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Articles 1 - 8 of 8
Full-Text Articles in Mathematics
Cohomology Of Frobenius Algebras And The Yang-Baxter Equation, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Enver Karadayi, Masahico Saito
Cohomology Of Frobenius Algebras And The Yang-Baxter Equation, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Enver Karadayi, Masahico Saito
Mathematics Faculty Works
A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.
Torsion In One-Term Distributive Homology, Alissa S. Crans, Józef H. Przytycki, Krzysztof K. Putyra
Torsion In One-Term Distributive Homology, Alissa S. Crans, Józef H. Przytycki, Krzysztof K. Putyra
Mathematics Faculty Works
The one-term distributive homology was introduced by J.H.Przytycki as an atomic replacement of rack and quandle homology, which was first introduced and developed by R.Fenn, C.Rourke and B.Sanderson, and J.S.Carter, S.Kamada and M.Saito. This homology was initially suspected to be torsion-free, but we show in this paper that the one-term homology of a finite spindle can have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their homology precisely. In addition, we show that any finite group can appear as the torsion subgroup of the first homology of some finite spindle. Finally, …
An Adaptive Total Variation Algorithm For Computing The Balanced Cut Of A Graph, Xavier Bresson, Thomas Laurent, David Uminsky, James H. Von Brecht
An Adaptive Total Variation Algorithm For Computing The Balanced Cut Of A Graph, Xavier Bresson, Thomas Laurent, David Uminsky, James H. Von Brecht
Mathematics Faculty Works
We propose an adaptive version of the total variation algorithm proposed in [3] for computing the balanced cut of a graph. The algorithm from [3] used a sequence of inner total variation minimizations to guarantee descent of the balanced cut energy as well as convergence of the algorithm. In practice the total variation minimization step is never solved exactly. Instead, an accuracy parameter is specified and the total variation minimization terminates once this level of accuracy is reached. The choice of this parameter can vastly impact both the computational time of the overall algorithm as well as the accuracy of …
A Method Based On Total Variation For Network Modularity Optimization Using The Mbo Scheme, Huiyi Hu, Thomas Laurent, Mason A. Porter, Andrea L. Bertozzi
A Method Based On Total Variation For Network Modularity Optimization Using The Mbo Scheme, Huiyi Hu, Thomas Laurent, Mason A. Porter, Andrea L. Bertozzi
Mathematics Faculty Works
The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called “communities.” One popular approach to finding communities is to maximize a quality function known as modularity to achieve some sort of optimal clustering of nodes. In this paper, we interpret the modularity function from a novel perspective: we reformulate modularity optimization as a minimization problem of an energy functional that consists of a total variation term and an $\ell_2$ balance term. By employing numerical techniques …
Twisted Alexander Polynomials Of 2-Bridge Knots, Jim Hoste, Patrick D. Shanahan
Twisted Alexander Polynomials Of 2-Bridge Knots, Jim Hoste, Patrick D. Shanahan
Mathematics Faculty Works
We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.
Polynomial Knot And Link Invariants From The Virtual Biquandle, Alissa S. Crans, Allison Henrich, Sam Nelson
Polynomial Knot And Link Invariants From The Virtual Biquandle, Alissa S. Crans, Allison Henrich, Sam Nelson
Mathematics Faculty Works
The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gr\"obner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.
Counting Links And Knots In Complete Graphs, Loren Abrams, Blake Mellor, Lowell Trott
Counting Links And Knots In Complete Graphs, Loren Abrams, Blake Mellor, Lowell Trott
Mathematics Faculty Works
We investigate the minimal number of links and knots in embeddings of complete partite graphs in S3. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links in an embedding of K4,4,1 is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.
Congruence Classes Of 2-Adic Valuations Of Stirling Numbers Of The Second Kind, Curtis Bennett, Edward Mosteig
Congruence Classes Of 2-Adic Valuations Of Stirling Numbers Of The Second Kind, Curtis Bennett, Edward Mosteig
Mathematics Faculty Works
We analyze congruence classes of S(n,k), the Stirling numbers of the second kind, modulo powers of 2. This analysis provides insight into a conjecture posed by Amdeberhan, Manna and Moll, which those authors established for k at most 5. We provide a framework that can be used to justify the conjecture by computational means, which we then complete for values of k between 5 and 20.