We use mock modular forms to compute generating functions for the critical values of modular L-functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler-Shimura theory for weakly holomorphic modular forms and mock modular forms. T...

Let introduce the discrete Sobolev-type inner product 〈f, g〉 = ∫ 1 −1 f(x)g(x)dμ(x) + M [f(1)g(1) + f(−1)g(−1)] + N [f ′(1)g′(1) + f ′(−1)g′(−1)],

Let μ be the Jacobi measure supported on the interval [−1, 1]. Let introduce the Sobolev-type inner product 〈f, g〉 = ∫ 1 −1 f(x)g(x) dμ(x) +Mf(1)g(1) +Nf ′(1)g′(1), where M,N ≥ 0. In this paper we prove that, for certain indices δ, there are functions whose Cesàro means of order δ in the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product ar...

we investigate some properties of approximately quasi inner generalized dynamics and quasi approximately inner generalized derivations on modules. in particular, we prove that if a is a c*-algebra, is the generator of a generalized dynamics on an a-bimodule m satisfying and there exist two sequences of self adjoint elements in a such that for all in a core for , , then is approximately quasi in...

We develop methods that construct an optimal set of vectors with a specified inner product structure, from a given set of vectors in a complex Hilbert space. The optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. Four special cases are considered. In the first, the constructed vectors are orthonormal. In the s...

We extend the reach of functional encryption schemes that are provably secure under simpleassumptions against unbounded collusion to include function-hiding inner product schemes.Our scheme is a private key functional encryption scheme, where ciphertexts correspond tovectors ~x, secret keys correspond to vectors ~y, and a decryptor learns 〈~x, ~y〉. Our schemeemploys asymmetr...

same initial elements. For the case of m= 12 this results in a reduction by a factor of about 80 in the number ofcases examined. Samples at m = 13 and 14 indicate that this pruning method will yield an additional improvement factor of about 2.5 to 3 for each increase in m. Using a characteristic vector representation, the test for a repeated spacing is improved by a factor of m2/8 over the stra...