Related papers: A primer on Script Geometry
Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader field of…
Fine-grained complexity theory is the area of theoretical computer science that proves conditional lower bounds based on the Strong Exponential Time Hypothesis and similar conjectures. This area has been thriving in the last decade, leading…
In this study, we delve into the discrete TC of surjective simplicial fibrations, aiming to unravel the interplay between topological complexity, discrete geometric structures, and computational efficiency. Moreover, we examine the…
We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward…
Digraphs are generalizations of graphs in which each edge is assigned with a direction or two directions. In this paper, we define discrete Morse functions on digraphs, and prove that the homology of the Morse complex and the path homology…
In this paper, we investigate vector fields on polyhedral complexes and their associated trajectories. We study vector fields which are analogue of the gradient vector field of a function in the smooth case. Our goal is to define a nice…
Dispersion is a fundamental concept in statistics, yet standard approaches - especially via stochastic orders - face limitations in the discrete setting. In particular, the classical dispersive order, well-established for continuous…
This paper is devoted to the study of a newly introduced tool, projectional coderivatives and the corresponding calculus rules in finite dimensions. We show that when the restricted set has some nice properties, more specifically, is a…
We find a necessary and sufficient condition for a doubling metric space to carry a (1,p)-Poincare inequality. The condition involves discretizations of the metric space and Poincare inequalities on graphs.
This paper defines the basis of a new hierarchical framework for segmentation algorithms based on energy minimization schemes. This new framework is based on two formal tools. First, a combinatorial pyramid encode efficiently a hierarchy of…
Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an effective way of handling such inference in the presence of epistemic uncertainty -- unfortunately, different approaches to…
Gauge invariance in discrete dynamical systems and its connection with quantization are considered. For a complete description of gauge symmetries of a system we construct explicitly a class of groups unifying in a natural way the space and…
The present paper develops a variational theory of discrete fields defined on abstract cellular complexes. The discrete formulation is derived solely from a variational principle associated to a discrete Lagrangian density on a discrete…
We introduce a new measure of complexity (called spectral complexity) for directed graphs. We start with splitting of the directed graph into its recurrent and non-recurrent parts. We define the spectral complexity metric in terms of the…
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial…
It is increasingly common for data to possess intricate structure, necessitating new models and analytical tools. Graphs, a prominent type of structure, can encode the relationships between any two entities (nodes). However, graphs neither…
In this paper, we study the construction and transformation of two-dimensional persistent graphs. Persistence is a generalization to directed graphs of the undirected notion of rigidity. In the context of moving autonomous agent formations,…
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but…
Piecewise-linear (PL) Morse theory and discrete Morse theory are used in shape analysis tasks to investigate the topological features of discretized spaces. In spite of their common origin in smooth Morse theory, various notions of critical…
Starting from the working hypothesis that both physics and the corresponding mathematics and in particular geometry have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this…