Related papers: Simplicial sets with a notion of smallness

Rough set theory is a new mathematical approach to imperfect knowledge. The notion of rough sets is generalized by using an arbitrary binary relation on attribute values in information systems, instead of the trivial equality relation. The…

We present a method to simplify expressions in the context of an equational theory. The basic ideas and concepts of the method have been presented previously elsewhere but here we tackle the difficult task of making it efficient in…

Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces. We show that in many cases, researchers using these tools have…

The local zero structure of a smooth map may qualitatively change, when the map is subjected to small perturbations. The changes may include births and/or deaths of zeros. The qualitative properties are defined as the invariances of an…

Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…

In this paper, we define a category DB, called the category of simplicial databases, whose objects are databases and whose morphisms are data-preserving maps. Along the way we give a precise formulation of the category of relational…

Some of the basic concepts of topology are explored through known physics problems. This helps us in two ways, one, in motivating the definitions and the concepts, and two, in showing that topological analysis leads to a clearer…

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We…

We define the notion of valuation on simplicial maps between geometric realizations of simplicial complexes in $\mathbb{R}^n$. Valuations on simplicial maps are analogous to valuations on sets. In particular, we define the Lefschetz…

The notion of soft sets is introduced as a general mathematical tool for dealing with uncertainty. In this paper, we consider the concepts of soft compactness, countably soft compactness and obtain some results. We study some soft…

The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global…

We generalize the very well known boundary operator of the ordinary singular homology theory, defined in many books about algebraic topology. We describe a variant of this ordinary simplicial boundary operator where the usual boundary…

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched…

Simplicial complexes form an important class of topological spaces that are frequently used in many application areas such as computer-aided design, computer graphics, and simulation. Representation learning on graphs, which are just 1-d…

ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated…

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…

The work lays the foundations of the theory of changeable sets. In author opinion, this theory, in the process of it's development and improvement, can become one of the tools of solving the sixth Hilbert problem least for physics of…

We illustrate the generative power of the lifting property (orthogonality of morphisms in a category) as means of defining natural elementary mathematical concepts by giving a number of examples in various categories, in particular showing…

In this note we show that a semisimplicial set with the weak Kan condition admits a simplicial structure, provided any object allows an idempotent self-equivalence. Moreover, any two choices of simplicial structures give rise to equivalent…

Text simplification reduces the language complexity of professional content for accessibility purposes. End-to-end neural network models have been widely adopted to directly generate the simplified version of input text, usually functioning…