Related papers: Constrained KP Models as Integrable Matrix Hierarc…
In this paper, the dispersionless D type Drinfeld-Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this herarchy are presented. These flows form an infinite…
We introduce a generalisation of the KP hierarchy, closely related to the cyclic quiver and the Cherednik algebra $H_k(\mathbb Z_m)$. This hierarchy depends on $m$ parameters (one of which can be eliminated), with the usual KP hierarchy…
The gauge equivalence between basic KP hierarchies is discussed. The first two Hamiltonian structures for KP hierarchies leading to the linear and non-linear $\Winf$ algebras are derived. The realization of the corresponding generators in…
The K-means algorithm is among the most commonly used data clustering methods. However, the regular K-means can only be applied in the input space and it is applicable when clusters are linearly separable. The kernel K-means, which extends…
We study branching problems for affine Kac--Moody algebras. Unlike the finite-dimensional case, an affine Kac--Moody algebra may contain proper subalgebras isomorphic to itself, such as winding subalgebras obtained by rescaling the loop…
We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups…
We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the…
Inspired by the squared eigenfunction symmetry constraint, we introduce a new $\ta_k$-flow by ``extending'' a specific $t_n$-flow of discrete KP hierarchy (DKPH). We construct extended discrete KPH (exDKPH), which consists of $t_n$-flow,…
We discuss a differential integrable hierarchy, which we call the (N, M)$--th KdV hierarchy, whose Lax operator is obtained by properly adding $M$ pseudo--differential terms to the Lax operator of the N--th KdV hierarchy. This new hierarchy…
We construct the hierarchy of a multi-component generalisation of modified KdV equation and find exact solutions to its associated members. The construction of the hierarchy and its conservation laws is based on the Drinfel'd-Sokolov…
We introduce controlled $KK$-theory groups associated to a pair $(A,B)$ of separable $C^*$-algebras. Roughly, these consist of elements of the usual $K$-theory group $K_0(B)$ that approximately commute with elements of $A$. Our main results…
In this paper we shall prove that the subalgebra generated over the integers by the divided powers of the Drinfeld generators $x_r^{\pm}$ of the Kac-Moody algebra of type $A_2^{(2)}$ is an integral form (strictly smaller than Mitzman's (see…
We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko (GLM) equation. This is a linear integral equation in which…
We show that the $m$-dimensional Euler--Manakov top on $so^*(m)$ can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety $\bar{\cal V}(k,m)$, and present its Lax representation…
In this paper, we give the form of the q-cmKP hierarchy generated by the gauge transformation operator $T_{n+k}$. We show a necessary and sufficient condition to reduce the generalised q-Wronskian solutions from the q-mKP hierarchy to…
A method is proposed to construct a new extended KP hierarchy, which includes two types of KP equation with self-consistent sources and admits reductions to k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It…
A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C*-algebras. We show how to build a (k+1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct…
In the context of deep learning with kernel machines, the deep Restricted Kernel Machine (DRKM) framework allows multiple levels of kernel PCA (KPCA) and Least-Squares Support Vector Machines (LSSVM) to be combined into a deep architecture…
We propose a formally completely integrable extension of heat hierarchy based on the space of symmetries isomorphic to the Weyl algebra $\mathcal{A}_1$. The extended heat hierarchy will be the basic model for the analysis of the extension…
The Kruithof iterative scaling process, which adjusts matrices to meet target row and column sums, is a longstanding problem that lacks a general closed form for its limit. While Nathanson derived the closed form for the Sinkhorn limit of…