On the resolution of points in generic position

ABSTRACT

It is shown that a Yang–Baxter system can be constructed from any entwining structure. It is also shown that,conversely,Yang–Baxter systems of certain types lead to entwining structures. Examples of Yang–Baxter systems associated to entwining structures are given,and a Yang–Baxter operator of Hecke type is defined for any bijective entwining map.     

Noncommutative Riemannian geometry on graphs

We show that arising out of noncommutative geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative Laplace-Beltrami operator on differential 1-forms, where we use the language of differential algebras to functorially interpret a graph as providing a ‘finite manifold structure’ on the set of vertices. We equip any graph with a canonical ‘Euclidean metric’ and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations.     

Low‐frequency line temperatures of the CMB (Cosmic Microwave Background)

Based on SU(2) Yang‐Mills thermodynamics we interprete Aracde2's and the results of earlier radio‐surveys on low‐frequency cosmic microwave background (CMB) line temperatures as a phase‐boundary effect. We explain the excess at low frequencies by evanescent, nonthermal photon fields of the CMB whose intensity is nulled by that of Planck distributed calibrator photons. The CMB baseline temperature thus is identified with the critical temperature of the deconfining‐preconfining transition.     

On bi-skew braces and brace blocks

L. N. Childs defined a bi-skew brace to be a skew brace such that if we swap the role of the two operations, then we find again a skew brace.In this paper, we give a systematic analysis of bi-skew braces. We study nilpotency and solubility, and connections between bi-skew braces and set-theoretic solutions of the Yang–Baxter equation. Further, we deal with Byott's conjecture in the case of bi-skew braces, and we use bi-skew braces as a tool to solve a classification problem proposed by L. Vendramin.In the final part, we investigate brace blocks, defined by A. Koch to be families of group operations on a given set such that any two of them yield a bi-skew brace. We provide a characterisation of brace blocks, illustrate how all known constructions in literature follow in a natural way from our characterisation, and give several new examples.     

Spectral-Parameter Dependent Yang–Baxter Operators and Yang–Baxter Systems From Algebra Structures

For any algebra, two families of colored Yang–Baxter operators are constructed, thus producing solutions to the two-parameter quantum Yang–Baxter equation. An open problem about a system of functional equations is stated. The matrix forms of these operators for two and three dimensional algebras are computed. A FRT bialgebra for one of these families is presented. Solutions for the one-parameter quantum Yang–Baxter equation are derived and a Yang–Baxter system constructed.     

杨乐不等式的探微

本文利用分析方法及不等式理论 ,建立了杨乐不等式及其推广与加强的探微形式 .     

A note on the deformed Hermitian Yang-Mills PDE

We prove a priori estimates for a generalised Monge–Ampère PDE with ‘non-constant coefficients’ thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob–Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins–Szèkelyhidi on toric varieties and use it to address a conjecture of Collins–Jacob–Yau.