In this paper we develope the fundamentals of the generalized symplectic geometry on the bundle of linear frames LM of an n-dimensional manifold M that follows upon taking the R-valued soldering 1-form θ on LM as a generalized symplectic potential. The development is centered around generalizations of the basic structure equation df = −Xf ω of standard symplectic geometry to LM when the symplec...

In this work, we use a notion of convexificator [25] together with the support function [3, 4, 15, 16, 41] to establish necessary optimality conditions for set valued bilevel optimization problems. Fortunately, the Lipschitz property of a set-valued mapping is conserved for its support function. An intermediate set-valued optimization problem is introduced to help us in our investigation.

1. Topological Vector Spaces 1 1.1. The Krein-Milman theorem 7 2. Banach Algebras 11 2.1. Commutative Banach algebras 14 2.2. ∗–Algebras (over complexes) 17 2.3. Exercises 20 3. The Spectral Theorem 21 3.1. Problems on the Spectral Theorem (Multiplication Operator Form) 26 3.2. Integration with respect to a Projection Valued Measure 27 3.3. The Functional Calculus 34 4. Unbounded Operators 37 4...

Recent work has shown that in some cases the phase information of synaptic signal is important in the learning and representation capabilities of networks. Modelling such information with complex valued activation signals is possible, and indeed complex back-propagation algorithms have been derived 3]. Cliiord algebras give a way to generalise complex numbers to many dimensions. This paper pres...

This paper is an attempt at developing a theory of algebraic systems that would correspond in a natural fashion to the N0-valued prepositional calculus^). For want of a better name, we shall call these algebraic systems MV-algebras where MV is supposed to suggest many-valued logics. It is known that the classical two-valued logic gives rise to the study of Boolean algebras and, as can be expect...

We propose a new concept of generalized di erentiation of setvalued maps that captures rst order information. This concept encompasses the standard notions of Fréchet di erentiability, strict di erentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between th...

Let A(D) be the disc algebra of all continuous complex-valued functions on the unit disc D holomorphic in its interior. Functions from A(D) act on the set of all contraction operators (‖A‖ 1) on Hilbert spaces. It is proved that the following classes of functions from A(D) coincide: (1) the class of operator Lipschitz functions on the unit circle T; (2) the class of operator Lipschitz functions...

2.1 Definition: Lipschitz Function A real valued function f : D ⊆ R → R is called Lipschitz continuous or is said to satisfy a Lipschitz condition if there exists a constant K ≥ 0 such that for all x1, x2 in D |f(x1)− f(x2)| ≤ K|x1 − x2| (3) The inequality is (trivially) satisfied if x1 = x2. Otherwise, for x1 6= x2, one can equivalently define a function to be Lipschitz if and only if |f(x1)−f...