Gentry’s bootstrapping technique is the most famous method of obtaining fully homomorphic encryption. In previous work I proposed a fully homomorphic encryption without bootstrapping which has the weak point in the plaintext [1],[15]. I also proposed a fully homomorphic encryption with composite number modulus which avoids the weak point by adopting the plaintext including the random numbers in...

Non-statistical weak measurements yield weak values that are outside the range of eigenvalues and are not rare, suggesting that weak values are a property of every preand-post-selected ensemble. They also extend the applicability and valid regime of weak values.

As Duncan Luce and other prominent scholars have pointed out on several occasions, testing algebraic models against empirical data raises difficult conceptual, mathematical, and statistical challenges. Empirical data often result from statistical sampling processes, whereas algebraic theories are nonprobabilistic. Many probabilistic specifications lead to statistical boundary problems and are s...

We use tropical geometry to analyze finite valued mechanisms. We geometrically characterize all mechanisms on arbitrary type spaces, derive geometric and algebraic criteria for weak monotonicity, incentive compatibility and revenue equivalence. As corollaries we obtain various results known in the literature and shed new light on their nature.

The objective of this paper is to introduce the fundamental algebro-geometric constructions over the extended tropical semi-ring. The study of tropical varieties, co-varieties and ideals over this extension eventually yields the theorem of the weak tropical Nullstellensatz and gives an algebraic interpretation of the tropical Nullstellensatz.

Using Poonen’s version of “weak vertical method” we produce new examples of “large” and “small” rings of algebraic numbers (including rings of integers) where Z and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur’s conjecture on topology of rational points does not hold.

In this paper we introduce a new direction in the investigations of the open problem of the representation of algebraic lattices by the weak congruence lattices. In some cases the representability of some lattice follows from the representability of another lattice. We give some results. AMS Mathematics Subject Classification (2010): 08A30

In this article we use our constructions from [BS05] to lay down some foundations for the application of A. Robinson's nonstandard methods to modern Algebraic Geometry. The main motivation is the search for another tool to transfer results from characteristic zero to positive characteristic and vice versa. We give applications to the resolution of singularities and weak factorization.

We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding infinitesimal objects. MSC 2000: 53C50, 53C35, 53C40