arXiv research

Quantum cellular automata form a homology theory.

This article introduces descriptive cellular homology on cell complexes, which is an extension of J.H.C. Whitehead's CW topology. A main result is that a descriptive cellular complex is a topology on fibres in a fibre bundle. An application of two forms of cellular homology is given in terms of the persistence of shape…

The paper computes the cellular homology of real flag manifolds.

In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincare formula and an equivariant Lefschetz theorem.

New stability theorem for nonorientable surfaces mapping class groups.

Geometric proof shows topological invariance of handle homology.

The study finds infinitely many distinct prime geodesics on Riemannian manifolds.

The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $\mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $\mathbb{D}^n$; (2) Every $n$-dimensional cell-like compactum can be embedded into $(2n+1)$-dimensional …

We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of…

In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a $\mathfrak{gl}_n$-web version of Hu--Mathas' graded cellular basis and has two major application…

Generalizes Milnor's axiomatic homology theorems to Polish spaces.

Spatial refinement of Bar-Natan homology constructed.

We study the string topology of a closed oriented Riemannian manifold M. We describe a compact moduli space of diagrams, and show how the cellular chain complex of this space gives algebraic operations on the singular chains of the free loop space LM of M. These operations are well-defined on the homology of a quotient…

Extends quantum annular homology to infinite sets.

Knot lattice homology invariant of smooth knot type in rational homology spheres.

In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are re…

We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we …

This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a local homeomorphism of $\mathds{R}^d$ and an invariant and isolated acyclic continuum, such as a cellular set or a fixed point. In this setting, we obtain a complete description of the first discrete h…

The study classifies cellular pseudomanifolds and their properties.

The paper examines topological features of ReLU networks and their relation to decision boundaries and training loss.

The paper studies Morse theory on manifolds with boundaries, constructing cellular structures and estimating critical points.

The paper extends the Cheeger-Müller theorem to spaces with conical singularities.

We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect…

Optimizes natural frequencies of cellular composites with various microstructures.

Machine learning attacks mimic cellular decision-making, revealing new defense mechanisms.

For leveled spatial graphs, we find a surface embedding that allows cellular embedding.

We give examples of harmonic cellular maps between negatively curved manifolds which are not diffeomorphisms but are homotopic to diffeomorphisms.

The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally …

Study immersions of punctured 4-manifolds for quantum automata applications.

The paper extends Gaussian processes to model complex interactions in cellular complexes.

ChemCPA predicts cellular responses to novel drugs using transfer learning.

Paper proposes RL algorithms for optimizing cellular network performance.

New findings on hyperbolicity of augmented links in thickened surfaces.

Solves challenges of drone communication in cellular networks.

This paper develops a discrete theory of real Riemann surfaces using quad-graphs and linear discretization.

TrajectoryNet models dynamic cellular trajectories using optimal transport.

Kernel-based algorithm optimizes cellular network configuration through multi-task learning.

We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We gi…

CURIE uses cellular automata to detect concept drift in data streams.

New method enhances graph neural networks using contractions and hourglass persistence.

Electron Cryo-Tomography (ECT) allows 3D visualization of subcellular structures at the submolecular resolution in close to the native state. However, due to the high degree of structural complexity and imaging limits, the automatic segmentation of cellular components from ECT images is very difficult. To complement an…

Classifies fake surfaces up to complexity 5.

The paper optimizes UAV path and power for QoS in cellular networks.

Motivation: Understanding functions of proteins in specific human tissues is essential for insights into disease diagnostics and therapeutics, yet prediction of tissue-specific cellular function remains a critical challenge for biomedicine. Results: Here we present OhmNet, a hierarchy-aware unsupervised node feature le…

We show that tree almost automorphism groups, including Neretin groups, satisfy the analogue of the $F_\infty$-finiteness condition in the world of totally disconnected groups: They possess a cellular action on a contractible cellular complex such that the stabilizers are open and compact and the restriction of the act…

The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe…

Designs a Cellular Automata rule for forming touching loop patterns.

In this brief sequel to a previous article, we recall the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by cou…