A functional equation related to generalized entropies and the modular group

We solve a functional equation connected to the algebraic characterization of generalized information functions. To prove the symmetry of the solution, we study a related system of functional equations, which involves two homographies. These transformations generate the modular group, and this fact plays a crucial role in solving the system. The method suggests a more general relation between conditional probabilities and arithmetic.Comment: Originally uploaded as an appendix to arXiv:1709.07807v1. Changes in v2: the introduction was extended to summarize in more detail previous results; there is a new lemma at the end of Section

Optimized Schwarz Waveform Relaxation for Advection Reaction Diffusion Equations in Two Dimensions

Optimized Schwarz Waveform Relaxation methods have been developed over the last decade for the parallel solution of evolution problems. They are based on a decomposition in space and an iteration, where only subproblems in space-time need to be solved. Each subproblem can be simulated using an adapted numerical method, for example with local time stepping, or one can even use a different model in different subdomains, which makes these methods very suitable also from a modeling point of view. For rapid convergence however, it is important to use effective transmission conditions between the space-time subdomains, and for best performance, these transmission conditions need to take the physics of the underlying evolution problem into account. The optimization of these transmission conditions leads to a mathematically hard best approximation problem of homographic type. We study in this paper in detail this problem for the case of linear advection reaction diffusion equations in two spatial dimensions. We prove comprehensively best approximation results for transmission conditions of Robin and Ventcel type. We give for each case closed form asymptotic values for the parameters, which guarantee asymptotically best performance of the iterative methods. We finally show extensive numerical experiments, and we measure performance corresponding to our analysisComment: 42 page

Topos and Stacks of Deep Neural Networks

Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs) correspond to Giraud's stacks. This invariance is supposed to be responsible of the generalization property, that is extrapolation from learning data under constraints. The fibers represent pre-semantic categories (Culioli, Thom), over which artificial languages are defined, with internal logics, intuitionist, classical or linear (Girard). Semantic functioning of a network is its ability to express theories in such a language for answering questions in output about input data. Quantities and spaces of semantic information are defined by analogy with the homological interpretation of Shannon's entropy (P.Baudot and D.B. 2015). They generalize the measures found by Carnap and Bar-Hillel (1952). Amazingly, the above semantical structures are classified by geometric fibrant objects in a closed model category of Quillen, then they give rise to homotopical invariants of DNNs and of their semantic functioning. Intentional type theories (Martin-Loef) organize these objects and fibrations between them. Information contents and exchanges are analyzed by Grothendieck's derivators

How the Learning Path and the Very Structure of a Multifloored Environment Influence Human Spatial Memory

International audienceFew studies have explored how humans memorize landmarks in complex multifloored buildings. They have observed that participants memorize an environment either by floors or by vertical columns, influenced by the learning path. However, the influence of the building's actual structure is not yet known. In order to investigate this influence, we conducted an experiment using an object-in-place protocol in a cylindrical building to contrast with previous experiments which used rectilinear environments. Two groups of 15 participants were taken on a tour with a first person perspective through a virtual cylindrical three-floored building. They followed either a route discovering floors one at a time, or a route discovering columns (by simulated lifts across floors). They then underwent a series of trials, in which they viewed a camera movement reproducing either a segment of the learning path (familiar trials), or performing a shortcut relative to the learning trajectory (novel trials). We observed that regardless of the learning path, participants better memorized the building by floors, and only participants who had discovered the building by columns also memorized it by columns. This expands on previous results obtained in a rectilinear building, where the learning path favoured the memory of its horizontal and vertical layout. Taken together, these results suggest that both learning mode and an environment's structure influence the spatial memory of complex multifloored buildings