REDUCTION OF TODA LATTICE HIERARCHY TO GENERALIZED KDV HIERARCHIES AND THE 2-MATRIX MODEL

Toda lattice hierarchy and the associated matrix formulation of the 2M-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which Abelianize the second KP Hamiltonian structure, we are able to obtain a unified formalism for the reduced SL(M + 1, M - k) KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded SL(M + 1, M - k) matrices in the diagonal gauge and we describe their bracket structure and field content. In particular, we provide explicit free field representations of the associated W(M, M - k) Poisson bracket algebras generalising the familiar nonlinear W-M+1 algebra. Discrete Backlund transformations for SL(M + 1, M - k) KdV are generated naturally from lattice translations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the SL(M + 1, 1) KdV hierarchy.

THE REDUCTIVE PERTURBATION METHOD AND THE KORTEWEG-DE VRIES HIERARCHY

By using the reductive perturbation method of Taniuti with the introduction of an infinite sequence of slow time variables tau(1), tau(3), tau(5), ..., we study the propagation of long surface-waves in a shallow inviscid fluid. The Korteweg-de Vries (KdV) equation appears as the lowest order amplitude equation in slow variables. In this context, we show that, if the lowest order wave amplitude zeta(0) satisfies the KdV equation in the time tau(3), it must satisfy the (2n+1)th order equation of the KdV hierarchy in the time tau(2n+1), With n = 2, 3, 4,.... AS a consequence of this fact, we show with an explicit example that the secularities of the evolution equations for the higher-order terms (zeta(1), zeta(2),...) of the amplitude can be eliminated when zeta(0) is a solitonic solution to the KdV equation. By reversing this argument, we can say that the requirement of a secular-free perturbation theory implies that the amplitude zeta(0) satisfies the (2n+1)th order equation of the KdV hierarchy in the time tau(2n+1) This essentially means that the equations of the KdV hierarchy do play a role in perturbation theory. Thereafter, by considering a solitary-wave solution, we show, again with an explicit, example that the elimination of secularities through the use of the higher order KdV hierarchy equations corresponds, in the laboratory coordinates, to a renormalization of the solitary-wave velocity. Then, we conclude that this procedure of eliminating secularities is closely related to the renormalization technique developed by Kodama and Taniuti.

The Role of the Korteweg-de Vries Hierarchy in the N-Soliton Dynamics of the Shallow Water Wave Equation

We apply a multiple-time version of the reductive perturbation method to study long waves as governed by the shallow water wave model equation. As a consequence of the requirement of a secularity-free perturbation theory, we show that the well known N-soliton dynamics of the shallow water wave equation, in the particular case of α = 2β, can be reduced to the N-soliton solution that satisfies simultaneously all equations of the Korteweg-de Vries hierarchy.

Higher order Painleve equations and their symmetries via reductions of a class of integrable models

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Two-matrix string model as constrained (2+1)-dimensional integrable system

We show that the 2-matrix string model corresponds to a coupled system of 2 + 1-dimensional KP and modified KP ((m)KP2+1) integrable equations subject to a specific symmetry constraint. The latter together with the Miura-Konopelchenko map for (m)KP2+1 are the continuum incarnation of the matrix string equation. The (m)KP2+1 Miura and Backhand transformations are natural consequences of the underlying lattice structure. The constrained (m)KP2+1 system is equivalent to a 1 + 1-dimensional generalized KP-KdV hierarchy related to graded SL(3,1). We provide an explicit representation of this hierarchy, including the associated W(2,1)-algebra of the second Hamiltonian structure, in terms of free currents.

Toda lattice field theories, discrete W algebras, Toda lattice hierarchies and quantum groups

In analogy with the Liouville case we study the sl3 Toda theory on the lattice and define the relevant quadratic algebra and out of it we recover the discrete W3 algebra. We define an integrable system with respect to the latter and establish the relation with the Toda lattice hierarchy. We compute the relevant continuum limits. Finally we find the quantum version of the quadratic algebra.

Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Solitons from dressing in an algebraic approach to the constrained KP hierarchy

The algebraic matrix hierarchy approach based on affine Lie sl(n) algebras leads to a variety of 1 + 1 soliton equations. By varying the rank of the underlying sl(n) algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy.The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine sl(n) algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time Bows which distinguishes them From the conventional structure of the Darboux-Backlund-Wronskian solutions of the constrained KP hierarchy.

A class of soliton solutions for the N=2 super mKdV/Sinh-Gordon hierarchy

Employing Hirota's method, a class of soliton solutions for the N = 2 super mKdV equations is proposed in terms of a single Grassmann parameter. Such solutions are shown to satisfy two copies of N = 1 supersymmetric mKdV equations connected by nontrivial algebraic identities. Using the super Miura transformation, we obtain solutions of the N = 2 super KdV equations. These are shown to generalize solutions derived previously. By using them KdV/sinh-Gordon hierarchy properties we generate the solutions of the N = 2 super sinh-Gordon as well.

Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Application of the analytic hierarchy process (AHP) in multi-criteria analysis of the selection of intelligent building systems

The availability of innumerable intelligent building (IB) products, and the current dearth of inclusive building component selection methods suggest that decision makers might be confronted with the quandary of forming a particular combination of components to suit the needs of a specific IB project. Despite this problem, few empirical studies have so far been undertaken to analyse the selection of the IB systems, and to identify key selection criteria for major IB systems. This study is designed to fill these research gaps. Two surveys: a general survey and the analytic hierarchy process (AHP) survey are proposed to achieve these objectives. The first general survey aims to collect general views from IB experts and practitioners to identify the perceived critical selection criteria, while the AHP survey was conducted to prioritize and assign the important weightings for the perceived criteria in the general survey. Results generally suggest that each IB system was determined by a disparate set of selection criteria with different weightings. ‘Work efficiency’ is perceived to be most important core selection criterion for various IB systems, while ‘user comfort’, ‘safety’ and ‘cost effectiveness’ are also considered to be significant. Two sub-criteria, ‘reliability’ and ‘operating and maintenance costs’, are regarded as prime factors to be considered in selecting IB systems. The current study contributes to the industry and IB research in at least two aspects. First, it widens the understanding of the selection criteria, as well as their degree of importance, of the IB systems. It also adopts a multi-criteria AHP approach which is a new method to analyse and select the building systems in IB. Further research would investigate the inter-relationship amongst the selection criteria.

Measuring building quality of shopping centres in Surabaya by Analytical Hierarchy Process (AHP)

Quality has been an important factor for shopping centers in competitive conditions. However, quality measurement has no standard. In Surabaya, only two regional shopping centers will be measured in this research. The objective is assessing quality of shopping centers building using Analytical Hierarchy Process (AHP) method and calculating the Building Quality Index. An overall ranking of Hierarchy priorities of quality criteria founded as a result from AHP analysis. Access and Circulation became the highest priority in affecting quality of shopping centers building according to respondents’ perception of quality. Weightened value as a result from comparison between two shopping centers as follows: Tunjungan Plaza get 0,732 point and Surabaya Plaza get 0,268 point. The first shopping center got higher weight than the second shopping center. The BQI for Tunjungan Plaza is 66% and for Surabaya Plaza is 64%.

Tea ceremony and Sukiya : negating social hierarchy

There is a widely held view that architecture is very strongly and even primarily determined by the society and culture rather than geographical, technical, economic or cli-matic factors. This is especially evident in societies where rituals, customs and tradition play a significant role in the design of built forms. One such society was that of Feudal Japan under the rule of samurai warriors. The strictly controlled hierarchical society of Feudal Japan, isolated from the rest of the world for over 250 years, was able to develop the art and architecture borrowed from neighboring older cultures of China and Korea into what is now considered uniquely Japanese. One such architecture is the Sukiya style tea houses where the ritual of tea ceremony took place. This ritual was developed by the tea masters who were Zen monks or the merchants who belonged to the lowest class in the hierarchical feudal society. The Sukiya style developed from 14th to 16th century and became an architectural space that negated all the rules imposed on commoners by the samurai rulers. The tea culture had a major influence on Japanese architecture, the concept of space and aesthetics. It extended into the design of Japanese gardens, clothes, presentation of food, and their manners in day to day life. The focus of this paper is the Japanese ritual of tea ceremony, the architecture of the tea house it inspired, the society responsible for its creation and the culture that promoted its popularity and its continuation into the 21st century.