The fact that many modal operators are part of an adjunction is probably folklore since the discovery of adjunctions. On the other hand, the natural idea of a minimal propositional calculus extended with a pair of adjoint operators seems to have been formulated only very recently. This recent research, mainly motivated by applications in computer science, concentrates on technical issues relate...

1. Cautionary example 2. Criterion for essential self-adjointness 3. Examples of essentially self-adjoint operators 4. Appendix: Friedrichs' canonical self-adjoint extensions 5. The following has been well understood for 70-120 years, or longer, naturally not in contemporary terminology. The differential operator T = d 2 dx 2 on L 2 [a, b] or L 2 (R) is a prototypical natural unbounded operator...

A conjunction T ties an implication operator A if the identity A(a,A(b, z)) = A(T (a, b), z) holds [A.A. Abdel-Hamid, N.N. Morsi, Associatively tied implications, Fuzzy Sets and Systems 136 (2003) 291–311]. We study the class of tied adjointness algebras (which are five-connective algebras on two partially ordered sets), in which the implications are tied by triangular norms. This class contain...

We prove essential self-adjointness of Dirac operators with Lorentz scalar potentials which grow sufficiently fast near the boundary $\partial\Omega$ spatial domain $\Omega\subset\mathbb R^d$. On way, we first consider general symmetric order differential systems, for identify a new, large class potentials, called ensuring self-adjointness. Furthermore, using supersymmetric structure operator i...

In this paper, Heir-equations method is applied to investigate nonclassical symmetries and new solutions of the Black-Scholes equation. Nonlinear self-adjointness is proved and infinite number of conservation laws are computed by a new conservation laws theorem.

Recently, Morsi [28] has developed a complete syntax for the semantical domain of all adjointness algebras. In [1], Abdel-Hamid and Morsi enrich adjointness algebras with one more conjunction, this time a t-norm T (T need not be commutative) that ties an implication A in the following sense: A(T(a, b), z) = A(a, A(b, z)). In this paper, we develop a new complete syntax for quite a general multi...

In this note, by means of the spectrum of the generating operator, we characterize the self-adjointness and closedness of the range of a normal and a self-adjoint Jordan *-derivation, respectively.

In Noncommutative Geometry, as in quantum theory, classically real variables are assumed to correspond to self-adjoint operators. We consider the relaxation of the requirement of self-adjointness to mere symmetry for operators Xi which encode space-time information.

in this paper, heir-equations method is applied to investigate nonclassical symmetries and new solutions of the black-scholes equation. nonlinear self-adjointness is proved and infinite number of conservation laws are computed by a new conservation laws theorem.