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Many complex networks display a mesoscopic structure with groups of nodes sharing many links with the other nodes in their group and comparatively few with nodes of different groups. This feature is known as community structure and encodes…
In this paper we obtain some noncommutative multiplier theorems and maximal inequalities on semigroups. As applications, we obtain the corresponding individual ergodic theorems. Our main results extend some classical results of Stein and…
We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients.…
A filtration of Soergel bimodules by virtual crossing bimodules extends to Rouquier's complexes associated with braid words. We show that these complexes are invariant up to filtered homotopy with respect to the second Reidemeister move,…
Link prediction is an elemental challenge in network science, which has already found applications in guiding laboratorial experiments, digging out drug targets, recommending friends in social networks, probing mechanisms in network…
The codewords of the homomorphism code $\operatorname{aHom}(G,H)$ are the affine homomorphisms between two finite groups, $G$ and $H$, generalizing Hadamard codes. Following the work of Goldreich--Levin (1989), Grigorescu et al. (2006),…
We describe completely the link invariants constructed using Markov traces on the Yokonuma-Hecke algebras in terms of the linking matrix and the HOMFLYPT polynomials of sublinks.
This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's…
We algorithmically compute integral Eilenberg-MacLane homology of all semigroups of order at most $8$ and present some particular semigroups with notable classifying spaces, refuting conjectures of Nico. Along the way, we give an…
We construct an action of a polynomial ring on the colored sl(2) link homology of Cooper-Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and…
In the 1970s, Williams developed an algorithm that has been used to construct and study modular links in the Lorenz template. We introduce an improved algorithm, which we call the bunch algorithm, to provide more insights into the geometry…
The aim of this paper is to investigate the homology groups of mathematical models of concurrency. We study the Baues-Wirsching homology groups of a small category associated with a partial monoid action on a set. We prove that these groups…
Given any unoriented link diagram, a group of new knot invariants are constructed. Each of them satisfies a generalized 4 term skein relation. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations…
Link invariants, for 3-manifolds, are defined in the context of the Rozansky-Witten theory. To each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold X. The invariants are evaluated for b_{1}(M)…
The inference of rankings plays a central role in the theory of social choice, which seeks to establish preferences from collectively generated data, such as pairwise comparisons. Examples include political elections, ranking athletes based…
We study the problem of identifying different behaviors occurring in different parts of a large heterogenous network. We zoom in to the network using lenses of different sizes to capture the local structure of the network. These network…
We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a…
The histogram is an analysis tool in widespread use within many sciences, with high energy physics as a prime example. However, there exists an inherent bias in the choice of binning for the histogram, with different choices potentially…
We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that…
We propose a novel and efficient method for link prediction in bipartite networks, using \textit{formal concept analysis} (FCA) and the Transformer encoder. Link prediction in bipartite networks finds practical applications in various…