Related papers: Topological aspects of Boolean functions
We show how locally smooth actions of compact Lie groups on a manifold $X$ can be used to obtain new upper bounds for the topological complexity $\TC(X)$, in the sense of Farber. We also obtain new bounds for the topological complexity of…
We survey results for Cameron-Liebler sets and low degree Boolean functions for Hamming graphs, Johnson graphs and Grassmann graphs from the point of view of association schemes. This survey covers selected results in finite geometry,…
The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains. The examples include biharmonic Steklov…
The characterization of the pointwise limits of the sequences of \'Swi\k{a}tkowski functions is given. Modifications of \'Swi\k{a}tkowski property with respect to different topologies finer than the Euclidean topology are discussed.
Topological data analysis refers to approaches for systematically and reliably computing abstract ``shapes'' of complex data sets. There are various applications of topological data analysis in life and data sciences, with growing interest…
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal…
Systems of Boolean equations of low degree arise in a natural way when analyzing block ciphers. The cipher's round functions relate the secret key to auxiliary variables that are introduced by each successive round. In algebraic…
Mathematical methods provide useful framework for the analysis and design of complex systems. In newer contexts such as biology, however, there is a need to both adapt existing methods as well as to develop new ones. Using a combination of…
Logical models have been successfully used to describe regulatory and signaling networks without requiring quantitative data. However, existing data is insufficient to adequately define a unique model, rendering the parametrization of a…
Topological field theory in three dimensions provides a powerful tool to construct correlation functions and to describe boundary conditions in two-dimensional conformal field theories.
In this paper we discuss two different existing algorithms for computing topological entropy and we perform one of them in order to compute the isentropes for cubic polynomials.
Recent development of network structure analysis shows that it plays an important role in characterizing complex system of many branches of sciences. Different from previous network centrality measures, this paper proposes the notion of…
Clustering is one of the most common tasks of Machine Learning. In this paper we examine how ideas from topology can be used to improve clustering techniques.
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
Towards better understanding of gate elimination, the only method known that can prove complexity lower bounds for explicit functions against unrestricted Boolean circuits, this work contributes: (1) formalizing circuit simplifications as a…
Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional…
We use an alternative definition of topological complexity to show that the topological complexity of the mapping telescope of a sequence $X_1\rightarrow X_2\rightarrow X_3\rightarrow...$ is bounded above by $2max{TC(X_i); i=1,2,...}$.
In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function $f$ is…
It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental…
Horn functions form a subclass of Boolean functions and appear in many different areas of computer science and mathematics as a general tool to describe implications and dependencies. Finding minimum sized representations for such functions…