We establish the existence of a stochastic integral in a nuclear space setting as follows. Let E, F, and G be nuclear spaces which satisfy the following conditions: the spaces are reflexive, complete, bornological spaces such that their strong duals also satisfy these conditions. Assume that there is a continuous bilinear mapping of E × F into G. If H is an integrable, E-valued predictable proc...

A Shallow Water Wave-like nonlinear differential equation is considered by using the generalized bilinear equation with the generalized bilinear derivatives 3,x D and 3,t D , which possesses the same bilinear form as the standard shallow water wave bilinear equation. By symbolic computation, four presented classes of rational solutions contain all rational solutions to the resulting Shallow Wat...

The category k-BFAlg of all k-algebras with a bilinear form, whose objects are pairs (R,b) where R is k-algebra and b:R×R→k mapping, equivalent to the unital A for which canonical homomorphism (k,1)→(A,1A) splitting monomorphism in k-Mod. Call left inverses this “weak augmentations” algebra. There isomorphism between weak augmentation (A,bA) form bA compatible multiplication A, i.e., such that ...

Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space test functions into given locally lattice measures. The endowed with topology uniform convergence on bounded sets. Their structure is characterized via regularization and function-valued seminorms under mild structural assumptions Many recent gen...

We consider the linear Schrödinger equation, in 1D, on a bounded interval, with Dirichlet boundary conditions and bilinear scalar control. The small-time local exact controllability around ground state was proved [5], under an appropriate nondegeneracy assumption. Here, we work weaker assumption prove projection, state, estimates control (depending linearly target) simultaneously several spaces...

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...

Cryptographic protocols depend on the hardness of some computational problems for their security. Joux briefly summarized known relations between assumptions related bilinear map in a sense that if one problem can be solved easily, then another problem can be solved within a polynomial time [6]. In this paper, we investigate additional relations between them. Firstly, we show that the computati...