Mathematics

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 are isomorphic for a transitive digraph. We also study the collapses defined by discrete gradient vector fields. Let G be a digraph and f a discrete Morse function. Assume the out-degree and in-degree of any zero-point of f on G are both 1. We prove that the original digraph G and its M -collapse G ~ have the same path homology groups.

In this paper, based on the embedded homology groups of hypergraphs defined in \cite{h1}, we define the product of hypergraphs and prove the corresponding Künneth formula of hypergraphs which can be generalized to the Künneth formula for the embedded homology of graded subsets of chain complexes with coefficients in a principal ideal domain.

Let X be a complex affine variety in C N , and let f: C N →C be a polynomial function whose restriction to X is nonconstant. For g: C N →C a general linear function, we study the limiting behavior of the critical points of the one-parameter family of f t :=f−tg as t→0 . Our main result gives an expression of this limit in terms of critical sets of the restrictions of g to the singular strata of (X,f) . We apply this result in the context of optimization problems. For example, we consider nearest point problems (e.g., Euclidean distance degrees) for affine varieties and a possibly nongeneric data point.

In this article, we define a new non-archimedian metric structure, called cophenetic metric, on persistent homology classes for all degrees. We then show that zeroth persistent homology together with the cophenetic metric and hierarchical clustering algorithms with a number of different metrics do deliver statistically verifiable commensurate topological information based on experimental results we obtained on different datasets. We also observe that the resulting clusters coming from cophenetic distance do shine in terms of internal and external evaluation measures such as silhouette score and the Rand index. Moreover, since the cophenetic metric is defined for all homology degrees, one can now display the inter-relations of persistent homology classes in all degrees via rooted trees.

Pirashvili's Dold-Kan type theorem for finite pointed sets follows from the identification in terms of surjections of the morphisms between the tensor powers of a functor playing the role of the augmentation ideal; these functors are projective. We give an unpointed analogue of this result: namely, we compute the morphisms between the tensor powers of the corresponding functor in the unpointed context. We also calculate the Ext groups between such objects, in particular showing that these functors are not projective; this is an important difference between the pointed and unpointed contexts. This work is motivated by our functorial analysis of the higher Hochschild homology of a wedge of circles.

A generalisation of the equivariant Dixmier-Douady invariant is constructed as a second-degree cohomology class within a new semi-equivariant Čech cohomology theory. This invariant obstructs liftings of semi-equivariant principal bundles that are associated to central exact sequences of structure groups in which each structure group is acted on by the equivariance group. The results and methods described can be applied to the study of complex vector bundles equipped with linear/anti-linear actions, such as Atiyah's Real vector bundles.

We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasi-convex functions. Given the measurement data, can one determine the dimension of this space? In this paper, we develop a method for inferring the intrinsic dimension of the data from measurements by quasi-convex functions, under natural generic assumptions. The dimension inference problem depends only on discrete data of the ordering of the measured points of space, induced by the sensor functions. We introduce a construction of a filtration of Dowker complexes, associated to measurements by quasi-convex functions. Topological features of these complexes are then used to infer the intrinsic dimension. We prove convergence theorems that guarantee obtaining the correct intrinsic dimension in the limit of large data, under natural generic assumptions. We also illustrate the usability of this method in simulations.

There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech--de Rham spectral sequence by making use of the {\it factor map} which connects the two de Rham complexes. As a consequence, it follows that the singular de Rham cohomology algebra of the irrational torus T θ is isomorphic to the tensor product of the original de Rham cohomology and the exterior algebra generated by a non-trivial flow bundle over T θ .

For primes p?? , K(K U p ) -- the algebraic K -theory spectrum of (KU ) ??p , Morava K -theory K(1) , and Smith-Toda complex V(1) , Ausoni and Rognes conjectured (alongside related conjectures) that L K(1) S 0 ????????uniti (KU ) ??p induces a map K( L K(1) S 0 )??v ?? 2 V(1)?�K(K U p ) h Z ? p ??v ?? 2 V(1) that is an equivalence. Since the definition of this map is not well understood, we consider K( L K(1) S 0 )??v ?? 2 V(1)??K(K U p )??v ?? 2 V(1) ) h Z ? p , which is induced by i and also should be an equivalence. We show that for any closed G< Z ? p , ? ??((K(K U p )??v ?? 2 V(1) ) hG ) is a direct sum of two pieces given by (co)invariants and a coinduced module, for K(K U p ) ??(V(1))[ v ?? 2 ] . When G= Z ? p , the direct sum is, conjecturally, K( L K(1) S 0 ) ??(V(1))[ v ?? 2 ] and, by using K( L p ) ??(V(1))[ v ?? 2 ] , where L p =((KU ) ??p ) hZ/((p??)Z) , the summands simplify. The Ausoni-Rognes conjecture suggests that in (??) h Z ? p ??v ?? 2 V(1)??K(K U p )??v ?? 2 V(1) ) h Z ? p , K(K U p ) fills in the blank; we show that for any G , the blank can be filled by (K(K U p ) ) dis O , a discrete Z ? p -spectrum built out of K(K U p ) .

The Python package ComCH is a lightweight specialized computer algebra system that provides models for well known objects, the surjection and Barratt-Eccles operads, parameterizing the product structure of algebras that are commutative in a derived sense. The primary examples of such algebras treated by ComCH are the cochain complexes of spaces, for which it provides effective constructions of Steenrod cohomology operations at all prime.