A threshold graph G on n vertices is defined by binary sequence of length n. In this paper we present an explicit formula for computing the characteristic polynomial of a threshold graph from its binary sequence. Applications include obtaining a formula for the determinant of adjacency matrix of a threshold graph and showing that no two nonisomorphic threshold graphs are cospectral.

For [Formula: see text], let [Formula: see text] be the dual of the Radford algebra of dimension [Formula: see text]. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter–Drinfeld modules over [Formula: see text]. Along the way, we describe the simple objects in [Formula: see text] and their projective envelopes. Then we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case [Formula: see text]. There are 18 possible cases. We present by generators and relations, the corresponding Nichols algebras on five of these eighteen cases. As an application, we characterize finite-dimensional Nichols algebras over indecomposable modules for [Formula: see text] and [Formula: see text], [Formula: see text], which recovers some results of the second and third author in the former case, and of Xiong in the latter. Cualquier destino, por largo y complicado que sea, consta en realidad de un solo momento: el momento en que el hombre sabe para siempre quién es. Jorge Luis Borges

For a connected, noncompact simple matrix Lie group G so that a maximal compact subgroup K has a three dimensional simple ideal, in this note we analyze the admissibility of the restriction of irreducible square integrable representations for the ambient group when they are restricted to certain subgroups that contain the three dimensional ideal. In this setting we provide a formula for the multiplicity of the irreducible factors. Also, for general G such that G/K is an Hermitian G-manifold we give a necessary and sufficient condition so that an arbitrary square integrable representation of the ambient group is admissible over the semisimple factor of K.

Introduction Cytokine release syndrome (CRS) is a common and potentially severe toxicity associated with haploidentical peripheral blood hematopoietic stem cell transplantation (Haplo-HSCT). CRS is characterized by immune activation and a high level of circulating inflammatory cytokines, particularly interleukin-6 (IL-6). While tocilizumab, an IL-6 receptor-targeted monoclonal antibody has been effectively used to treat CRS in the context of immunotherapy, it has not been evaluated during the acute Haplo-HSCT period. We hypothesized that prophylactic tocilizumab can prevent CRS after Haplo-HSCT. We report the results of a pilot trial to evaluate its use. Methods Patients ≥18 years undergoing Haplo-HSCT of peripheral blood in a single center with any underlying diagnosis were included. The conditioning was myeloablative and was administered as follows: cyclophosphamide 350 mg / m2 and fludarabine 25 mg / m2 (days -5-3) or rabbit antithymocyte globulin (2.5 mg / kg days -4, -3), plus melphalan 200 mg / m2 PO (days -2-1). Prophylaxis of graft versus host disease (GVHD) consisted of cyclophosphamide (50 mg / kg / day, days +3 and +4), cyclosporine and mycophenolate mofetil on day +5. The cells were infused on day 0. Tocilizumab was administered on day -1 in a single dose of 4 mg / kg IV. CRS was rated according to Lee et al. The primary outcome included the incidence and severity of CRS, the adverse effects associated with tocilizumab, hospitalization and graft rates and the incidence of GVHD. Results Ten patients have enrolled, with an average age of 26 years (range 19-32). The underlying diagnoses were acute lymphoblastic leukemia (n = 7), acute myeloid leukemia (n = 1) and non-Hodgkin lymphoma (n = 2). There were no infusion reactions associated with tocilizumab. Six patients (60%) developed CRS, all grades 1-2, with a median day of onset on day +2 (range 1-3) and a median fever duration of 2 days (range 2-3), often accompanied by nausea, diarrhea, dehydration and transaminitis. All patients were hospitalized; hospitalization was not related to CRS in three patients. 4 patients are alive without relapse (40%), 2 patients alive with relapse (20%), 3 patients died due to transplant-related mortality (30%), one patient dies from relapse (10%). No graft failure occurred and mixed chimerism was observed in a patient. Acute GVHD with steroid response has been diagnosed in 2 cases (20%, grade I / II). The median follow-up is 8.3 months (range 0-20). Conclusion What we observe about the prophylactic use of tocilizumab for Haplo-HSCT of peripheral blood is that its administration is feasible, Grades 1-2 CRS was observed in 60% of patients, grades 3-4 were not observed, however the use of tocilizumab did not prevent hospitalizations. More feasibility data is required before starting a definitive randomized trial to evaluate the use of tocilizumab for Haplo-HSCT peripheral blood. Disclosures Gomez-Almaguer: Janssen: Consultancy, Speakers Bureau; Teva: Consultancy, Speakers Bureau; Takeda: Consultancy, Speakers Bureau; Celgene: Consultancy, Speakers Bureau; Amgen: Consultancy, Speakers Bureau. OffLabel Disclosure: Tocilizumab for Prophylaxis of Cytokine Release Syndrome