Note on Hermitian Jacobi Forms

We compare the spaces of Hermitian Jacobi forms (HJF) of weight $k$ and indices $1,2$ with classical Jacobi forms (JF) of weight $k$ and indices $1,2,4$. Using the embedding into JF, upper bounds for the order of vanishing of HJF at the origin is obtained. We compute the rank of HJF as a module over elliptic modular forms and prove the algebraic independence of the generators in case of index 1. Some related questions are discussed.Comment: 24 pages; title changed, abstract changed, some proofs expanded and new results adde

Simultaneous nonvanishing of Dirichlet LL-functions and twists of Hecke-Maass L-functions

We prove that given a Hecke-Maass form $f$ for $\text{SL}(2, \mathbb{Z})$ and a sufficiently large prime $q$, there exists a primitive Dirichlet character $\chi$ of conductor $q$ such that the $L$-values $L(\tfrac{1}{2}, f \otimes \chi)$ and $L(\tfrac{1}{2}, \chi)$ do not vanish. We expect the same method to work for any large integer $q$

Linear Relations Among Poincare Series via Harmonic Weak Maass Forms

We discuss the problem of the vanishing of Poincar\'e series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan's mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincar\'e series and the connection to Ramanujan's mock theta functions

Hemolytic Disease of the Fetus and Newborn

Hemolytic disease of the fetus and newborn (HDFN) also called as “erythroblastosis fetalis” is characterized by the increased rate of red blood cells (RBCs) destruction. Hemolysis should always be investigated even if the anemia is mild and apparently trivial. The principle clues which suggest hemolytic anemia includes: increased number of reticulocytes and/or circulating nucleated RBCs, unconjugated hyperbilirubinemia, a positive direct antiglobulin test and characteristic changes in red cells in the blood films. Based on etiology, hemolysis in newborn can be immune or non-immune mediated. The immune-mediated hemolysis due to blood group incompatibility between the mother and the fetus is the main cause of HDFN