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This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. The main feature of our approach is simplicity, requiring ...

We present a compendium of Hodge decompositions of vector fields on tetrahedral meshes embedded in the 3D Euclidean space. After describing the foundations of the Hodge decomposition in the continuous setting, we describe how to implement a five-...

We unify the resource-theoretic and the cohomological perspective on quantum contextuality. At the center of this unification stands the notion of the contextual fraction. For both symmetry and parity based contextuality proofs, we establish ...

We provide a cohomological framework for contextuality of quantum mechanics that is suited to describing contextuality as a resource in measurement-based quantum computation. This framework applies to the parity proofs first discussed by Mermin, as well ...

Algorithms for persistent homology are well-studied where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under Z2 coefficients for a (monotone) sequence of general simplicial maps ...

Following Baryshnikov-Coffman-Kwak, we use cyclic network automata (CNA) to generate a decentralized protocol for dynamic coverage problems in a sensor network, with only a small fraction of sensors awake at every moment. This paper gives a rigorous ...

Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combinatorial structure of Q(I) and obtain a homeomorphic cellular complex P(I) with fewer cells. We introduce formulas for a diagonal ...

In this paper, we provide a graph-based representation of the homology (information related to the different "holes" the object has) of a binary digital volume. We analyze the digital volume AT-model representation [8] from this point of view and the ...

When integrating a differential equation numerically, it can be important for the solution method to reflect the geometric properties of the original model. These include

conservation laws and first integrals, symmetries, and symplectic or variational ...

We attempt to understand a cohomological approach to lower bounds in Boolean circuits (of [Fri05]) by studying a very restricted case; in this case Boolean complexity is described via the kernel (or nullspace) of a fairly simple linear transformation ...

Let P_K (n,d) be the set of polynomials in n variables of degree at most d over the field K of characteristic zero. We show that there is a number c _n,d such that if f ∈ P_K (n,d) then the algebraic de Rham cohomology group H_dR ⁱ (Kⁿ \Var(f)) has ...

If a GQ \S^\prime of order ( s, s ) is contained in a GQ \S of order ( s, s ²) as a subquadrangle, then for each point X of \S{\setminus}\S^\prime the set of points \O_X of \S^\prime collinear with X form an ovoid of \S^\prime . Thas and Payne ...

We define a contravariant functorKfrom the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graphX, an abelian groupB, and a nonnegative integerj, an element of Hom(Kj(X),...

Let {\cal D}_{n, k} be the family of linear subspaces of \Bbb {R}^n given by all equations of the form \epsilon_1x_{i_1}=\epsilon_2 x_{i_2} = \cdots = \epsilon_k x_{i_k} for 1 \leq i_1 \lt \cdots \lt i_k \leq n and (\epsilon_1,\ldots , \epsilon_k)\...