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We propose a new concept of generalized differentiation of set-valued maps that captures the first order information. This concept encompasses the standard notions of Frechet differentiability, strict differentiability, calmness and…

We consider a notion of relative homology (and cohomology) for surfaces with two types of boundaries. Using this tool, we study a generalization of Kitaev's code based on surfaces with mixed boundaries. This construction includes both…

A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and…

This paper has been withdrawn. With the advancement of statistical theory and computing power, data sets are providing a greater amount of insight into the problems of today. Statisticians have an ever increasing number of tools to attack…

Demixing is the problem of identifying multiple structured signals from a superimposed observation. This work analyzes a general framework, based on convex optimization, for solving demixing problems. We present a new solution to determine…

Compositional generalization, the ability of an agent to generalize to unseen combinations of latent factors, is easy for humans but hard for deep neural networks. A line of research in cognitive science has hypothesized a process,…

Different graph generalizations have been recently used in an ad-hoc manner to represent multilayer networks, i.e. systems formed by distinct layers where each layer can be seen as a network. Similar constructions have also been used to…

In [6] the notion of a g-cell structure was introduced as a generalization of the construction proposed by Debski and Tymchatyn to realize a certain class of topological spaces as quotient spaces of inverse limits. In [2], cell maps are…

We give some basic properties of strongly topologically transitive, supermixing, and hypermixing maps on general topological spaces. Then we present some other results for which our mappings need to be continuous.

Natural language processing (NLP) researchers develop models of grammar, meaning and communication based on written text. Due to task and data differences, what is considered text can vary substantially across studies. A conceptual…

In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the…

In the analysis of bibliometric networks, researchers often use mapping and clustering techniques in a combined fashion. Typically, however, mapping and clustering techniques that are used together rely on very different ideas and…

In this paper we invastigate the notion of generalized (I,J) - Luzin set. This notion generalize the standard notion of Luzin set and Sierpinski set. We find set theoretical conditions which imply the existence of generalized (I,J) - Luzin…

Compositional generalization, the ability to recognize familiar parts in novel contexts, is a defining property of intelligent systems. Although modern models are trained on massive datasets, they still cover only a tiny fraction of the…

The extent to which neural networks are able to acquire and represent symbolic rules remains a key topic of research and debate. Much current work focuses on the impressive capabilities of large language models, as well as their often…

In classification, it is usual to observe that models trained on a given set of classes can generalize to previously unseen ones, suggesting the ability to learn beyond the initial task. This ability is often leveraged in the context of…

This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. We give a unified model for all continuous maps on a metric space, by representing a map through a general…

A concise discussion of the axiomatic approach to the concept of parallel transport is presented. Attention is drawn to a bijective map between the sets of connections and (axiomatically defined) parallel transports. The transports along…

Digital Geometry software should reflect the generality of the underlying mathe- matics: mapping the latter to the former requires genericity. By designing generic solutions, one can effectively reuse digital geometry data structures and…

Interactions and relations between objects may be pairwise or higher-order in nature, and so network-valued data are ubiquitous in the real world. The "space of networks", however, has a complex structure that cannot be adequately described…