Although there is a vast literature on the properties of p(n), typically motivated by work of Ramanujan, some of the simplest questions remain open. For example, little is known about p(n) modulo 2 and 3. Most results concerning the congruence properties of p(n) have been proved using properties of (1.1). Theorems typically depend on q-series identities, the theory of modular equations, or the ...

The triplet excited state behavior of c84 has been probed by the triplet-triplet energy transfer method using pulse radiolytically generated biphenyl triplets in benzene. The triplet excited state has a weak absorption in the UV (difference absorption maxima at 3 10 and 345 nm) but no significant absorption in the visible. The rate constants for energy transfer from 3BP* and 3C60* are 4 x lo9 a...

a modular $k$-coloring, $kge 2,$ of a graph $g$ without isolated vertices is a coloring of the vertices of $g$ with the elements in $mathbb{z}_k$ having the property that for every two adjacent vertices of $g,$ the sums of the colors of the neighbors are different in $mathbb{z}_k.$ the minimum $k$ for which $g$ has a modular $k-$coloring is the modular chromatic number of $g.$ except for some s...

Let $\\rho$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic automorphic of $\\operatorname{GL}\_n$ unitary type. Under very mild hypotheses on $\\rho$, we prove vanishing (Bloch–Kato) adjoint Selmer group $\\rho$. We obtain definitive results for groups associated non-CM Hilbert modular forms and elliptic curves over totally real fields.

A certain sequence of weight 1/2 modular forms arises in the theory of Borcherds products for modular forms for SL2(Z). Zagier proved a family of identities between the coefficients of these weight 1/2 forms and a similar sequence of weight 3/2 modular forms, which interpolate traces of singular moduli. We obtain the analogous results for modular forms arising from Borcherds products for Hilber...

2 Modular forms and Automorphic forms 2 2.1 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The Adele group of GL2 over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Hecke act...