Enterococcal spondylitis (ES) is a disease of commercial broiler chickens, with a worldwide distribution. Symmetrical hind limb paralysis typical of ES results from infection of the free thoracic vertebra (FTV) by pathogenic strains of Enterococcus cecorum . To determine the pathogenesis of ES, birds with natural and experimental ES were studied over time. In natural disease, case birds (n = 150) from an affected farm and control birds (n = 100) from an unaffected farm were evaluated at weeks 1-6. In control birds, intestinal colonization by E. cecorum began at week 3. In case birds, E. cecorum was detected in intestine and spleen at week 1, followed by infection of the FTV beginning at week 3. E. cecorum isolates recovered from intestine, spleen, and FTV of case birds had matching genotypes, confirming that intestinal colonization with pathogenic strains precedes bacteremia and infection of the FTV. Clinical intestinal disease was not required for E. cecorum bacteremia. In 1- to 3-week-old case birds, pathogenic E. cecorum was observed within osteochondrosis dissecans (OCD) lesions in the FTV. To determine whether OCD of the FTV was a risk factor for ES, 214 birds were orally infected with E. cecorum, and the FTV was evaluated histologically at weeks 1-7. Birds without cartilage clefts of OCD in the FTV did not develop ES; while birds with OCD scores ≥3 were susceptible to lesion development. These findings suggest that intestinal colonization, bacteremia, and OCD of the FTV in early life are crucial to the pathogenesis of ES.

We show that for fields that are of characteristic 0 or algebraically closed of characteristic greater than 5, that certain classes of Leibniz algebras are 2-recognizeable. These classes are solvable, strongly solvable and supersolvable. These same results hold in Lie algebras and in general for groups.Key Words: 2-recognizeable, strongly solvable, supersolvable, Leibniz algebras I. PRELIMINARIES A property of algebras is called n-recognizeable if whenever all the n generated subalgebras of algebra L have the property, then L also has the property. An analogous definition holds for classes of groups. In Lie algebras, nilpotency is 2-recognizeable due to Engel's theorem and the same holds for Leibniz algebras. For Lie algebras, solvability, strong solvability and supersolvability are 2-recognizeable when they are taken over a field of characteristic 0 or an algebraically closed field of characteristic greater than 5. These results are shown in [7] and [12] using different methods. The purpose of this work is to extend these results to Leibniz algebras. Corresponding results in group theory are shown in [8] and [9].The definition of Leibniz algebra can be given in terms of the left multiplications being derivations. A theme in this work is that assumptions will be given in terms of the left multiplications. Thus, that nilpotency is 2-recognizeable in Leibniz algebras follows from all left multiplications being nilpotent, Engel's theorem. This result, shown in several places, can be cast as in Jacobson's refinement to Engel's theorem for Lie algebras, see [6], a result that we use.

Bousquet-Mélou & Eriksson's lecture hall theorem generalizes Euler's celebrated distinctodd partition theorem. We present an elementary and transparent proof of a refined version of the lecture hall theorem using a simple bijection involving abacus diagrams.