Phase classification has become a prototypical benchmark for data-driven analysis of condensed matter physics. The type and complexity of the phase transition dictate the level of complexity of the algorithm one has to employ. This topic has been broadly explored, offering a menu of both supervised and unsupervised techniques ranging from simple clustering [1-3] to more complex machine learning methods [4-7]. The phase classification problem is most commonly posed like so: we allow our model to view a dataset that is both relevant and straightforwardly obtainable in the scenario we wish to study. We introduce this data set to a model that has no prior knowledge of underlying physics.

In this article, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain sets of transformations, in a non-expansive way. If the set of transformations acting is a group, then we obtain the so-called GENEOs. We then study the spaces of measurements, whose domains are subject to the action of certain self-maps, and the space of P-GENEOs between these spaces. We define pseudo-metrics on them and show some properties of the resulting spaces. In particular, we show how such spaces have convenient approximation and convexity properties.

We characterize homotopical equivalences between causal DAG models, exploiting the close connections between partially ordered set representations of DAGs (posets) and finite Alexandroff topologies. Alexandroff spaces yield a directional topological space: the topology is defined by a unique minimal basis defined by an open set for each variable x, specified as the intersection of all open sets containing x. Alexandroff spaces induce a (reflexive, transitive) preorder. Alexandroff spaces satisfying the Kolmogorov T0 separation criterion, where open sets distinguish variables, converts the preordering into a partial ordering. Our approach broadly is to construct a topological representation of posets from data, and then use the poset representation to build a conventional DAG causal model. We illustrate our framework by showing how it unifies disparate algorithms and case studies proposed previously. Topology plays two key roles in causal discovery. First, topological separability constraints on datasets have been used in several previous approaches to infer causal structure from observations and interventions. Second, a diverse range ofgraphical models used to represent causal structures can be represented in a unified way in terms of a topological representation of the induced poset structure. We show that the homotopy theory of Alexandroff spaces can be exploited to significantly efficiently reduce the number of possible DAG structures, reducing the search space by several orders of magnitude.

By- Amith Singhee At the dawn of the Information Age in the 1970s, the role of Information Technology (IT) was limited to'computing plumbing' - to keep the networks and computers working. In the 90s and 2000s, it evolved into an enterprise shared servicesmodel that was essential for operational efficiency, cost takeout and decision support. Today, IT is witnessing another shift that increasingly requires the Chief Information Officer organization to act as a partner in defining business strategy and driving topline growth via IT-driven business transformation. To realize this, the IT delivery platform that includes infrastructure, applications, processes and roles of people -needs to be scalable and adaptable tokeep pace with the rapidly changing business and operational needs, and, hence, transform to a hybrid cloud IT architecture. The transformation will involve four phases: Advice for Cloud, Move to Cloud, Build for Cloud and Manage on Cloud.

TLDR: Neural Networks are powerful but complex and opaque tools. Using Topological Data Analysis, we can describe the functioning and learning of a convolutional neural network in a compact and understandable way. The implications of the finding are profound and can accelerate the development of a wide range of applications from self-driving everything to GDPR. Neural networks have demonstrated a great deal of success in the study of various kinds of data, including images, text, time series, and many others. One issue that restricts their applicability, however, is the fact that it is not understood in any kind of detail how they work.

The concept of a cognitive robotic companion inspires some of the best science fiction but one day may be science fact following the work of the four-year COGNIRON project funded since January 2004 by the IST's Future and Emerging Technologies initiative. But what could a cognitive robot companion do? The example that's often used is a robot that's able to fulfil your needs, like passing you a drink or helping in everyday tasks," says Dr Raja Chatila, research director at the Systems Architecture and Analysis Laboratory of the French Centre National de la Recherche Scientifique (LAAS-CNRS), and COGNIRON project coordinator. "That might seem a bit trivial, but let me ask you a question: In the 1970s, what was the use of a personal computer?" he asks. In fact, it was then impossible to imagine how PCs would change the world's economics, politics and society in just 30 years. The eventual uses, once the technology developed, were far from trivial. COGNIRON set out on the same principle, ...

Such maps specify the topology of important landmarks and situations (states), and routes or transitions (arcs) between them. They are concerned less with the physical location of landmarks, and more with topological relationships between situations. Typically, they are less complex and support much more ecient planning than metric maps. Topological maps are built on lowerlevel abstractions that allow the robot to move along arcs (perhaps by wall-or road-following), to recognize properties of locations, and to distinguish signicant locations as states; they are exible in allowing a more general notion of state, possibly including information about the non-geometrical aspects of the robot's situation. There are two typical strategies for deriving topological maps: one is to learn the topological map directly; the other is to rst learn a geometric map, then to derive a topological model from it through some process of analysis. A nice example of the second approach is provided by Thrun and B--ucken (1996a, 1996b; Thrun, 1999), who use occupancy-grid techniques to build the initial map. This strategy is appropriate when the primary cues for decomposition and abstraction of the map are geometric. However, in many cases, the nodes of a topological map are dened in terms of other sensory data (e.g., labels on a door or whether or not the robot is holding a bagel). Learning a geometric map rst also relies on the odometric abilities of a robot; if they are weak and the space is large, it is very dicult to derive a consistent map.

Subset space logic (SSL, henceforth) was presented in early In this paper, we consider public announcement logic (PAL, 90s as a bimodal logic to formalize reasoning about sets and henceforth) in several different geometric models, and prove points (Moss and Parikh 1992). The language of SSL has its completeness of those models. Moreover, we also consider two modal operators K and . A subset space model is a some applications of our ideas in different fields varying triple S 〈S, σ, v〉 where S is a nonempty set, σ is a collection from game theory to epistemic logic. What makes our of subsets (not necessarily a topology), v is a valuation work novel is the fact that PAL has never been investigated function. Semantics of SSL for modal operators is given in geometric and topological models with further applications.