{j=1}^m\{t\in\mathbb{Z}\setminus\{0\}:(N_j,B_j,tL_j)\text{ is a Hadamard triple}\},$$ Our main result is that, for every $$0\le s\le \kappa_\omega:=\limsup_{R\to\infty}\frac{\sum_{r=1}^R\log \#D_r}{\sum_{r=1}^R\log q_r},$$ there exist continuum many countable sets $\Lambda\subset\mathbb{Z}$ such that $t\Lambda$ is a spectrum of $\mu$ for every $t\in\mathcal{T}_*$ and $\dim_{Be}\Lambda=s$.

We give a unified analytic account of correlation emergence and the Epps effect in two coupled limit order books. The Epps effect is the empirical reduction in measured cross-asset log-return correlation at short aggregation scales, or equivalently the recovery of measured correlation as the aggregation interval increases. The model starts from a fixed-grid discrete random walk for order flow in operational time, with creation, cancellation and diffusion. A pair-trader coupling between the books is introduced at the level of order creation, and calendar time is imposed through separate observation clocks. We clarify how the operational-time model reduces to coupled reaction–diffusion equations with a moving reaction boundary defining the model log-mid-price. Using a regularised local-response representation of the coupling, we derive approximate closed-form expressions for realised correlations as a function of aggregation time. Here the Epps effect is shown to arise from two distinct mechanisms: asynchronous observation clocks (subordination), finite coupling response times, and their combination.

Physics-informed neural networks (PINNs) are meshless and carry moving geometry and topology change through resampling of collocation points; the finite-element method (FEM) is the workhorse for boundary-fitted discretisations. Coupling the two across a shared interface promises the best of both, yet existing PINN-FEM schemes are validated only empirically. We put the coupling on a domain-decomposition footing: viewing each solver as a Steklov-Poincaré (trace-to-flux) operator, we transfer the classical Dirichlet-Neumann (DN) divergence diagnosis and its Robin-Neumann (RN) cure, including a closed-form, sweep-free interface impedance, and prove a PINN-specific contraction theorem: a trained network realises only a perturbed Steklov operator with a per-step training residual, and RN still contracts, with no shared-eigenbasis hypothesis, to a floor set by the achieved training loss. Because a PINN has no stiffness matrix, we introduce a Fourier-mode interface probe that recovers the network's resolvable Steklov eigenvalues to within 0.5% and doubles as a diagnostic of the network's spectral cap. The theory predicts measured PINN-FEM contraction rates to within 7% on 1D and 2D Poisson couplings, and a two-slab analogue of the large-added-mass regime shows RN's per-mode impedance matching winning decisively where tuned scalar relaxation saturates. We demonstrate the framework on a Stokes/rigid-disc problem with Alart-Curnier contact: the meshless PINN fluid absorbs the topology change at contact by collocation exclusion alone, no remeshing and no cut cells, and the static-equilibrium contact reaction matches the submerged weight to 0.4% under mesh refinement. We quantify remaining limitations: the warm-started PINN drifts off the Stokes manifold over long horizons, and matched FEM-FEM benchmarks attribute pre-impact squeeze-film signatures to PINN under-resolution.

In this work, by construct an asymptotic coupling by reflection, we first explore the uniform-in-time estimate on probability distance for two measure-valued processes induced by a McKean-Vlasov SDE with common noise and an interacting particle system, where the drift terms are dissipative merely in the long distance. As direct applications of this estimate, we establish the uniform-in-time error estimates for the numerical solutions derived via backward/tamed/adaptive Euler-Maruyama methods. Moreover, as another direct application, the uniform-in-time conditional propagation of chaos is quantified.

The classical Corrádi-Hajnal theorem states that for any multiple $n$ of $3$, if $G$ is a graph with $n$ vertices and $\delta(G) \geq 2n/3$, then $G$ can be partitioned into $n/3$ vertex-disjoint copies of the triangle [Acta Math. Acad. Sci. Hung., 14:423-439, 1964]. Balogh, Molla and Sharifzadeh obtained a smaller lower bound by adding the independence number condition [Random Struct. Algorithms, 49:669-693, 2016]. In this paper, we study perfect tilings in digraphs subject to conditions on the independence number and the degree. The independence number, $\alpha(D)$, of $D$ is the maximum integer $k$ such that $D$ has an independent set of cardinality $k$. We show that if $D$ is an $n$-vertex digraph with $\alpha(D)\leq o(1)n$ and $\delta(D) \geq (1+o(1))n$, then $D$ has a perfect $T_3$-tiling, where $T_3$ denotes a transitive triangle. This minimum degree condition is asymptotically best possible. Moreover, our result implies the theorem of Balogh, Molla, and Sharifzadeh concerning perfect triangle tilings.

This paper investigates the nonlinear asymptotic stability and optimal decay rates of entropy waves for the Landau equation with physically realistic Coulomb interactions under general perturbations. We consider the infinite channel domain $\mathbb{R} \times \mathbb{T}^2$ in three dimensions, which possesses both one-dimensional and high-dimensional characteristics, thereby posing two primary analytical challenges: (i) for the one-dimensional Landau equation with Coulomb potentials, the absence of a spectral gap in the linearized operator has obstructed the derivation of wave pattern stability results with explicit time decay rates; (ii) in the study of contact discontinuities, the multidimensional case fundamentally differs from the one-dimensional setting due to lack of a key structural condition. We develop effective analytical approaches to treat those difficulties. To overcome the weak dissipation caused by the spectral gap deficiency, we implement a time-velocity interpolation technique to enhance dissipation and simultaneously construct coupled diffusion waves to compensate for the loss of time decay. To address the missing structural condition in higher dimensions, a novel transformation is introduced to recover the two-sided structural condition within the perturbation system. By developing a derivative-level transformation and a refined energy framework, we restore the necessary structural condition for derivatives, establish the optimal decay of the solution, and prove the stretched exponential decay of its non-zero modes. In contrast to previous methods that rely on artificial viscosity or the Navier–Stokes approximation, our approach directly leverages the intrinsic physical dissipation of the equation and its coupling with the microscopic kinetic component, ensuring broader applicability.

We prove that many virtual properties are closed under taking graph products, including: virtually RFRS, virtually (compact) special, virtually CAT(0) cube, and virtually normally poly-free. Our proof uses Januszkiewicz and Świątkowski's strong commensurability theorem for graph products, for which we provide an elementary proof.

Let $(M,g)$ be a $C^{\infty}$ smooth compact connected Riemannian manifold with boundary $\partial M$. Consider the Dirichlet problem: $\Delta_gu=0,\ u|_{\partial M}=f$ for $f\in C^{\infty}(\partial M)$. The Dirichlet-to-Neumann (DN) operator $\Lambda_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $\Lambda_gf=\left.\frac{\partial u}{\partial\nu}\right|_{\partial M}$, where $\nu$ is the unit outer normal to the boundary and $u$ is the unique solution to the Dirichlet problem. Let $g_\partial$ be the Riemannian metric on $\partial M$ induced by $g$. The Calderón problem is as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_\partial,\Lambda_g)$? In the two-dimensional case the surface $(M,g)$ is determined by the DN data uniquely up to conformal equivalence. Knowledge of the DN data is equivalent to knowledge of the Hilbert transform ${\mathcal H}_\Omega:C^\infty(\Gamma)\to C^\infty(\Gamma)$ on the boundary curve $\Gamma=\partial\Omega$ of a planar domain $\Omega$. We study properties of the Hilbert transform. In particular, we obtain an integral formula for ${\mathcal H}_\Omega$ for a simply connected $\Omega$ which generalizes the classical integral formula for the Hilbert transform on the unit circle. This formula is the base of our algorithm for reconstructing a simply connected planar domain from the DN data. Several numerical reconstructions are presented.

Valid treatment effect inference in survival studies is fundamental yet challenging when the treatment assignments and outcomes are confounded by many baseline covariates. To this end, in this paper we propose a high-dimensional survival causal inference (HSCI) framework that delivers valid inference under a sparse high-dimensional Cox proportional hazards outcome model and a high-dimensional logistic propensity score working model. To mitigate the nuisance estimation bias, we develop a Neyman near-orthogonal score for the treatment effect and implement it with cross-fitting. Under doubly robust nuisance-rate conditions, we establish the root-n asymptotic normality and consistent variance estimation. We also extend the framework to inference on high-dimensional survival covariate effects. Simulation examples confirm that HSCI reduces sharply the bias relative to the regularized Cox estimators and maintains valid confidence interval coverage across different dimensionality, censoring, and misspecified propensity-model settings. An application to diffuse large-B-cell lymphoma data further showcases its value for high-dimensional biomedical survival studies.

Designing rapid transportation routes requires balancing efficiency and reachability. Shortest-path models ensure direct, cost-efficient routes but ignore coverage, while centrality-based approaches maximize accessibility but do not enforce operational constraints. We study the problem of selecting a shortest path that maximizes reachability, measured as the number of nodes within a fixed distance of the path. To do this, we introduce the $k$-Step-Central Shortest Path problem and analyse its structural properties. We show that optimal solutions on unweighted graphs can be found in polynomial time and propose an algorithm with a novel pruning rule. We also prove that the problem becomes NP-hard when edge weights are introduced. Additionally, we show that our algorithm can be used to solve the NP-hard problem of finding the closeness-central shortest path in a graph. We demonstrate the efficiency and scalability of our algorithm on synthetic and real-world networks with up to 2,000 nodes. Our results show that improving reachability can substitute for route expansion: increasing the reach of transit lines drastically increases their coverage with shorter routes. This suggests that investments in active transport infrastructure that improve reachability can be more effective than extending primary routes, providing a data-driven basis for allocating resources in network design.

Let $G$ be a connected affine algebraic group defined over a field of positive characteristic. We prove that the action of $G$ on a smooth projective variety can be lifted to its associated Artin-Schreier root stacks, whenever $G$ has no non-trivial characters. The existence of a $G$-linearization on a certain tautological invertible sheaf on such Artin-Schreier root stacks is also shown.

We consider the existence of vortex solutions to the complex sine-Gordon II (CSG2) equation, which can be viewed as an analogy of the Ginzburg-Landau (GL) equation. Using the nontrivial kernels $\eta_{\pm}$ of the linearized CSG2 equation at the standard degree-2 vortex solution $\Psi_2$, we show that it bifurcating to a one-parameter family of symmetry-breaking solutions $\Psi_{2,\alpha}$. Explicit formulas of these solutions are also available, from which we propose a new bilinear system for this equation. Our method can be generalized to higher degree case. Nondegenracy and stability of degree-1 solution are also proved. Finally, we formally discuss the Lyapunov-Schmidt reduction procedure for the multivortex solutions of the CSG2 and GL equation.

In this paper, we introduce a convolution based short time offset linear canonical transform (STOLCT) and investigate its fundamental mathematical properties. Specifically, we establish its continuity, orthogonality relations, inversion formulas, range theorem, and convolution theorem. We further explore several important applications of STOLCT, including the Poisson summation formula, the Paley Wiener criterion, and a sampling theorem. In addition, numerical simulations and graphical analyses are presented to compare signal reconstruction performance under different scenarios. A comparative study between STOLCT and STLCT is conducted with respect to their reconstruction formulas, demonstrating the effectiveness and potential advantages of the proposed transform.

Let $n\geq 2$, and let $\mathcal{S}(1,n)$ be the Lie superalgebra of zero-superdivergence superderivations of $\mathbb{C}[t^{\pm1}]\otimes\Lambda(n)$. Its derived algebra $\mathcal{S}^\prime(1,n):=[\mathcal{S}(1,n),\mathcal{S}(1,n)]$ is well known as a superconformal algebra. In this paper, we first study Shen-Larsson modules over $\mathcal{S}^\prime(1,n)$. These modules, introduced by G. Shen and T. A. Larsson, are constructed from modules over the Weyl superalgebra $K_{1,n}$ and the special linear Lie superalgebra $\mathfrak{sl}(1,n)$. We establish necessary and sufficient conditions for the simplicity of Shen-Larsson modules and investigate their simple subquotients in the non-simple case. Then as an application, building on the classification of simple cuspidal $\mathcal{S}^\prime(1,n)$-modules by C. Martínez, O. Mathieu and E. Zelmanov, we obtain an explicit construction of all simple cuspidal modules over $\mathcal{S}^\prime(1,n)$.

Let $G = (V,E)$ be an edge-colored graph, and let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$. We show that for a sufficiently large $n$ if $G$ is an edge-colored triangle-free graph of order $n$ that satisfies $\delta^c(G)\geq (n+7)/5$, then $G$ contains a rainbow cycle of length four, which improves a bound of Ding et al. and is best possible. In addition, we show that given $k$, there is $n_0$ such that for $n\geq n_0$, if $G$ is an edge-colored triangle-free graph with $\delta^c(G)> n/5+3$, then $G$ contains a rainbow cycle of length $4k$.

We study the sample complexity of learning in average-reward weakly-coupled Markov decision processes (WCMDPs) and Restless Bandits (RBs) under a generative model. Naive reduction to a tabular MDP leads to high complexity bounds as the state-action space is exponentially large in the number of arms $N$. By exploiting the weakly coupled structure, we show that near-optimal policies can be learned with sample and computational complexities that are polynomial in $N$. Specifically, we analyze the plug-in approach, which applies an efficient planning algorithm to an empirical model estimated from data. For fully heterogeneous WCMDPs, we establish the first finite-sample PAC guarantee with polynomial complexity and an $O(1/\sqrt{N})$ optimality gap. For homogeneous RBs, we further prove that a smaller optimality gap is achievable under mild structural assumptions. A primary technical contribution is a novel Lyapunov-based analysis framework. Unlike classical approaches that rely on the difficult-to-control bias function, our framework uses an explicitly constructed Lyapunov function along with a drift transfer technique between the true and empirical models. A key step of independent interest in our framework is a fine-grained perturbation analysis for the underlying linear programming (LP) relaxation, which provides a general tool for analyzing LP-based policies and weakly-coupled systems.

In hierarchical organizations, authenticating data from multiple users can be complex and resource-intensive. Hierarchical Identity-Based Signature with Designated Aggregator (HIBS-DA) provides an efficient solution by allowing users at different levels to generate signatures that can be combined into a single, compact signature. We first introduce the HIBS-DA framework and present the {\em{first}} lattice-based construction of HIBS-DA. Our scheme allows users at different hierarchical levels to generate individual signatures that can be aggregated into a single, compact signature, reducing communication and verification costs. The proposed construction is secure, correct, and resistant to forgery, making it suitable for large-scale environments such as universities, corporations, and government agencies.