M. Lacey and C. Thiele proved in [26] (Annals of Math. (1997)) and [27] (Annals of Math. (1999)) that the bilinear Hilbert transform maps L1 × L2 → L boundedly when 1 p1 + 1 p2 = 1 p with 1 < p1, p2 ≤ ∞ and 2 3 < p < ∞. Whether the L estimates hold in the range p ∈ (1/2, 2/3] has remained an open problem since then. In this paper, we prove that the bilinear Hilbert transform does not satisfy an...

Abstract. For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respect to (the growth of) the underlying scale. These results strengthen previous works of L...

In this paper, we introduce a symbolic model to analyse protocols that use a bilinear pairing between two cyclic groups. This model consists in an extension of the Abadi-Rogaway logic and we prove that the logic is still computationally sound: symbolic indistinguishability implies computational indistinguishability provided that the Bilinear Decisional DiffieHellman assumption holds and that th...

We study the simultaneous convexification of graphs of bilinear functions gk(x;y) = yTAkx over x ∈ Ξ = {x ∈ [0, 1]n |Ex ≥ f} and y ∈ ∆m = { y ∈ R+ ∣∣1Ty ≤ 1}. We propose a constructive procedure to obtain a linear description of the convex hull of the resulting set. This procedure can be applied to derive convex and concave envelopes of certain bilinear functions, to study unary expansions of i...

In this paper the existence of a quadratic control Lyapunov function for bilinear systems is considered. The existence of a control Lyapunov function ensures the existence of a control law which ensures the global asymptotic stability of the closed loop control system. In this paper we will derive conditions for the existence of a control Lyapunov function for bilinear systems. These conditions...

We are interested in approximation of a multivariate function f(x1, . . . , xd) by linear combinations of products u (x1) · · ·u(xd) of univariate functions u(xi), i = 1, . . . , d. In the case d = 2 it is the classical problem of bilinear approximation. In the case of approximation in the L2 space the bilinear approximation problem is closely related to the problem of singular value decomposit...

Value function approximation methods have been successfully used in many applications, but the prevailing techniques often lack useful a priori error bounds. We propose a new approximate bilinear programming formulation of value function approximation, which employs global optimization. The formulation provides strong a priori guarantees on both robust and expected policy loss by minimizing spe...

By means of the three-wave method one can solve some nonlinear partial differential equations (NLPDEs) in their bilinear forms. When an NLPDE has no bilinear closed form we can not use this method. We modify the idea of three-wave method to obtain some analytic solutions for the (2+1)-dimensional Breaking soliton equation by obtaining a bilinear closed form for it. By comparison of this method ...

There are a large number of problems of physical interest that can be formulated in the abstract setting in which the Lax-Milgram lemma is applied to an equation expressed in terms of an appropriate bilinear form to prove existence of a unique solution. The selection of the solution space and the bilinear form controls the PDE and boundary conditions that the solution will satisfy. We will give...

The reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators which can be seen as more general variable coefficient counterparts of the bilinear Hilbert transform and other singular bilinear multipliers operators. The un...