Solutions to higher Hamiltonians in the Toda hierarchies

A method is presented for constructing the general solution to higher Hamiltonians (nonquadratic in the momenta) of the Toda hierarchies of integrable models associated with a simple Lie group G. The method is representation independent and is based on a modified version of the Lax operator. It constitutes a generalization of the method used to construct the solutions of the Toda molecule models. The SL(3) and SL(4) cases are discussed in detail. © 1990 American Institute of Physics.

Toda and Volterra lattice equations from discrete symmetries of KP hierarchies

The discrete models of the Toda and Volterra chains are being constructed out of the continuum two-boson KP hierarchies. The main tool is the discrete symmetry preserving the Hamiltonian structure of the continuum models. The two-boson currents of KP hierarchy are being associated with sites of the corresponding chain by successive actions of discrete symmetry.

On discrete symmetries of the multi-boson KP hierarchies

We show that the multi-boson KP hierarchies possess a class of discrete symmetries linking them to discrete Toda systems. These discrete symmetries are generated by the similarity transformation of the corresponding Lax operator. This establishes a canonical nature of the discrete transformations. The spectral equation, which defines both the lattice system and the corresponding Lax operator, plays a key role in determining pertinent symmetry structure. We also introduce the concept of the square root lattice leading to a family of new pseudo-differential operators with covariance under additional Bäcklund transformations.

Affine Lie algebraic origin of constrained KP hierarchies

An affine sl(n + 1) algebraic construction of the basic constrained KP hierarchy is presented. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and it is shown that these approaches are equivalent. The model is recognized to be the generalized non-linear Schrödinger (GNLS) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-Bäcklund transformations and interpolate between GNLS and multi-boson KP-Toda hierarchies. Our construction uncovers the origin of the Toda lattice structure behind the latter hierarchy. © 1995 American Institute of Physics.

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.

Constrained KP models as integrable matrix hierarchies

We formulate the constrained KP hierarchy (denoted by cKP K+1,M) as an affine sl(M + K+ 1) matrix integrable hierarchy generalizing the Drinfeld-Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld-Sokolov hierarchy, we are able to find several new universal results valid for the cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac-Moody current algebra. An explicit example is given for the case sl(M + K + 1), for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple non-regular element E of sl(M + K+ 1) and the content of the center of the kernel of E. © 1997 American Institute of Physics.

Some comments on the bi(tri)-Hamiltonian structure of generalized AKNS and DNLS hierarchies

We give the correct prescriptions for the terms involving ∂ -1 xδ(x - y), in the Hamiltonian structures of the AKNS and DNLS systems, necessary for the Jacobi identities to hold. We establish that the sl(2) and sl(3) AKNS systems are tri-Hamiltonians and construct two compatible Hamiltonian structures for the sl(n) AKNS system. We give a method for the derivation of the recursion operator for the sl(n + 1) DNLS system, and apply it explicitly to the sl(2) case, showing that such a system is tri-Hamiltonian. © 1998 Elsevier Science B.V.

Supersymmetry and the KdV equations for integrable hierarchies with a half-integer gradation

Supersymmetry is formulated for integrable models based on the sl(2 1) loop algebra endowed with a principal gradation. The symmetry transformations which have half-integer grades generate supersymmetry. The sl(2 1) loop algebra leads to N=2 supersymmetric mKdV and sinh-Gordon equations. The corresponding N=1 mKdV and sinh-Gordon equations are obtained via reduction induced by twisted automorphism. Our method allows for a description of a non-local symmetry structure of supersymmetric integrable models. © 2003 Elsevier B.V. All rights reserved.

Backlund transformation for integrable hierarchies: example - mKdV hierarchy

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Relating network rigidity, time scale hierarchies, and expression noise in gene networks

Fluctuation-dissipation theorems can be used to predict characteristics of noise from characteristics of the macroscopic response of a system. In the case of gene networks, feedback control determines the "network rigidity," defined as resistance to slow external changes. We propose an effective Fokker-Planck equation that relates gene expression noise to topology and to time scales of the gene network. We distinguish between two situations referred to as normal and inverted time hierarchies. The noise can be buffered by network feedback in the first situation, whereas it can be topology independent in the latter.

Dominance hierarchies and social status ascent opportunity: Anticipatory behavioral and physiological adjustments in a Neotropical cichlid fish

In this work we characterized the social hierarchy of non-reproductive individuals of Cichlasoma dimerus (Heckel, 1840). independently for both sexes, and its relationship to the opportunity for social status ascent. Female and male individuals who were located on the top rank of the social hierarchy, ascended in social status when the opportunity arose, therefore indicating that dominance is directly correlated with social ascent likelihood. Dominance was positively correlated with size in males but not in females, suggesting for the latter a relationship with intrinsic features such as aggressiveness or personality rather than to body and/or ovarian size. Physiological and morphometrical variables related to reproduction, stress and body color were measured in non-reproductive fish and correlated with dominance and social ascent likelihood. Dominance was negatively correlated with plasma cortisol levels for both sexes. No correlation with dominance was found for androgen plasma levels (testosterone and 11-ketotestosterone). No correlation was detected between dominance and the selected morphological and physiological variables measured in females, suggesting no reproductive inhibition in this sex at a physiological level and that all females seem to be ready for reproduction. In contrast, social hierarchy of non-reproductive males was found to be positively correlated with follicle stimulating hormone (FSH) pituitary content levels and gonadosomatic indexes. This suggests an adaptive mechanism of non reproductive males, adjusting their reproductive investment in relation to their likelihood for social status ascent, as perceived by their position in the social hierarchy. This likelihood is translated into a physiological signal through plasma cortisol levels that inhibit gonad investment through pituitary inhibition of FSH, representing an anticipatory response to the opportunity for social status ascent. (C) 2012 Elsevier Inc. All rights reserved.

The effects of aging and musical experience on the representation of tonal hierarchies

Two experiments explored the representation of the tonal hierarchy in Western music among older (aged 60 to 80) and younger (aged 15 to 22) musicians and nonmusicians. A probe tone technique was used: 4 notes from the major triad were presented, followed by 1 note chosen from the 12 notes of the chromatic scale. Whereas musicians had a better sense of the tonal hierarchy than nonmusicians, older adults were no worse than younger adults in differentiating the notes according to musical principles. However, older adults were more prone than younger adults to classify the notes by frequency proximity (pitch height) when proximity was made more salient, as were nonmusicians compared with musicians. With notes having ambiguous pitch height, pitch height effects disappeared among older adults but not nonmusicians. Older adults seem to have internalized tonal structure, but they sometimes fail to inhibit less musically relevant information.