The algebraic structures of social organizations: the operad of cooperative games arxiv.org/abs/2507.01969

The algebraic structures of social organizations: the operad of cooperative games

The main goal of this paper is to settle a conceptual framework for cooperative game theory in which the notion of composition/aggregation of games is the defining structure. This is done via the mathematical theory of algebraic operads: we start by endowing the collection of all cooperative games with any number of players with an operad structure, and we show that it generalises all the previous notions of sums, products and compositions of games considered by Owen, Shapley, von Neumann and Morgenstern, and many others. Furthermore, we explicitly compute this operad in terms of generators and relations, showing that the Möbius transform map induces a canonical isomorphism between the operad of cooperative games and the operad that encodes commutative triassociative algebras. In other words, we prove that any cooperative game is a linear combination of iterated compositions of the 2-player bargaining game and the 2-player dictator games. We show that many interesting classes of games (simple, balanced, capacities a.k.a fuzzy measures and convex functions, totally monotone, etc) are stable under compositions, and thus form suboperads. In the convex case, this gives by the submodularity theorem a new operad structure on the family of all generalized permutahedra. Finally, we focus on how solution concepts in cooperative game theory behave under composition: we study the core of a composite and describe it in terms of the core of its components, and we give explicit formulas for the Shapley value and the Banzhaf index of a compound game.

arXiv.org

Investigations of Complex Systems' Dynamics Based on a Reduced Amount of Information -- Part 3: Dynamical Phenomena Indicator -- The Fastest and Most Universal Approach for Analyzing the Dynamics of Networks of Coupled Nonlinear Systems arxiv.org/abs/2507.02069

Investigations of Complex Systems' Dynamics Based on a Reduced Amount of Information -- Part 3: Dynamical Phenomena Indicator -- The Fastest and Most Universal Approach for Analyzing the Dynamics of Networks of Coupled Nonlinear Systems

Recently, we have demonstrated that our approach is a highly effective tool while analysing complex phenomena existing in networks of coupled nonlinear systems. In the present article we present the results of our investigations into a specific aspect of the presented method. We prove its effectiveness while applying for fast investigations of complex systems and easy detection of different uncommon dynamical phenomena states. We also extend our method introducing new Dynamical Phenomena Indicator (DPI), designed especially for effective detection of complex dynamical phenomena states in the wide range of the parameters of complex networks of coupled nonlinear systems. Contrary to commonly applied methods, the proposed approach allows for identification of complex dynamical phenomena long before stabilization of the system. The method bases on early signalized tendency of the system to split its dynamics to separately synchronized subsystems. The most important fact is that proposed approach is highly universal and can be applied for both, symmetrical and non-symmetrical topologies of coupling as well as networks of identical and non-identical oscillators. Moreover, since DPI values are obtained from the current state of dynamical system given by values of the system variables, proposed method of fast searching has a huge potential for experimental application. Following this reasoning the presented results can be treated both as exemplary numerical investigations and analysis of experimentally obtained results.

arXiv.org

Classification of spin$^c$ manifolds with generalized positive scalar curvature arxiv.org/abs/2507.02090

Classification of spin$^c$ manifolds with generalized positive scalar curvature

Suppose $M$ is a closed $n$-dimensional spin$^c$ manifold with spin$^c$ structure $σ$ and associated spin$^c$ line bundle $L$. If one fixes a Riemannian metric $g$ on $M$ and a connection $\nabla_L$ on $L$, the generalized scalar curvature $R^{\text{gen}}$ of $(M,L)$ is $R_g - 2|Ω_L|_{\text{op}}$, where $|Ω_L|_{\text{op}}$ is the pointwise operator norm of the curvature $2$-form $Ω_L$ of $\nabla_L$, acting on spinors. In a previous paper, we showed that positivity of $R^{\text{gen}}$ is obstructed by the non-vanishing of the index of the spin$^c$ Dirac operator on $(M,g,L,\nabla_L)$, and that in some cases, the vanishing of this index guarantees the existence of a pair $(g,\nabla_L)$ with positive generalized scalar curvature. Building on this and on surgery techniques inspired by those that have been developed in the theory of positive scalar curvature on spin manifolds, we show that if $\dim M = n \ge 5$, if the fundamental group $π$ of $M$ is in a large class including surface groups and finite groups with periodic cohomology, and if $M$ is totally non-spin (meaning that the universal cover is not spin), then $(M,L)$ admits positive generalized scalar curvature if and only if the generalized $α$-invariant of $(M,L)$ vanishes in the $K$-homology group $K_n(Bπ)$. We also develop an analogue of Stolz's sequence for computing the group of concordance classes of positive generalized scalar curvature metrics, and connect this to the analytic surgery sequence of Roe and Higson. Finally, we give a number of applications to moduli spaces of positive generalized scalar curvature metrics.

arXiv.org

Cross-Attention Message-Passing Transformers for Code-Agnostic Decoding in 6G Networks arxiv.org/abs/2507.01038 .SP .IT .LG

Cross-Attention Message-Passing Transformers for Code-Agnostic Decoding in 6G Networks

Channel coding for 6G networks is expected to support a wide range of requirements arising from heterogeneous communication scenarios. These demands challenge traditional code-specific decoders, which lack the flexibility and scalability required for next-generation systems. To tackle this problem, we propose an AI-native foundation model for unified and code-agnostic decoding based on the transformer architecture. We first introduce a cross-attention message-passing transformer (CrossMPT). CrossMPT employs two masked cross-attention blocks that iteratively update two distinct input representations-magnitude and syndrome vectors-allowing the model to effectively learn the decoding problem. Notably, our CrossMPT has achieved state-of-the-art decoding performance among single neural decoders. Building on this, we develop foundation CrossMPT (FCrossMPT) by making the architecture invariant to code length, rate, and class, allowing a single trained model to decode a broad range of codes without retraining. To further enhance decoding performance, particularly for short blocklength codes, we propose CrossMPT ensemble decoder (CrossED), an ensemble decoder composed of multiple parallel CrossMPT blocks employing different parity-check matrices. This architecture can also serve as a foundation model, showing strong generalization across diverse code types. Overall, the proposed AI-native code-agnostic decoder offers flexibility, scalability, and high performance, presenting a promising direction to channel coding for 6G networks.

arXiv.org

Computational Insights into Orthotropic Fracture: Crack-Tip Fields in Strain-Limiting Materials under Non-Uniform Loads arxiv.org/abs/2507.01150

Computational Insights into Orthotropic Fracture: Crack-Tip Fields in Strain-Limiting Materials under Non-Uniform Loads

A finite element framework is presented for analyzing crack-tip phenomena in transversely isotropic, strain-limiting elastic materials. Mechanical response is characterized by an algebraically nonlinear constitutive model, relating stress to linearized strain. Non-physical strain singularities at the crack apex are mitigated, ensuring bounded strain magnitudes. This methodology significantly advances boundary value problem (BVP) formulation, especially for first-order approximate theories. For a transversely isotropic elastic solid with a crack, the governing equilibrium equation, derived from linear momentum balance and the nonlinear constitutive model, is reduced to a second-order, vector-valued, quasilinear elliptic BVP. This BVP is solved using a robust numerical scheme combining Picard-type linearization with a continuous Galerkin finite element method for spatial discretization. Numerical results are presented for various loading conditions, including uniform tension, non-uniform slope, and parabolic loading, with two distinct material fiber orientations. It is demonstrated that crack-tip strain growth is substantially lower than stress growth. Nevertheless, strain-energy density is found to be concentrated at the crack tip, consistent with linear elastic fracture mechanics principles. The proposed framework provides a robust basis for formulating physically meaningful, rigorous BVPs, critical for investigating fundamental processes like crack propagation, damage, and nucleation in anisotropic, strain-limiting elastic solids under diverse loading conditions.

arXiv.org
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