The number of apparent double points of a smooth, irreducible projective variety X of dimension n in P is the number of secant lines to X passing through the general point of P. This classical notion dates back to Severi. In the present paper we classify smooth varieties of dimension at most three having one apparent double point. The techniques developed for this purpose allow to treat a wider...

We show that there exists k ∈ N and 0 < ǫ ∈ R such that for every field F of characteristic zero and for every n ∈ N, there exists explicitly given linear transformations T1, . . . , Tk : F n → F satisfying the following: For every subspaceW of F of dimension less or equal n 2 , dim(W+ k

A population of oligonucleotides co-purified with the dsRNA genomic segments of the killer virus of Saccharomyces cerevisiae during electrophoresis through agarose gels. These smaller RNA molecules must be separated from the viral genome in order to determine the structure of the dsRNA molecules. Sequence analysis of these isolated oligonucleotides showed that the population contained tRNA-like...

In this lecture we will see a fuller development of the LWE based homomorphic encryption scheme. The multiplication operation in the naive LWE− HE scheme changes the form of the cipher text. In this lecture we see a dimensionality reduction trick to restore the form of the original ciphertext. Finally we conclude with reductions between LWE assumptions that relax the constraints on the probabil...

If the total degree of a polynomial in n ≥ 2 variables of dimension n s bounded by a double exponential function in n, we show that its Mahler measure is bounded from below by an absolute constant > 1. 2000 Mathematics Subject Classification 11G50, 11J81, 14G40.