The Arens products are the standard way of extending the product from a Banach algebra A to its bidual A′′. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if A is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known ...

• A quadratic space is a pair (V,B) where V is a finite-dimensional vector space over a field K and B : V × V → K is a symmetric bilinear form. • For quadratic spaces (V,B) and (V ′, B′), a map τ : V → V ′ is an isometry iff τ is a linear isomorphism and ∀~v, ~ w ∈ V : B(~v, ~ w) = B′(τ(~v), τ(~ w)). In this case V and V ′ are isometric. • The quadratic form determined by B and basis β := {~v1,...

The h-deformed quantum plane is a counterpart of the q-deformed one in the set of quantum planes which are covariant under those quantum deformations of GL(2) which admit a central determinant. We have investigated the noncommutative geometry of the h-deformed quantum plane. There is a 2-parameter family of torsion-free linear connections, a 1-parameter sub-family of which are compatible with a...

For an elliptic curve E over any field K, the Weil pairing en is a bilinear map on n-torsion. For K of characteristic p > 0, the map en is degenerate if and only if n is divisible by p. In this paper, we consider E over the dual numbers K[ǫ] and define a non-degenerate “Weil pairing on ptorsion” which shares many of the same properties of the Weil pairing. We also show that the discrete logarit...

Bilinear systems offer a promising approach for nonlinear control because a broad class of nonlinear problems can be reformulated and approximated in bilinear form. System identification is a technique to obtain such a bilinear approximation for a nonlinear system from input-output data. Recent discrete-time bilinear model identification methods rely on Input-Output-to-State Representations (IO...

Bilinear systems offer a promising approach for nonlinear control because a broad class of nonlinear problems can be reformulated in bilinear form. In this paper system identification is shown to be a technique to obtain such a bilinear approximation of a nonlinear system. Recent discrete-time bilinear model identification methods rely on Input-Output-to-State Representations. These IOSRs are e...

We introduce a new identification scheme based on the Gap Diffie-Hellman problem. Our identification scheme makes use of the fact that the computational Diffie-Hellman problem is hard in the additive group of points of an elliptic curve over a finite field, on the other hand, the decisional Diffie-Hellman problem is easy in the multiplicative group of the finite field mapped by a bilinear map. ...

Basic properties in Perron-Frobenius theory are strict positivity, primitivity, and irreducibility. Whereas for nonnegative matrices, these properties are equivalent to elementary graph properties which can be checked in polynomial time, we show that for Kraus maps the noncommutative generalization of stochastic matrices checking strict positivity (whether the map sends the cone to its interior...

In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework to exhaustively search for optimal formulae for evaluating bilinear maps, such as Strassen or Karatsuba formulae. The main contribution of this work is a new criterion to aggressively prune useless branches in the exhaustive search, thus leading to the computation of new optimal formulae, in particular for the short p...

We propose an identity-based group signature scheme from bilinear maps. The scheme has the security properties of group signatures. Our scheme is based on the CDHP assumption and bilinear maps between groups. The size of the group pubic key and the length of the signature are independent on the number of group members. Furthermore, a group member can sign many message using the same key pair.