Let V be a non-degenerate inner product space over Q of signature (n, 2), and let D be the space of oriented negative 2-planes in V (R). In [2], Borcherds constructed certain meromorphic modular forms Ψ(F ) on D with respect to arithmetic subgroups ΓM of G = O(V ) by regularizing the theta integral of vector valued elliptic modular forms f of weight 1 − n 2 for SL2(Z) with poles at the cusp, cf...

Let f be a weight 2 normalized newform on the congruence subgroup Γ0(N) with integral Fourier coefficients. There are two important numerical invariants attached to f : its congruence number and its modular degree. By definition, the congruence number of f is the largest integer Df such that there exists a weight 2 cusp form on Γ0(N), with integral coefficients, which is orthogonal to f with re...

In the present paper, we investigate a connection between two fuzzy inner product one of which arises from Felbin's fuzzy norm and the other is based on Bag and Samanta's fuzzy norm. Also we show that, considering a fuzzy inner product space, how one can construct another kind of fuzzy inner product on this space.

we define a new function-valued inner product on l2(g), called ?-bracket product, where g is a locally compact abelian group and ? is a topological isomorphism on g. we investigate the notion of ?-orthogonality, bessel's inequality and ?-orthonormal bases with respect to this inner product on l2(g).

In this paper, we introduce the concept of family of local atoms in a 2-inner product space and then this concept is generalized to an atomic system. Besides, a characterization of an atomic system lead to obtain a new frame. Actually this frame is a generalization of previous works.

We study quadratic periods on quaternionic Shimura varieties and formulate an integral refinement of Shimura's conjecture regarding Petersson inner products automorphic forms that are related by the Jacquet-Langlands correspondence. The main result is this implied another (Conjecture D below) integrality theta lifts between certain unitary groups.

We analyze the coarse geometry of the Weil-Petersson metric on Teichmüller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the pants graph. We show that in dimension 3 the pants graph is strongly rela...