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What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only partial answers to these questions. To make these problems more precise and easier to tackle, we describe an abstract algebraic definition of classical computation by generalizing traditional models to semigroups. This way implementations are morphic relations between semigroups. The mathematical abstraction also allows the investigation of different computing paradigms (e.g. cellular automata, reversible computing) in the same framework. While semigroup theory helps in clarifying foundational issues about computation, at the same time it has several open problems that require extensive computational efforts. This mutually beneficial relationship is the central tenet of the described research.

Let G be a discrete group. It is well known that the (co)homology groups of G have both topological and algebraic definitions. Now, consider a subgroup H of G. In the literature, there are two versions of relative (co)homology groups for the pair (G, H), one generalises in a natural way the topological definition, while the other one generalises in a natural way the algebraic definition. In this article, we give a topological definition for the latter one, we give simple examples that show that these theories do not coincide in general, and we give a sufficient condition on the subgroup H in order that both relative group (co)homologies of the pair (G, H) coincide.

In their daily practices, meteorologists make extensive use of the geostrophic wind properties to explain many weather phenomena such as the meaning and direction of the horizontal winds that take place around the low atmospheric pressures. The biggest challenge that faces the public who is interested in information disseminated by meteorologists is to know exactly what means the geostrophic wind. Besides the literal definitions scattered in very little scientific work, there is unfortunately no book which gives importance to the algebraic definition of the geostrophic wind. Our work shows that to better understand the behavior of natural phenomena, it is essential to combine the theories with based observations. Obviously, observations cannot be relevant without a theory that guides the observers. Conversely, no theory can be validated without experimental verification. Synoptic observations show that in the “free atmosphere!” the wind vectors are very nearly parallel to isobars, and the flow is perpendicular to the horizontal pressure gradient force, at least at any given instant. This kind of information recommends great caution when making geostrophic approximations. Our work also shows that for tornadoes, there is no need to move away from the surface of the oceans to observe the geostrophic balance. Undoubtedly, identification and interpretation of earth’s atmosphere dynamics’ and thermodynamics’ similarities between rogue waves and oceans’ surface geostrophic wind will be an easy exercise to researchers who will give importance to result provided by this paper.

Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences. Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories. The definition has an intrinsic non-symmetry. Suppose that we have a triangulated category in which idempotents split and either for which the endomorphism rings of all objects are artinian, or which is the category of compact objects in an algebraic compactly generated triangulated K-category. Then we show that the non-symmetry in the algebraic definition of the degeneration is inessential in the sense that the two possible choices which can be made in the definition lead to the same concept.

I argue that a version of the quantum-corrected Ryu-Takayanagi formula holds in any quantum error-correcting code. I present this result as a series of theorems of increasing generality, with the final statement expressed in the language of operator-algebra quantum error correction. In AdS/CFT this gives a "purely boundary" interpretation of the formula. I also extend a recent theorem, which established entanglement-wedge reconstruction in AdS/CFT, when interpreted as a subsystem code, to the more general, and I argue more physical, case of subalgebra codes. For completeness, I include a self-contained presentation of the theory of von Neumann algebras on finite-dimensional Hilbert spaces, as well as the algebraic definition of entropy. The results confirm a close relationship between bulk gauge transformations, edge-modes/soft-hair on black holes, and the Ryu-Takayanagi formula. They also suggest a new perspective on the homology constraint, which basically is to get rid of it in a way that preserves the validity of the formula, but which removes any tension with the linearity of quantum mechanics. Moreover they suggest a boundary interpretation of the "bit threads" recently introduced by Freedman and Headrick., 40 pages plus appendix, 11 figures, many subscripts on subscripts. v2: Minor corrections and improvements, section 6.3 revised more substantially for clarity, section 6.4 added to discuss some limitations

International audience; We study the algebraic structure of a programming language with higher-order store, in the style of ML references. Instead of working directly on the operational semantics of the language, we consider its fully abstract game semantics defined by Abramsky, Honda and McCusker one decade ago. This alternative description of the language is nice and conceptual, except on one significant point: the interactive behavior of the higher-order memory cell is reflected in the model by a strategy $\mathtt{cell}$ whose definition remains slightly enigmatic. The purpose of our work is precisely to clarify this point, by providing a neat algebraic definition of the strategy. This conceptual reconstruction of the memory cell is based on the idea that a general reference behaves essentially as a linear feedback (or trace operator) in an ambient category of Conway games and strategies. This analysis leads to a purely axiomatic proof of soundness of the model, based on a natural refinement of the replication modality of tensor logic.

What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only partial answers to these questions. In order to make these problems more precise, we describe an abstract algebraic definition of classical computation, generalizing traditional models to semigroups. The mathematical abstraction also allows the investigation of different computing paradigms (e.g. cellular automata, reversible computing) in the same framework. Here we summarize the main questions and recent results of the research of finite computation., Comment: 12 pages, 3 figures, will be presented at CANDAR'16 and final version published by IEEE Computer Society

Topology is the branch of mathematics that studies shapes and maps among them. From the algebraic definition of topology a new set of algorithms have been derived. These algorithms are identified with “computational topology” or often pointed out as Topological Data Analysis (TDA) and are used for investigating high-dimensional data in a quantitative manner. Persistent homology appears as a fundamental tool in Topological Data Analysis. It studies the evolution of k−dimensional holes along a sequence of simplicial complexes (i.e. a filtration). The set of intervals representing birth and death times of k−dimensional holes along such sequence is called the persistence barcode. k−dimensional holes with short lifetimes are informally considered to be topological noise, and those with a long lifetime are considered to be topological feature associated to the given data (i.e. the filtration). In this paper, we derive a simple method for separating topological noise from topological features using a novel measure for comparing persistence barcodes called persistent entropy. Ministerio de Economía y Competitividad MTM2015-67072-P

International audience; We tackle the problem of graph transformation with particular focus on node cloning. We propose a new approach to graph rewriting, called polarized node cloning, where a node may be cloned together with either all its incident edges or with only its outgoing edges or with only its incoming edges or with none of its incident edges. We thus subsume previous works such as the sesqui-pushout, the heterogeneous pushout and the adaptive star grammars approaches. We first define polarized node cloning algorithmically, then we propose an algebraic definition. We use polarization annotations to declare how a node must be cloned. For this purpose, we introduce the notion of polarized graphs as graphs endowed with some annotations on nodes and we define graph transformations with polarized node cloning by means of sesqui-pushouts in the category of polarized graphs.