Bridging the gap between finite-and infinite-dimensional theories in fuzzy Banach spaces, this work is centered on a countably infinite set of Strongly Linearly Independent (SLI) fuzzy numbers. The linear span of this SLI set is isomorphic to the canonical sequence space R" (R) 1'p for p E [1, cc). Through this powerful correspondence, a complete norm and consistent vector operations on the fuzzy number space are defined by inheriting the structure of its sequence space counterpart. This framework provides a solid analytical foundation for this class of fuzzy numbers, yields a Hilbert space for p = 2, and outlines a path for expansion towards a fuzzy calculus.
We study the Laplacian of the undirected De Bruijn graph over an alphabet A of order k. While the eigenvalues of this Laplacian were found in 1998 by Delorme and Tillich [10], an explicit description of its eigenvectors has remained elusive. In this work, we find these eigenvectors in closed form and show that they yield a natural and canonical basis for the cut- and cycle-spaces of undirected De Bruijn graphs. Remarkably, we find that the cycle basis we construct is a basis for the cycle space of both the undirected and the directed De Bruijn graph. This is done by developing an analogue of the Fourier transform on the De Bruijn graph, which acts to diagonalize the undirected Laplacian. Moreover, we show that the cycle-space of De Bruijn graphs, when considering all possible orders of k simultaneously, contains a rich algebraic structure, that of a graded Hopf algebra. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
The internet is a general-purpose network grounded on openness, decentralisation and interoperability. Such features have allowed innovation to flourish, lowering barriers to communication, participation and cooperation, thus empowering end-users. ‘General purpose’ means that the purpose for which the internet is used is not predefined by the operator but can be autonomously decided by the end-user. Accordingly, the network neutrality (NN) principle mandates non-discriminatory treatment of internet traffic to preserve the general-purpose nature of the internet, unleashing end-users’ creativity. This paper starts by exploring the NN debate, stressing that the NN rationale is to preserve an open and decentralised internet architecture, empowering end-users and protecting their rights. Subsequently, it argues that the combination of reduced data caps and zero-rating (ZR) schemes may create artificial scarcity, raise the price of the open internet and jeopardise the achievement of the NN rationale. It provides a taxonomy of ZR models and emphasises that several ZR practices might impose on the internet a centralised configuration that characterises less innovative networks, such as the Minitel. The phenomenon that I define as ‘Minitelisation’ of the internet consists of the shift from a user-centric, general-purpose network to one with predefined purposes, thereby creating passive consumers of predetermined services, rather than active internet users.
We introduce the Method of Ellipcenters (ME) for unconstrained minimization. At the cost of two gradients per iteration and a line search, we compute the next iterate by setting it as the centre of an elliptical interpolation. The idea behind the ellipse built in each step is to emulate the original level curve of the objective function constrained to a suitable two-dimensional affine space, which is determined by the current iterate and two appropriate gradient vectors. We present the method for general unconstrained minimization and carry out a convergence analysis for the case where the objective function is quadratic. In this context, ME enjoys linear convergence with the rate being at least as good as the linear rate of the steepest descent (gradient) method with optimal step. In our experiments, however, ME was much faster than the gradient method with optimal step size. Moreover, ME seems highly competitive in comparison to several well-established algorithms. The efficiency in terms of both time and number of iterations is stressed even more for ill-conditioned problems.
We investigate the heterogeneity in the long-run effects of two colonial roads in Brazil. These roads connected the coast to a thriving gold mining region in the highlands. The oldest road was heavily used at the beginning of the gold rush, but tax evasion concerns led to the construction of an alternative road. By the middle of the gold rush, trade flows through the newer road vastly surpassed those through the older road. By comparing the patterns of how modern economic density increases with proximity to each of these colonial roads, we estimate that the persistent effects of the newer colonial road are larger than those of the older colonial road. Our results suggest that the past importance for trade of a historic road determines whether the effects of the road persist in the long run.
