Regular Conjugacy Classes in the Weyl Group and Integrable Hierarchies

Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to grade one regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra \G\otimes{\bf C}[\lambda,\lambda^{-1}] are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group {\bf W}(\G) of the simple Lie algebra \G. A representative w\in {\bf W}(\G) of a regular conjugacy class can be lifted to an inner automorphism of \G given by $\hat w=\exp\left(2i\pi {\rm ad I_0}/m\right)$, where $I_0$ is the defining vector of an $sl_2$ subalgebra of \G.The grading is then defined by the operator $d_{m,I_0}=m\lambda {d\over d\lambda} + {\rm ad} I_0$ and any grade one regular element $\Lambda$ from the Heisenberg subalgebra associated to $[w]$ takes the form $\Lambda = (C_+ +\lambda C_-)$, where $[I_0, C_-]=-(m-1) C_-$ and $C_+$ is included in an $sl_2$ subalgebra containing $I_0$. The largest eigenvalue of ${\rm ad}I_0$ is $(m-1)$ except for some cases in $F_4$, $E_{6,7,8}$. We explain how these Lie algebraic results follow from known results and apply them to construct integrable systems.If the largest ${\rm ad} I_0$ eigenvalue is $(m-1)$, then using any grade one regular element from the Heisenberg subalgebra associated to $[w]$ we can construct a KdV system possessing the standard \W-algebra defined by $I_0$ as its second Poisson bracket algebra. For \G a classical Lie algebra, we derive pseudo-differential Lax operators for those non-principal KdV systems that can be obtained as discrete reductions of KdV systems related to $gl_n$. Non-abelian Toda systems are also considered.Comment: 44 pages, ENSLAPP-L-493/94, substantial revision, SWAT-95-77. (use OLATEX (preferred) or LATEX

Similarity reduction of the modified Yajima-Oikawa equation

We study a similarity reduction of the modified Yajima-Oikawa hierarchy. The hierarchy is associated with a non-standard Heisenberg subalgebra in the affine Lie algebra of type A_2^{(1)}. The system of equations for self-similar solutions is presented as a Hamiltonian system of degree of freedom two, and admits a group of B\"acklund transformations isomorphic to the affine Weyl group of type A_2^{(1)}. We show that the system is equivalent to a two-parameter family of the fifth Painlev\'e equation.Comment: latex2e file, 18 pages, no figures; (v2)Introduction is modified. Some typos are correcte

Dressing Technique for Intermediate Hierarchies

A generalized AKNS systems introduced and discussed recently in \cite{dGHM} are considered. It was shown that the dressing technique both in matrix pseudo-differential operators and formal series with respect to the spectral parameter can be developed for these hierarchies.Comment: 16 pages, LaTeX Report/no: DFTUZ/94/2

Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves

We define complexes of vector bundles on products of moduli spaces of framed rank r torsion-free sheaves on the complex projective plane. The top non-vanishing Chern classes of the cohomology of these complexes yield actions of the r-colored Heisenberg and Clifford algebras on the equivariant cohomology of the moduli spaces. In this way we obtain a geometric realization of the boson-fermion correspondence and related vertex operators.Comment: 36 pages; v2: Definition of geometric Heisenberg operators modified; v3: Minor typos correcte

On Separation of Variables for Integrable Equations of Soliton Type

We propose a general scheme for separation of variables in the integrable Hamiltonian systems on orbits of the loop algebra $\mathfrak{sl}(2,\Complex)\times \mathcal{P}(\lambda,\lambda^{-1})$. In particular, we illustrate the scheme by application to modified Korteweg--de Vries (MKdV), sin(sinh)-Gordon, nonlinear Schr\"odinger, and Heisenberg magnetic equations.Comment: 22 page

Generalized Drinfeld-Sokolov Reductions and KdV Type Hierarchies

Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al, reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra $\ell(gl_n)$, graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions of $n$ into the sum of equal numbers $n=pr$ or to equal numbers plus one $n=pr+1$. We prove that the reduction belonging to the grade $1$ regular elements in the case $n=pr$ yields the $p\times p$ matrix version of the Gelfand-Dickey $r$-KdV hierarchy, generalizing the scalar case $p=1$ considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even for $p=1$.Comment: 43 page

Unheard voices: A qualitative study of LGBT+ older people experiences during the first wave of the COVID-19 pandemic in the UK

This paper reports findings from a qualitative study into the immediate impact of social distancing measures on the lives of lesbian, gay, bisexual and trans (LGBT+) older people (≥60 years) living in the UK during the first lockdown of the COVID-19 pandemic. It draws on in-depth interviews with 17 older people and 6 key informants from LGBT+ community-based organisations, exploring the strategies used to manage their situations, how they responded and adapted to key challenges. Five themes emerged related to: 1) risk factors for LGBT+ older people and organisations, including specific findings on trans experiences,;2) care practices in LGBT+ lives,;3) strengths and benefits of networking 4) politicisation of ageing issues and their relevance to LGBT+ communities; and 5) learning from communication and provision in a virtual world. The findings illuminate adaptability and many strengths in relation to affective equality and reciprocal love, care and support among LGBT+ older people. It is vital UK that the government recognises and addresses the needs and concerns of LGBT+ older people during emergencies. What is known: The coronavirus (COVID-19) pandemic, and the wider governmental and societal response, brought health inequalities into sharp focus, exposing the structural disadvantage and discrimination faced by many marginalised communities in the UK and globally. LGBT+ older people are known to experience health inequalities compounded by anticipated or poor experiences of accessing health and social care services. What this paper adds: An exploration of LGBT+ older peple, their communities and social networks and how these were adapted in the COVID-19 context. Trans older people have been affected in very specific ways. The findings illuminate adaptability and many strengths in relation to affective equality and reciprocal love, care and support among LGBT+ older people. It is vital UK that the government recognises and addresses the needs and concerns of LGBT+ older people during emergencies

BRCA1/2 mutation testing in breast cancer patients: a prospective study of the long-term psychological impact of approach during adjuvant radiotherapy

This study assessed psychological distress during the first year after diagnosis in breast cancer patients approached for genetic counseling at the start of adjuvant radiotherapy and identified those vulnerable to long-term high distress. Of the approached patients some chose to receive a DNA test result (n = 58), some were approached but did not fulfill criteria for referral (n = 118) and some declined counseling and/or testing (n = 44). The comparative group consisted of patients not eligible for genetic counseling (n = 182) and was therefore not approached. Patients actively approached for genetic counseling showed no more long-term distress than patients not eligible for such counseling. There were no differences between the subgroups of approached patients. Predictors for long-term high distress or an increase in distress over time were pre-existing high distress and a low quality of life, having children, and having no family members with breast cancer. It is concluded that breast cancer patients can be systematically screened and approached for genetic counseling during adjuvant radiotherapy without imposing extra psychological burden. Patients vulnerable to long-term high distress already displayed high distress shortly after diagnosis with no influence of their medical treatment on their level of distress at long-term

An approximate inverse to the extended born modeling operator

Given a correct (data-consistent) velocity model, reverse time migration (RTM) correctly positions reflectors but generally with incorrect amplitudes and wavelets. Iterative least-squares migration (LSM) corrects the amplitude and wavelet by fitting data in the sense of Born modeling, that is, replacing migration by Born inversion. However, LSM also requires a correct velocity model, and it may require many migration/demigration cycles. We modified RTM in the subsurface offset domain to create an asymptotic (high-frequency) approximation to extended LSM. This extended Born inversion operator outputs extended reflectors (depending on the subsurface offset and position in the earth) with correct amplitude and phase, in the sense that similarly extended Born modeling reproduces the data to good accuracy. Although the theoretical justification of the inversion property relies on ray tracing and stationary phase, application of the weight operators does not require any computational ray tracing. The computational expense of the extended Born inversion operator is roughly the same as that of extended RTM, and the inversion (data-fit) property holds even when the velocity is substantially incorrect. The approximate inverse operator differes from extended RTM only in the application of data- and model-domain weight operators, and takes the form of an adjoint in the sense of weighted inner products in data and model space. Because the Born modeling operator is approximately unitary with respect to the weighted inner products, a weighted version of conjugate gradient iteration dramatically accelerates the convergence of extended LSM. An approximate LSM may be extracted from the approximate extended LSM by averaging over subsurface offset