The moving firefighter problem (MFP) is a more realistic variant of the classic firefighter problem (FP), where firefighters require time for both travel and defense. Unfortunately, the only known exact solution for the MFP does not scale. In this paper, we establish that the MFP is NP-complete on trees of maximum degree three and present four alternative methods to find exact solutions for the case of arbitrary trees with a single initial fire and one firefighter. The first method is a dynamic programming algorithm, while the other three methods are based on mathematical programming: an integer quadratically constrained program (IQCP), and two distinct integer linear programs (ILP and E-ILP). Our mathematical programming formulations exploit the inherent properties of tree topologies to significantly improve scalability by reducing the number of decision variables and constraints relative to those for arbitrary graphs. We present a comprehensive experimental analysis of the performance and scalability of the proposed solutions.
We characterize the limiting root distribution mu of a sequence of polynomials {pd}infinity d=1 with nonnegative roots and degree d, in terms of their coefficients. Specifically, we relate the asymptotic behavior of the ratio of consecutive coefficients of pd to Voiculescu's S-transform S mu of mu. In the framework of finite free probability, we interpret these ratios of coefficients as a new notion of finite S-transform, which converges to S mu in the large d limit. It also satisfies several analogous properties to those of the S-transform in free probability, including multiplicativity and monotonicity. The proof of the main theorem is based on various ideas and new results relating finite free probability and free probability. In particular, we provide a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation. This new insight has several applications that strengthen the connection between free and finite free probability. Most notably, we generalize the approximation of (R) d to (R) and prove a finite approximation of the Tucci-Haagerup-M & ouml;ller limit theorem in free probability, conjectured by two of the authors. We also provide finite analogues of the free multiplicative Poisson law, the free max-convolution powers and some free stable laws. (c) 2026 Published by Elsevier Inc.
In Differential Geometry, it is well--known that given a principal $G$--bundle with a principal connection, for every unitary finite--dimensional linear representation of $G$, one can induce a linear connection and a hermitian structure on associated vector bundles, which are compatible. Furthermore, the gauge group acts on the space of principal connections and on the space of linear connections defined in associated vector bundles. This paper aims to present the non--commutative counterpart of all these classical facts using the theory of quantum bundles and quantum connections.
We study the problem of community detection in a general version of the block spin Ising model featuring M groups, a model inspired by the Curie-Weiss model of ferromagnetism in statistical mechanics. We solve the general problem of identifying any number of groups with any possible coupling constants. Up to now, the problem was only solved for the specific situation with two groups of identical size and identical interactions, see [1, 2]. Our results can be applied to the most realistic situations, in which there are many groups of different sizes and different interactions. In addition, we give an explicit algorithm that permits the reconstruction of the structure of the model from a sample of observations based on the comparison of empirical correlations of the spin variables, thus unveiling easy applications of the model to real-world voting data and communities in biology.
In this note, we revisit the fundamental question of the strong law of large numbers and central limit theorem for processes in continuous time with conditional stationary and independent increments. For convenience, we refer to them as Markov additive processes, or MAPs for short. Historically used in the setting of queuing theory, MAPs have often been written about when the underlying modulating process is an ergodic Markov chain on a finite state space, cf. (Asmussen in Applied probability and queues, Springer-Verlag, New York, 2003; Asmussen and Albrecher in Ruin probabilities, Hackensack, 2010), not to mention the classical contributions of Pacheco and Prabhu (Markov additive processes of arrivals, CRC, Boca Raton, 1995), Prabhu (Stochastic storage processes, Springer-Verlag, New York, 1998). Recent works have addressed the strong law of large numbers when the underlying modulating process is a general Markov processes; cf. as reported (Kyprianou et al. Entrance laws at the origin of self-similar Markov processes in high dimensions, 2019; Yaran and Çağlar in ALEA Lat Am J Probab Math Stat 22:991–1010, 2025) . We add to the latter with a different approach based on an ergodic theorem for additive functionals and on the semimartingale structure of the additive part. This approach also allows us to deal with the setting that the modulator of the MAP is either positive or null-recurrent. The methodology additionally inspires a CLT-type result.
