Related papers: Lifting Cocycles: From Heuristic to Theory
In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in…
We survey both old and new developments in the theory of algorithms in real algebraic geometry -- starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for…
Topological data analysis has emerged as a powerful tool for analyzing large-scale data. An abstract simplicial complex, in principle, can be built from data points, and by using tools from homology, topological features could be…
We study the existence of first integrals in nonholonomic systems with symmetry. First we define the concept of $\mathcal{M}$-cotangent lift of a vector field on a manifold $Q$ in order to unify the works [Balseiro P., Arch. Ration. Mech.…
A classical approach to investigate a closed projective scheme $W$ consists of considering a general hyperplane section of $W$, which inherits many properties of $W$. The inverse problem that consists in finding a scheme $W$ starting from a…
Co-clustering is a data mining technique used to extract the underlying block structure between the rows and columns of a data matrix. Many approaches have been studied and have shown their capacity to extract such structures in continuous,…
We present a framework for learning Hamiltonian systems using data. This work is based on a lifting hypothesis, which posits that nonlinear Hamiltonian systems can be written as nonlinear systems with cubic Hamiltonians. By leveraging this,…
We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus >=2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we…
Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical…
In this paper we propose a heuristic technique for distributing points on the surface of a unit n-dimensional Euclidean sphere, generated as the orbit of a finite cyclic subgroup of orthogonal matrices, the so called cyclic group codes.…
We use rewriting systems to spell out cup-products in the (twisted) cohomology groups of a product of surface groups. This allows us to detect a non-trivial obstruction bounding from below the effective topological complexity of an…
For a morphism $f : X \to Y$ of schemes, we give a tropical criterion for which points of $Y$ (valued in a field, discrete valuation ring, number ring, or Dedekind domain) lift to $X$. Our criterion extends the firmaments of Abramovich to a…
Let P be a closed triangulated manifold, dim P=n. We consider the group of simplicial 1-chains C_1(P) and the homology group H_1(P). We also use some nonnegative weighting function L on C_1(P). For any homological class from H_1(P) method…
Topological landscape is introduced for networks with functions defined on the nodes. By extending the notion of gradient flows to the network setting, critical nodes of different indices are defined. This leads to a concise and…
The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally…
Topologically ordered systems are characterized by topological invariants that are often calculated from the momentum space integration of a certain function that represents the curvature of the many-body state. The curvature function may…
Thresholding--the pruning of nodes or edges based on their properties or weights--is an essential preprocessing tool for extracting interpretable structure from complex network data, yet existing methods face several key limitations.…
Cylindrical algebraic decomposition(CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. When using CAD, there is often a choice for the ordering placed on the variables.…
Feature extraction and matching are among central problems of computer vision. It is inefficent to search features over all locations and scales. Neurophysiological evidence shows that to locate objects in a digital image the human visual…
Topology Optimization (TO), which maximizes structural robustness under material weight constraints, is becoming an essential step for the automatic design of mechanical parts. However, existing TO algorithms use the Finite Element Analysis…