Coboundary Operator Research Articles

A refinement of Morse–Novikov cohomology on manifolds with boundary and the cohomology of the space of solutions of kerΔ𝜃

In this paper, we consider the Morse–Novikov coboundary operator [Formula: see text] where [Formula: see text], and which is given by [Formula: see text], and for which the associated Morse–Novikov cohomology groups are denoted by [Formula: see text]. The ultimate objective of this paper is to uncover more geometric and topological insights about [Formula: see text] when the boundary of [Formula: see text] is nonvanishing. To begin, we employ a different approach to prove that the concrete realizations of the Morse–Novikov cohomology groups of [Formula: see text] intersect at the origin, and then we use the long exact sequence and cohomological algebra as powerful tools to decompose the absolute and relative Morse–Novikov cohomologies into interior and boundary portions, respectively. On the other hand, we use a novel reasoning based on cohomological algebra to determine the cohomology of [Formula: see text], and as a result the absolute Morse–Novikov cohomology of [Formula: see text] can be characterized in terms of two successive degrees in just one degree of the cohomology of [Formula: see text]. Consequently, the cohomology of [Formula: see text] can be recovered from the interior and boundary sections.

Schottky cohomology for rank two bosonic vertex operator algebra

Using the Schottky procedure of forming a genus [Formula: see text] Riemann surface by attaching handles to the Riemann sphere, we construct coboundary operators and corresponding cohomology for the double complexes of rational functions associated to a particular example of the rank two bosonic vertex operator algebra [Formula: see text].

Reductive cohomology associated with vertex algebras

We review the notion of the reduction cohomology of vertex algebras. The algebraic conditions leading to the chain property for complexes of vertex operator algebra n-point functions (with their convergence assumed) with a coboundary operator defined through reduction formulas are studied. Algebraic, geometrical, and cohomological meanings of reduction formulas and chain condition are clarified. The reduction cohomology for vertex operator algebras associated to Jacobi forms is computed. A counterpart of the Bott-Segal theorem for Riemann surfaces in terms of the reductions cohomology is proven.

Ripser: efficient computation of Vietoris\u2013Rips persistence barcodes

We present an algorithm for the computation of Vietoris–Rips persistence barcodes and describe its implementation in the software Ripser. The method relies on implicit representations of the coboundary operator and the filtration order of the simplices, avoiding the explicit construction and storage of the filtration coboundary matrix. Moreover, it makes use of apparent pairs, a simple but powerful method for constructing a discrete gradient field from a total order on the simplices of a simplicial complex, which is also of independent interest. Our implementation shows substantial improvements over previous software both in time and memory usage.

The simplest cohomological invariants for vertex algebras

For the double complex structure of grading-restricted vertex algebra cohomology defined in [6,7], we introduce a multiplication of elements of double complex spaces. We show that the orthogonality and bi-grading conditions applied on double complex spaces, provide in relation among mappings and actions of co-boundary operators. Thus, we endow the double complex spaces with structure of bi-graded differential algebra. We then introduce the simplest cohomology classes for a grading-restricted vertex algebra, and show their independence on the choice of mappings from double complex spaces. We prove that its cohomology class does not depend on mappings representing of the double complex spaces. Finally, we show that the orthogonality relations together with the bi-grading condition bring about generators and commutation relations for a continual Lie algebra.

On Numerical Invariant of Graph

Let G = (V (G), E(G)) be a finite graph with n vertices, where V (G) = {v i, … , v n} denote vertex set of G and E(G) denote an edge set G. In this paper, we have introduced the graph G p , q for G. The motivation for introducing the graph G p , q are as follows: Many mathematicians studied the properties of graph G by using boundary operator and co-boundary operator which consider only the relation between vertices and an edges of a graph G. There is no place for complete subgraph (whose vertices greater than 2) of a graph G in boundary operator and co-boundary operator of G. Hence to study the relation between complete subgraph of G and also to study the properties of G, we have introduced an edge-boundary operator and an edge-co-boundary operator on G p,q .

On computational Poisson geometry II: Numerical methods

<p style='text-indent:20px;'>We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and modular vector fields, compute the image under the coboundary and trace operators, the Lie bracket of differential 1–forms, gauge transformations, and normal forms of Lie–Poisson structures on <inline-formula><tex-math id="M1">\begin{document}$ {\mathbf{R}^{{3}}} $\end{document}</tex-math></inline-formula>. The complexity of each of our methods is calculated, and we include experimental verifications on examples in dimensions two and three.</p>

The last formula of Jean-Louis Koszul

Since the first international conference in France on the Information Geometry, GSI2013, Jean-Louis Koszul’s interest in Information Geometry went increasing. Motivated by the impact of the cohomology of Koszul-Vinberg algebras on the Information Geometry and moved by some issues raised by Albert Nijenhuis, Jean-Louis Koszul undertook another rewriting of the Brut formula of the coboundary operator of the KV complex. The source of other motivations of Jean-Louis Koszul was the relationships between the theory of KV cohomology and the theory of deformation of locally flat manifolds.In 2015 Jean-Louis Koszul sent me his Last formula of the KV boundary operator. In that Last formula Jean-Louis Koszul dealt with the case where spaces of coefficients are trivial modules of KV algebras. A part of the present work is devoted to extending the Last formula of Jean-Louis Koszul to KV cochain complexes whose spaces of coefficients are non-trivial two-sided modules of KV algebras. At another side, I also aim to highlight other significant impacts of the theory of KV cohomology of Koszul-Vinberg algebras. In particular I will use the KV cohomology to widely revisit the theory of statistical models of measurable sets. The reader will see why the source of the theory of statistical models is of homological nature.I also intend to highlight several impacts of the KV cohomology on the quantitative differential topology. I am particularly concerned with problems regarding the existence of Riemannian foliations, the existence of symplectic foliations as well as the existence of multi-dimensional webs. The homological theory of statistical models is presented as branches of rooted trees whose roots are weakly Jensen random cohomology classes.

Poisson cohomology of broken Lefschetz fibrations

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary operator of an arbitrary Jacobian Poisson structure in 4 dimensions.

Simplicial degree in complex networks. Applications of topological data analysis to network science

Network Science provides a universal formalism for modelling and studying complex systems based on pairwise interactions between agents. However, many real networks in the social, biological or computer sciences involve interactions among more than two agents, having thus an inherent structure of a simplicial complex. We propose new notions of higher-order degrees of adjacency for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces, which as far as we know were lacked in the literature. We introduce multi-parameter boundary and coboundary operators in an oriented simplicial complex and also a novel multi-combinatorial Laplacian is defined, which generalises the graph and combinatorial Laplacian. To illustrate the potential applications of these theoretical results, we perform a structural analysis of higher-order connectivity in simplicial-complex networks by studying the associated distributions with these simplicial degrees in 17 real-world datasets coming from different domains such as coauthor networks, cosponsoring Congress bills, contacts in schools, drug abuse warning networks, e-mail networks or publications and users in online forums. We find rich and diverse higher-order connectivity structures and observe that datasets of the same type reflect similar higher-order collaboration patterns. Furthermore, we show that if we use what we have called the maximal simplicial degree (which counts the distinct maximal communities in which our simplex and all its strict sub-communities are contained), then its degree distribution is, in general, surprisingly different from the classical node degree distribution.

Derivations and Deformations of δ-Jordan Lie Supertriple Systems

Let T be a δ-Jordan Lie supertriple system. We first introduce the notions of generalized derivations and representations of T and present some properties. Also, we study the low-dimensional cohomology and the coboundary operator of T, and then we investigate the deformations and Nijenhuis operators of T by choosing some suitable cohomologies.

On Hom-Lie Superalgebras

In this paper, first we show that $$({\mathfrak {g}},[\cdot ,\cdot ],\alpha )$$ is a hom-Lie superalgebra if and only if $$(\wedge {\mathfrak {g}}^{*}, \alpha ^{*}, d)$$ is an $$(\alpha ^{*},\alpha ^{*})$$ -differential graded commutative superalgebra. Then, we revisit representations of hom-Lie superalgebras, and show that there are a series of coboundary operators. We also introduce the notion of an omni-hom-Lie superalgebra associated to a vector space and an even invertible linear map. We show that regular hom-Lie superalgebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie superalgebra. The underlying algebraic structure of the omni-hom-Lie superalgebra is a hom-Leibniz superalgebra.

Coboundary operators for infinite frameworks

Coboundary operators for infinite frameworks

Wave Functions, Creation and Annihilation Operators of Quantum Physical System

In this article we proposed derivation of the wave function of a quantum system from the functional representation of the state of this system. There is considered representation of creation and annihilation operators by boundary and co-boundary operators of chain and co-chain complexes on the physical space.

Sewn sphere cohomologies for vertex algebras

We define sewn elliptic cohomologies for vertex algebras by sewing procedure for coboundary operators.

Coboundary operators for infinite frameworks

Coboundary operators for infinite frameworks

On Traces of Fourier Integral Operators on Submanifolds

Given a smooth embedding of manifolds and a Fourier integral operator on the ambient manifold, the trace of this operator on the submanifold (i.e., its composition with the boundary and coboundary operators, which is an operator on the submanifold) is considered. Conditions under which such a trace is also a Fourier integral operator are determined, and its amplitude in canonical local coordinates is calculated. The results are applied to quantized canonical transformations.

Tractor calculus, BGG complexes, and the cohomology of cocompact Kleinian groups

For a compact, oriented, hyperbolic n-manifold (M,g), realised as M=Γ\\Hn where Γ is a torsion-free cocompact subgroup of SO(n,1), we establish and study a relationship between differential geometric cohomology on M and algebraic invariants of the group Γ. In particular for F an irreducible SO(n,1)-module, we show that the group cohomology with coefficients H•(Γ,F) arises from the cohomology of an appropriate projective BGG complex on M. This yields the geometric interpretation that H•(Γ,F) parameterises solutions to certain distinguished natural PDEs of Riemannian geometry, modulo the range of suitable differential coboundary operators. Viewed in another direction, the construction shows one way that non-trivial cohomology can arise in a BGG complex, and sheds considerable light on its geometric meaning. We also use the tools developed to give a new proof that H1(Γ,S0kRn+1)≠0 whenever M contains a compact, orientable, totally geodesic hypersurface. All constructions use another result that we establish, namely that the canonical flat connection on a hyperbolic manifold coincides with the tractor connection of projective differential geometry.

Subalgebras of the Gerstenhaber algebra of differential operators

We construct a family of subalgebras of the Gerstenhaber algebra of differential operators. The subalgebras are labeled by subsets of the additive group [Formula: see text] that are closed under addition. Each subalgebra is invariant under the Hochschild coboundary operator.

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.