Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model

The spin-1/2 highest weight representations of the dynamical 6-vertex and the standard 8-vertex Yang-Baxter algebra on a finite chain are considered in this paper. For the antiperiodic dynamical 6-vertex transfer matrix defined on chains with an odd number of sites, we adapt the Sklyanin's quantum separation of variable (SOV) method and explicitly construct SOV representations from the original space of representations. We provide the complete characterization of eigenvalues and eigenstates proving also the simplicity of its spectrum. Moreover, we characterize the matrix elements of the identity on separated states by determinant formulae. The matrices entering in these determinants have elements given by sums over the SOV spectrum of the product of the coefficients of separate states. This SOV analysis is not reduced to the case of the elliptic roots of unit and the results here derived define the required setup to extend to the dynamical 6-vertex model the approach recently developed in [1]-[5] to compute the form factors of the local operators in the SOV framework, these results will be presented in a future publication. For the periodic 8-vertex transfer matrix, we prove that its eigenvalues have to satisfy a fixed system of equations. In the case of a chain with an odd number of sites, this system of equations is the same entering in the SOV characterization of the antiperiodic dynamical 6-vertex transfer matrix spectrum. This implies that the set of the periodic 8-vertex eigenvalues is contained in the set of the antiperiodic dynamical 6-vertex eigenvalues. A criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate to any of such eigenvalues one nonzero eigenstate of the periodic 8-vertex transfer matrix by using the SOV results. Moreover, a preliminary discussion on the degeneracy of the periodic 8-vertex spectrum is also presented.Comment: 36 pages, main modifications in section 3 and one appendix added, no result modified for the dynamical 6-vertex transfer matrix spectrum and the matrix elements of identity on separate states for chains with an odd number of site

Bedbugs and Infectious Diseases

Bedbugs (Cimex lectularius or Cimex hemipterus) are cosmopolite hematophagus insects, and recent outbreaks have been reported in all major occidental countries. Although they are suspected of transmitting more than 40 infectious agents, no report has yet definitively demonstrated that capacity

Under-Age Children Returning From Jihadist Group Operation Areas: How Can We Make a Diagnosis and Construct a Narrative With a Fragmentary Anamnesis?

International audienceIntroduction: Since 2011, the French government estimates that about 500 French children have been born in or taken by their parents to areas where terrorist operations prevail. Since May 2017, 75 children who returned to France have benefited from a dedicated health care system. Method: This article is the result of clinical interviews conducted with 53 patients evaluated and taken care of at Avicenne Hospital in Bobigny. To our knowledge, no studies have been published on this subject. Results: A total of 32 evaluations have been completed, all of which indicated the need for care for these children. Of these children, 64% are under 5 years old, and 59% were born in France. Their clinical profiles are heterogeneous and fluctuate with time. Discussion: The multiple adverse events experienced by these children and the uniqueness of children born to families suspected by authorities of having participated in activities related to terrorism make this situation unprecedented. How can we make a diagnosis of PTSD without the help of a precise anamnesis? How can we help these children form a structuring narrative that avoids the pitfalls inherent to generalized fascination