We develop a unified framework for automatic debiased machine learning (autoDML) for inference on a broad class of statistical parameters. The framework applies to any smooth functional of a nonparametric M-estimand, defined as the minimizer of a population risk over an infinite-dimensional linear space. Examples include counterfactual regression, quantile, and survival functions, as well as conditional average treatment effects. Rather than requiring manual derivation of influence functions, our approach automates the construction of debiased estimators using three ingredients: the gradient and Hessian of the loss function and a linear approximation of the target functional. Estimation reduces to solving two risk minimization problems, one for the M-estimand and one for a Riesz representer. The framework accommodates Neyman-orthogonal loss functions that depend on nuisance parameters and extends to vector-valued M-estimands through joint risk minimization. We characterize the efficient influence function and construct efficient autoDML estimators via one-step correction, targeted minimum loss estimation, and sieve-based plug-in methods. Under quadratic risk, these estimators satisfy double robustness for linear functionals. We further show that they are robust to mild misspecification of the M-estimand model, incurring only second-order bias. We illustrate the method by estimating long-term survival probabilities under a semiparametric two-parameter beta-geometric failure model.

Conventional double-spending attack models ignore the revenue losses stemming from the orphan blocks. On the other hand, selfish mining literature usually ignores the chance of the attacker to double-spend at no-cost in each attack cycle. In this paper, we give a rigorous stochastic analysis of an attack where the goal of the adversary is to double-spend while mining selfishly. To do so, we first combine stubborn and selfish mining attacks, \textit{i.e.}, construct a strategy where the attacker acts stubborn until its private branch reaches a certain length and then switches to act selfish. We provide the optimal stubbornness for each parameter regime. Next, we provide the maximum stubbornness that is still more profitable than honest mining and argue a connection between the level of stubbornness and the $k$-confirmation rule. We show that, at each attack cycle, if the level of stubbornness is higher than $k$, the adversary gets a free shot at double-spending. At each cycle, for a given stubbornness level, we rigorously formulate how great the probability of double-spending is. We further modify the attack in the stubborn regime in order to conceal the attack and increase the double-spending probability.

Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on Følner sequences for amenable groups, entropy defined using a random ordering of the group, and Bowen's sofic entropy for sofic groups. In this work we pursue these notions across an analogy between ergodic theory and representation theory. We arrive at new quantities associated to unitary representations of groups and representations of other C*-algebras. Our main results show that these new quantities can often be evaluated as Fuglede--Kadison determinants. The resulting determinantal formulas offer various non-commutative generalizations of Szegő's limit theorem for Toeplitz determinants. They make contact with Arveson's theory of subdiagonal subalgebras, and also with some entropy formulas in the ergodic theory of actions by automorphisms of compact Abelian groups.

Voting is a cornerstone of collective participatory decision-making in contexts ranging from political elections to decentralized autonomous organizations (DAOs). Despite the proliferation of internet voting protocols promising enhanced accessibility and efficiency, their evaluation and comparison are complicated by a lack of standardized criteria and unified definitions of security and maturity. Furthermore, socio-technical requirements by decision makers are not structurally taken into consideration when comparing internet voting systems. This paper addresses this gap by introducing a trust-centric maturity scoring framework to quantify the security and maturity of seventeen internet voting systems. A comprehensive trust model analysis is conducted for selected internet voting protocols, examining their security properties, trust assumptions, technical complexity, and practical usability. In this paper we propose the Internet Voting Maturity Framework (IVMF) which supports nuanced assessment that reflects real-world deployment concerns and aids decision-makers in selecting appropriate systems tailored to their specific use-case requirements. The framework is general enough to be applied to other systems, where the aspects of decentralization, trust, and security are crucial, such as digital identity, Ethereum layer-two scaling solutions, and federated data infrastructures. Its objective is to provide an extendable toolkit for policy makers and technology experts alike that normalizes technical and non-technical requirements on a univariate scale.

We show that perfect revivals of Rabi oscillations are possible, under certain conditions, in the population inversion of a trapped ion. Based on this property, we find that Schrödinger cat states of the atomic motion are naturally generated by the unitary dynamics. Using a pair of symmetric and antisymmetric Schrödinger cat states of the motion, together with an electronic excited or ground state, we find that the interaction leads to four orthonormal maximally entangled states of the two partitions, which we identify as Bell states. We also study a quadratic Kerr-type evolution that is possible for short interaction times.

The inherent trade-off between ultra-low thermal conductivity and high mechanical rigidity in natural materials limits their utility in advanced applications. Inspired by the unique architecture of layered honeycomb structures, this study introduces a new class of metamaterials designed to overcome these constraints. By systematically exploring unit cell configurations and stacking arrangements, we demonstrate that a zigzag internal geometry, analogous to rhombohedral graphene stacking, optimizes thermal insulation while maintaining relatively high mechanical rigidity. Our finite element simulations predict that these layered structures can achieve a thermal conductivity of 12.5 mW/(m.K) using zirconia as the constructing material, theoretically outperforming state-of-the-art ceramic aerogels while maintaining robust mechanical stability. This novel approach paves the way for designing next-generation super-insulating materials with customizable mechanical properties, enabling innovative applications in extreme environments, lightweight aerospace structures, and advanced thermal management systems.

In this article, we examine the Standard Model extended by a $Y=0$ real Higgs triplet, the $Δ$SM. It contains a $CP$-even neutral Higgs ($Δ^0$) and two charged Higgs bosons ($Δ^\pm$), which are quasi-degenerate in mass. We first study the theoretical constraints from vacuum stability and perturbative unitarity and then calculate the Higgs decays, including the loop-induced modes such as di-photons ($γγ$) and $Zγ$. In the limit of a small mixing between the SM Higgs and $Δ^0$, the latter decays dominantly to $WW$ and can have a sizable branching ratio to di-photon. The model predicts a positive definite shift in the $W$ mass, which agrees with the current global electroweak fit. At the Large Hadron Collider, it leads to a $(i)$ stau-like signature from $pp\to Δ^+Δ^-\to τ^+τ^-ν\barν$, $(ii)$ multi-lepton final states from $pp\to γ^*\to Δ^+Δ^-\to W^+W^-ZZ$ and $pp\to W^{*} \to Δ^\pmΔ^0\to W^\pm Z W^+W^-$ as well as $(iii)$ associated di-photon production from $pp\to W^{*} \to Δ^\pm(Δ^0\toγγ)$. Concerning $(i)$, the reinterpretation of the recent supersymmetric tau partner search by ATLAS and CMS excludes $m_{Δ^\pm}<110$ GeV at 95% CL. From $(ii)$, some of the signal regions of multi-lepton searches lead to bounds close to the predicted cross-section, but electroweak scale masses are still allowed. For $(iii)$, the recast of the associated di-photon searches by ATLAS and a combined log-likelihood fit of signal and background to data find that out of the 25 signal regions, 10 provide relevant limits on Br$(Δ^0\toγγ)$ at the per cent level. Interestingly, 6 signal regions show excesses at around 152 GeV, leading to a preference for a non-zero di-photon branching ratio of about 0.7% with the corresponding significance amounting to about $4σ$.

We review 2d CFT in the bootstrap approach, and sketch the known exactly solvable CFTs with no extended chiral symmetry: Liouville theory, (generalized) minimal models, limits thereof, and loop CFTs, including the $O(n)$, Potts and $PSU(n)$ CFTs. Exact solvability relies on local conformal symmetry, and on the existence of degenerate fields. We show how these assumptions constrain the spectrum and correlation functions. We discuss how crossing symmetry equations can be solved analytically and/or numerically, leading to analytic expressions for structure constants in terms of the double Gamma function. In the case of loop CFTs, we sketch the corresponding statistical models, and derive the relation between statistical and CFT variables. We review the resulting combinatorial description of correlation functions, and discuss what remains to be done for solving the CFTs.

This paper is concerned with one-dimensional 2 x 2 systems of conservation laws with a flux f=f(x, U) that is discontinuous with respect to the spatial variable. No monotonicity assumption is imposed on the mapping x \to f(x,U). We introduce a Kruzhkov-type entropy condition and establish the global existence of entropy solutions for large data. The proof relies on a wave-front tracking approximation. The main technical novelty consists in the introduction of adapted Riemann invariant coordinates, specifically designed to account for the discontinuities of the flux, which yield a uniform-in-time bound on the total variation of the approximate solutions U^n(t). We also outline several alternative approaches that may lead to existence results under possibly weaker assumptions. As an application, we propose second-order vehicular traffic models on inhomogeneous roads featuring abrupt ''collective'' changes in the speed law or road capacity.

We prove sphere packing density bounds in hyperbolic space (and more generally irreducible symmetric spaces of noncompact type), which were conjectured by Cohn and Zhao and generalize Euclidean bounds by Cohn and Elkies. We work within the Bowen-Radin framework of packing density and replace the use of the Poisson summation formula in the proof of the Euclidean bound by Cohn and Elkies with an analogous formula arising from methods used in the theory of mathematical quasicrystals.

Transverse electron transport in magnetic materials - manifested in effects such as the anomalous Hall and Nernst effects - holds promise for spintronic and thermoelectric applications. While recent advances have focused on enhancing such transport through topological single crystals via intrinsic mechanisms linked to Berry curvature, practical limitations remain due to their mechanical fragility and narrow material scope. Here, we demonstrate a distinct approach for transverse transport enhancement based on composite formation. Using both theoretical modeling and experiments, we show that disordered mixtures of two ferromagnetic materials can exhibit significantly stronger transverse electron deflection than either constituent alone. This enhancement originates from meandering electron pathways created by the disordered mixture of two materials and does not rely on long-range crystalline order. The identified requirements for this mechanism can be broadly satisfied across different material systems, offering a universal and tunable strategy to engineer large transverse responses in structurally robust platforms.

We consider the problem of estimating assortment probabilities, which is common in operations management applications, including product bundling, advertising, etc. Existing approaches typically model each assortment as a category and apply multinomial models to estimate the choice probabilities; while computationally convenient, these methods do not exploit independence structures in the joint distribution and may therefore be statistically inefficient when the total number of items is large. Using the representation from Bahadur (1959), we relate the sparsity of the generalized correlation coefficients to the independence structure of the binary components. We formulate the problem as estimating a high-dimensional vector of generalized correlation coefficients, together with low or moderate-dimensional nuisance parameters corresponding to the marginal probabilities. We develop a regularized adversarial estimator that attains the optimal rate under standard regularity conditions while remaining computationally feasible. The framework naturally extends to settings with covariates. We apply the proposed estimators to causal inference with multiple binary treatments and show substantial finite-sample improvements over non-adaptive methods. Numerical studies corroborate the theoretical results.

The properties of alloys that undergo to chemical order-disorder transformations depend heavily on the degree of ordering in the crystal lattice. In the literature, it is well established that the ordering in a magnetic alloy such as Fe-rich Fe_xAl_1-x (x>0.5) leads to reducing its magnetization and even to a transition from the ferromagnetic (FM) to paramagnetic (PM) state at x<0.7. Studying the ordering kinetics in thin (50 nm) Fe_xAl_1-x films with a non-stoichiometric composition (0.5<x<0.7), we demonstrate the opposite behavior: When the alloy is aged at a high temperature Ta>600 °C, the ordering process is accompanied by an increase in magnetization and related properties. For example, we find the further enhancement of the giant anomalous Hall (AH) effect found recently in Fe_xAl_1-x alloys. Based on both experimental data and theoretical modeling, we argue that these properties are enhanced due to the nucleation and growth of the B2-Fe_0.5Al_0.5 phase. Growing B2 nanocrystals enable segregation and clustering of excess Fe in the alloy. It has been revealed that the PM phase, which is formed in the aged samples and contains Fe-enriched superparamagnetic clusters, contributes to the AH resistivity even more than the FM phase in the as-grown sample. Our findings open a route for improving the properties of functional alloys.

In this paper we propose an end-to-end algorithm for indirect data-driven control for bilinear systems with stability guarantees. We consider the case where the collected i.i.d. data is affected by probabilistic noise with possibly unbounded support and leverage tools from statistical learning theory to derive finite sample identification error bounds. To this end, we solve the bilinear identification problem by solving a set of linear and affine identification problems, by a particular choice of a control input during the data collection phase. We provide a priori as well as data-dependent finite sample identification error bounds on the individual matrices as well as ellipsoidal bounds, both of which are structurally suitable for control. Further, we integrate the structure of the derived identification error bounds in a robust controller design to obtain an exponentially stable closed-loop. By means of an extensive numerical study we showcase the interplay between the controller design and the derived identification error bounds. Moreover, we note appealing connections of our results to indirect data-driven control of general nonlinear systems through Koopman operator theory and discuss how our results may be applied in this setup.

The ubiquity of top-rich final states in the context of beyond the Standard Model (BSM) searches has led to their status as extensively studied signatures at the LHC. Over the past decade, numerous endeavours have been undertaken in the literature to develop methods for efficiently distinguishing boosted top quark jets from QCD jets. Although cut-based strategies for boosted top tagging, which rely on substructure information from fat jets resulting from the hadronic decay of boosted top quarks, were introduced in the literature as early as 2008, recent years have witnessed a surge in the utilization of machine learning-based approaches for the classification of top-jets from QCD jets. The review focuses on the present status of boosted top tagging and its application for BSM searchers.

Given a sequence \( S = (s_1, s_2, \ldots, s_k) \) of positive integers satisfying \( s_1 \leq s_2 \leq \dots \leq s_k \), an \( S \)-packing coloring of a graph \( G \) is a partition of \( V(G) \) into \( k \) subsets \( V_1, V_2, \dots, V_k \) such that, for each \( 1 \leq i \leq k \), the distance between any two distinct vertices \( x, y \in V_i \) is at least \( s_i + 1 \). Yang and Wu established that every $3$-irregular subcubic graph admits a \( (1,1,3) \)-packing coloring. Later, Mortada and Togni introduced the concept of an \( i \)-saturated subcubic graph, defined as a subcubic graph in which every vertex of degree three has at most \( i \) neighbors of degree three for \( 0 \leq i \leq 3 \). They further demonstrated that all $1$-saturated subcubic graphs are \( (1,1,2) \)-packing colorable. In this paper, we present new concise proofs of these results using a novel tool.

It is well known that there is no direct one-to-one relation between $p$-values and likelihood ratios or Bayes factors, since their relation crucially involves the sample size $n$. We investigate their (asymptotic) relation in a coin-tossing context where the hypotheses of interest address the success probability of the coin, and where detailed computations are possible. This leads to useful insights in the nature of $p$-values and likelihood ratios. Our results imply, for instance, that under mild conditions, a $p$-value of 0.05 cannot correspond to a likelihood ratio larger than 7.5, for any hypothesis versus a null hypothesis that the success probability has a specific value. We also show it is unlikely one can obtain a large likelihood ratio by tossing a fair coin until the number of heads deviates from the mean by several standard deviations.

In this paper we classify all singular irreducible symplectic surfaces, i.e., compact, connected complex surfaces with canonical singularities that have a holomorphic symplectic form $σ$ on the smooth locus, and for which every finite quasi-étale covering has the algebra of reflexive forms spanned by the reflexive pull-back of $σ$. We moreover prove that the Hilbert scheme of two points on such a surface $X$ is an irreducible symplectic variety, at least in the case where the smooth locus of $X$ is simply connected.

We use the concept of electric potential energy from physics, the mathematical field of graph theory, and the notion of majorization to study maximal evenness in a broader mathematical context than what was previously possible, so that we can go beyond the well-known one-dimensional maximally even sets into higher dimensional and more geometrically complex territory. We investigate musical connections between certain generalizations of maximally even sets, one of the oldest Puerto Rican musical traditions of African origin called bomba, and with certain scales ranging from the familiar to the esoteric.

We consider maximal operators acting on vector valued functions, that is, functions taking values on $\mathbb{C}^d,$ that incorporate matrix weights in their definitions. We show vector valued estimates, in the sense of Fefferman--Stein inequalities, for such operators. These are proven using an extrapolation result for convex body valued functions due to Bownik and Cruz-Uribe. Finally, we show an $\mathrm{H}^1$-$\mathrm{BMO}$ duality for matrix valued functions and we apply the previous vector valued estimates to show upper bounds for biparameter paraproducts. For the reader's convenience, we include an appendix explaining how to adapt the extrapolation for real convex body valued functions of Bownik and Cruz-Uribe to the setting of complex convex body valued functions that we treat.

Reformulating the incompressible Stokes equations with the velocity sought in H(curl) has recently emerged as a promising approach for the design of helicity-preserving schemes in magnetohydrodynamics and pressure-robust finite element methods on polygonal meshes. A key challenge in this setting, however, is the treatment of Navier slip boundary conditions. In this paper, we overcome this difficulty by recasting the slip condition as a Robin boundary condition and proving well-posedness of the resulting continuous problem. We further identify the geometric and regularity assumptions on the domain and the exact solution under which the classical Stokes solution is recovered. Finally, we study a conforming finite element Galerkin discretization, establishing stability and a priori error estimates. Numerical experiments validate the optimal convergence rates predicted by the theory.

We design Snowflake, a quantum error correction decoder that, for the surface code under circuit-level noise, is roughly 25% more accurate than the Union-Find decoder, with a better mean runtime scaling: subquadratic as opposed to cubic in the code distance. Our decoder runs in a streaming fashion and has a distributed, local implementation. In designing Snowflake, we propose a new method for general stream decoding that eliminates the processing overhead due to window overlap in existing windowing methods.

Post-clustering inference in single-cell RNA sequencing (scRNA-seq) analysis presents significant challenges in controlling Type I error during differential expression analysis. Data fission, a promising approach that aims to split data into two independent parts, relies on strong parametric assumptions of non-mixture distributions that are inherently violated in clustered data. To address this limitation, we introduce conditional data fission, an extension designed to decompose each mixture component into two independent parts. However, we demonstrate that applying such conditional data fission to mixture distributions requires prior knowledge of the clustering structure to ensure valid post-clustering inference. This arises from the need to accurately estimate component-specific scale parameters, which are critical for performing decomposition while maintaining independence. We theoretically quantify how biases in estimating these parameters lead to inflated Type I error rates due to deviations from independence. Given that mixture components are typically unknown in practice, our results underscore the fundamental difficulty of applying data fission in real-world settings, despite its prior proposal as a solution for post-clustering inference.

We give improved approximations for two metric Traveling Salesman Problem (TSP) variants. In Ordered TSP (OTSP) we are given a linear ordering on a subset of nodes $o_1, \ldots, o_k$. The TSP solution must have that $o_{i+1}$ is visited at some point after $o_i$ for each $1 \leq i < k$. This is the special case of Precedence-Constrained TSP ($PTSP$) in which the precedence constraints are given by a single chain on a subset of nodes. In $k$-Person TSP Path (k-TSPP), we are given pairs of nodes $(s_1,t_1), \ldots, (s_k,t_k)$. The goal is to find an $s_i$-$t_i$ path with minimum total cost such that every node is visited by at least one path. We obtain a $3/2 + e^{-1} < 1.878$ approximation for OTSP, the first improvement over a trivial $α+1$ approximation where $α$ is the current best TSP approximation. We also obtain a $1 + 2 \cdot e^{-1/2} < 2.214$ approximation for k-TSPP, the first improvement over a trivial $3$-approximation. These algorithms both use an adaptation of the Bridge Lemma that was initially used to obtain improved Steiner Tree approximations [Byrka et al., 2013]. Roughly speaking, our variant states that the cost of a cheapest forest rooted at a given set of terminal nodes will decrease by a substantial amount if we randomly sample a set of non-terminal nodes to also become terminals such provided each non-terminal has a constant probability of being sampled. We believe this view of the Bridge Lemma will find further use for improved vehicle routing approximations beyond this paper.

We prove the non-degeneracy of solutions to a fractional and singular Liouville equation defined on the whole real line in presence of a singular term. We use conformal transformations to rewrite the linearized equation as a Steklov eigenvalue problem posed in a bounded domain, which is defined either by an intersection or a union of two disks. We conclude by proving the simplicity of the corresponding eigenvalue.

Let $spex(n,H_{minor})$ denote the maximum spectral radius of $n$-vertex $H$-minor free graphs. The problem on determining this extremal value can be dated back to the early 1990s. Up to now, it has been solved for $n$ sufficiently large and some special minors, such as $\{K_{2,3},K_4\}$, $\{K_{3,3},K_5\}$, $K_r$ and $K_{s,t}$. In this paper, we find some unified phenomena on general minors. Every graph $G$ on $n$ vertices with spectral radius $ρ\geq spex(n,H_{minor})$ contains either an $H$ minor or a spanning book $K_{γ_H}\nabla(n-γ_H)K_1$, where $γ_H=|H|-α(H)-1$. Furthermore, assume that $G$ is $H$-minor free and $Γ^*_s(H)$ is the family of $s$-vertex irreducible induced subgraphs of $H$, then $G$ minus its $γ_H$ dominating vertices is $Γ^*_{α(H)+1}(H)$-minor saturate, and it is further edge-maximal if $Γ^*_{α(H)+1}(H)$ is a connected family. As applications, we obtain some known results on minors mentioned above. We also determine the extremal values for some other minors, such as flowers, wheels, generalized books and complete multi-partite graphs. Our results extend some conjectures on planar graphs, outer-planar graphs and $K_{s,t}$-minor free graphs. To obtain the results, we combine stability method, spectral techniques and structural analyses. Especially, we give an exploration of using absorbing method in spectral extremal problems.

For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Note that if $H$ is the triangle, then $H$-colorings are equivalent to $3$-colorings. In this paper we are interested in the case that $H$ is the five-vertex cycle $C_5$. A minimal obstruction to $C_5$-coloring is a graph that does not have a $C_5$-coloring, but every proper induced subgraph thereof has a $C_5$-coloring. In this paper we are interested in minimal obstructions to $C_5$-coloring in $F$-free graphs, i.e., graphs that exclude some fixed graph $F$ as an induced subgraph. Let $P_t$ denote the path on $t$ vertices, and let $S_{a,b,c}$ denote the graph obtained from paths $P_{a+1},P_{b+1},P_{c+1}$ by identifying one of their endvertices. We show that there is only a finite number of minimal obstructions to $C_5$-coloring among $F$-free graphs, where $F \in \{ P_8, S_{2,2,1}, S_{3,1,1}\}$ and explicitly determine all such obstructions. This extends the results of Kamiński and Pstrucha [Discr. Appl. Math. 261, 2019] who proved that there is only a finite number of $P_7$-free minimal obstructions to $C_5$-coloring, and of Dębski et al. [ISAAC 2022 Proc.] who showed that the triangle is the unique $S_{2,1,1}$-free minimal obstruction to $C_5$-coloring. We complement our results with a construction of an infinite family of minimal obstructions to $C_5$-coloring, which are simultaneously $P_{13}$-free and $S_{2,2,2}$-free. We also discuss infinite families of $F$-free minimal obstructions to $H$-coloring for other graphs $H$.

This paper studies the numerical approximation of evolution equations by nonlinear parametrizations $u(t)=Φ(\param(t))$ with time-dependent parameters $\param(t)$, which are to be determined in the computation. The motivation comes from approximations in quantum dynamics by multiple Gaussians and approximations of various dynamical problems by tensor networks and neural networks. In all these cases, the parametrization is typically irregular: the derivative $Φ'(\param)$ can have arbitrarily small singular values and may have varying rank. We derive approximation results for a regularized approach in the time-continuous case as well as in time-discretized cases. For the latter, there is a nontrivial interplay between the regularization parameter and the time stepsize, depending also on the defect size and local bounds of the second derivative of the parametrization map $Φ$. When this is appropriately taken into account, the approach can be successfully applied in irregular situations, even though it runs counter to the basic principle in numerical analysis to avoid solving ill-posed subproblems when aiming for a stable algorithm. The paper also includes two theoretical case studies: regularized parametric steepest descent for optimization and regularized parametric Strang splitting for the time-dependent Schrödinger equation. Numerical experiments with sums of Gaussians for approximating quantum dynamics and with neural networks for approximating the flow map of a system of ordinary differential equations illustrate and complement the theoretical results.

Coordination in multiplayer games enables players to avoid the lose-lose outcome that often arises at Nash equilibria. However, designing a coordination mechanism typically requires the consideration of the joint actions of all players, which becomes intractable in large-scale games. We develop a novel coordination mechanism, termed reduced rank correlated equilibria, which reduces the number of joint actions to be considered and thereby mitigates computational complexity. The idea is to approximate the set of all joint actions with the actions used in a set of pre-computed Nash equilibria via a convex hull operation. In a game with n players and each player having m actions, the proposed mechanism reduces the number of joint actions considered from O(m^n) to O(mn). We demonstrate the application of the proposed mechanism to an air traffic queue management problem. Compared with the correlated equilibrium-a popular benchmark coordination mechanism-the proposed approach is capable of solving a problem involving four thousand times more joint actions while yielding similar or better performance in terms of a fairness indicator and showing a maximum optimality gap of 0.066% in terms of the average delay cost. In the meantime, it yields a solution that shows up to 99.5% improvement in a fairness indicator and up to 50.4% reduction in average delay cost compared to the Nash solution, which does not involve coordination.

We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact orientable surface $S$. The surface $S$ is presented via a triangulation or a handle structure, and the 1-manifolds are given in normal form via their normal coordinates. The running time is bounded above by a polynomial function of the number of triangles in the triangulation (or the number of handles in the handle structure), and the logarithm of the weight of $C_1$ and $C_2$. This algorithm represents an improvement over previous work, since its running time depends polynomially on the size of the triangulation of $S$ and it can deal with closed surfaces, unlike many earlier algorithms. Another algorithm, with similar bounds on its running time, can determine whether $C_1$ and $C_2$ are isotopic. We also present a closely related algorithm that can be used to place a standard 1-manifold into normal form.