Singularities of Type-Q ABS Equations

The type-Q equations lie on the top level of the hierarchy introduced by Adler, Bobenko and Suris (ABS) in their classification of discrete counterparts of KdV-type integrable partial differential equations. We ask what singularities are possible in the solutions of these equations, and examine the relationship between the singularities and the principal integrability feature of multidimensional consistency. These questions are considered in the global setting and therefore extend previous considerations of singularities which have been local. What emerges are some simple geometric criteria that determine the allowed singularities, and the interesting discovery that generically the presence of singularities leads to a breakdown in the global consistency of such systems despite their local consistency property. This failure to be globally consistent is quantified by introducing a natural notion of monodromy for isolated singularities.Comment: contribution to the SIDE-9 special issue of SIGM

Multi-quadratic quad equations: integrable cases from a factorised-discriminant hypothesis

We give integrable quad equations which are multi-quadratic (degree-two) counterparts of the well-known multi-affine (degree-one) equations classified by Adler, Bobenko and Suris (ABS). These multi-quadratic equations define multi-valued evolution from initial data, but our construction is based on the hypothesis that discriminants of the defining polynomial factorise in a particular way that allows to reformulate the equation as a single-valued system. Such reformulation comes at the cost of introducing auxiliary (edge) variables and augmenting the initial data. Like the multi-affine equations listed by ABS, these new models are consistent in multidimensions. We clarify their relationship with the ABS list by obtaining Backlund transformations connecting all but the primary multi-quadratic model back to equations from the multi-affine class.Comment: 21 pages, 2 figures, 3 table

Elliptic Solutions of ABS Lattice Equations

Elliptic N-soliton-type solutions, i.e. solutions emerging from the application of N consecutive B\"acklund transformations to an elliptic seed solution, are constructed for all equations in the ABS list of quadrilateral lattice equations, except for the case of the Q4 equation which is treated elsewhere. The main construction, which is based on an elliptic Cauchy matrix, is performed for the equation Q3, and by coalescence on certain auxiliary parameters, the corresponding solutions of the remaining equations in the list are obtained. Furthermore, the underlying linear structure of the equations is exhibited, leading, in particular, to a novel Lax representation of the Q3 equation.Comment: 42 pages, 3 diagram

Economics of intelligent selection of wireless access networks in a market-based framework : a game-theoretic approach

The Digital Marketplace is a market-based framework where network operators offer communications services with competition at the call level. It strives to address a tussle between the actors involved in a heterogeneous wireless access network. However, as with any market-like institution, it is vital to analyze the Digital Marketplace from the strategic perspective to ensure that all shortcomings are removed prior to implementation. In this paper, we analyze the selling mechanism proposed in the Digital Marketplace. The mechanism is based on a procurement first-price sealed-bid auction where the network operators represent the sellers/bidders, and the end-user of a wireless service is the buyer. However, this auction format is somewhat unusual as the winning bid is a composition of both the network operator’s monetary bid and their reputation rating. We create a simple economic model of the auction, and we show that it is mathematically intractable to derive the equilibrium bidding behavior when there are N network operators, and we make only generic assumptions about the structure of the bidding strategies. We then move on to consider a scenario with only two network operators, and assume that network operators use bidding strategies which are linear functions of their costs. This results in the derivation of the equilibrium bidding behavior in that scenario