Linear first order systems of partial differential equations of the form ∇f = M∇g, where M is a constant matrix, are studied on vector spaces over the fields of real and complex numbers, respectively. The Cauchy– Riemann equations belong to this class. We introduce a bilinear ∗-multiplication on the solution space, which plays the role of a nonlinear superposition principle, that allows for alg...

A q-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear q-difference equations. These bilinear equations are translated to the language of wave functions, which turn out to satisfy a system of linear q-difference equations. These linear q-difference equations are used to formulate the Lax for...

The Groth-Sahai proof system (EUROCRYPT 2008, SIAM Journal of Computing 41(5) [GS12]) provides e cient non-interactive witness-indistinguishable (NIWI) and zero-knowledge (NIZK) proof systems for languages over bilinear groups and is a widely-used versatile tool to design e cient cryptographic schemes and protocols. We revisit randomization of the prover in the GS proof system. We nd an unnotic...

This paper deals with the design of a reduced-order H∞ filter for a stochastic bilinear system with a prescribed H∞ norm criterion. The considered system is bilinear in control and with multiplicative noises in the dynamics and in the measurement equations. The problem is transformed into the search of a unique gain matrix by using Sylvester-like constraints. The approach is based on the resolu...

In this paper, we will discuss the problem of optimal model order reduction of bilinear control systems with respect to the generalization of the well-known H2norm for linear systems. We revisit existing first order necessary conditions for H2-optimality based on the solutions of generalized Lyapunov equations arising in bilinear system theory and present an iterative algorithm which, upon conv...

In this paper, the generalized (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (shortly CBS) equations are investigated. We employ the Hirota’s bilinear method to obtain the bilinear form of CBS equations. Then by the idea of extended homoclinic test approach (shortly EHTA), some exact soliton solutions including breather type solutions are presented. Keywords—EHTA, (2+1)-dimensional CBS equat...