where the residual vector R contains high-order derivatives of the dependent variable u(t, x1, x2, . . . , xn) with respect to independent variables. Equation (1) is an abstract representation of a generic Partial Differential Equation (PDE) in the residual form where the linear operator L accounts for a linear combination of the k partial derivative in the i direction, i.e. ∂/∂xi . When discre...

Let (H ; 〈·, ·〉) be a complex Hilbert space and T : H → H a bounded linear operator on H. Recall that T is a normal operator if T T = TT . Normal operator T may be regarded as a generalisation of self-adjoint operator T in which T ∗ need not be exactly T but commutes with T [5, p. 15]. An equivalent condition with normality that will be extensively used in the following is that ‖Tx‖ = ‖T ∗x‖ fo...

We consider suspension semi-flows of angle-multiplying maps on the circle. Under a Cgeneric condition on the ceiling function, we show that there exists an anisotropic Sobolev space[3] contained in the L space such that the Perron-Frobenius operator for the time-t-map acts on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum exp...

In this paper we show that the output of a nonlinear system with inputs in (L2 [0, T]; R ' ) whose state satisfies a nonlinear differential equation with standard smoothness conditions can be written as the composition of a nonlinear map with a linear Hilbert-Schmidt operator acting on the input. The result also extends to abstract semi-linear infinite dimensional systems. The approach is via t...

Here, the driving process B is a (semi)martingale (for instance, a Brownian motion), and φ is a stochastic process satisfying suitable measurability and integrability conditions. This observation has been used by various authors as the starting point of a theory of stochastic integration for vector-valued processes. Let X be a Banach space. In [5] we characterized the class of functions φ : (0,...

This paper introduces the logical system HOL2P that extends classical higher order logic (HOL) with type operator variables and universal types. HOL2P has explicit term operations for type abstraction and type application. The formation of type application terms t [T ] is restricted to small types T that do not contain any universal types. This constraint ensures the existence of a set-theoreti...

Abstract Let ???? be a complex Hilbert space and A non-zero positive bounded linear operator on . The main aim of this paper is to discuss general method develop -operator seminorm -numerical radius inequalities semi-Hilbertian operators using the existing corresponding Among many other we prove that if S , T X ? ???? ( ), i.e., -adjoint exist, then <m:math xmlns:m="http://www.w3.org/1998/Math/...

In this paper, we treat requests in XQuery in a system of mediation "everyXML".These requests use the operators of an algebra of semi-structured data which already exists and to which. we add a procedural aspect that treats the data of type document XML A mixed approach based on relational operators and models of data under shape of graphs, using timekeepers indexed in the process of reconstruc...