In this paper we review the definition of the monodromy of an angle valued map based on linear relations as proposed in [3]. This definition provides an alternative treatment of the Jordan cells, topological persistence invariants of a circle valued maps introduced in [2]. We give a new proof that homotopic angle valued maps have the same monodromy, hence the same Jordan cells, and we show that...

We generalize the notion of complete binary relation on complete lattice to residuated lattice valued ordered sets and show its properties. Then we focus on complete fuzzy tolerances on fuzzy complete lattices and prove they are in one-to-one correspondence with extensive isotone Galois connections. Finally, we prove that fuzzy complete lattice, factorized by a complete fuzzy tolerance, is agai...

We prove that certain Lipschitz properties of the inverse F-1 of a set-valued map F are inherited by the map (f+F)~x when / has vanishing strict derivative. In this paper, we present an inverse mapping theorem for set-valued maps F acting from a complete metric space I toa linear space Y with a (translation) invariant metric. We prove that, for any function f: X -> Y with "vanishing strict deri...

The most often applied non-numerical uncertainty degrees are those taking their values in complete lattices, but also their weakened versions may be of interest. In what follows, we introduce and analyze possibilistic distributions and measures taking values in finite upper-valued possibilistic lattices, so that only for finite sets of such values their supremum is defined. For infinite sets of...

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued ma...

This is an overview of the recent results of interaction of Boolean valued analysis and vector lattice theory.

We develop a numerical framework for the quantum analogue of the “classical” lattice Boltzmann method (LBM), with the Maxwell-Boltzmann distribution replaced by the Fermi-Dirac function. To accommodate the spin density matrix, the distribution functions become 2× 2-matrix valued. We show that the efficient, commonly used BGK approximation of the collision operator is valid in the present settin...

Lattice-valued possibilistic measures, conceived and developed in more detail by G.*De Cooman in 1997 [3], enabled to apply the main ideas on which the real-valued possibilistic measures are founded also to the situations often occurring in the real world around, when the degrees of possibility, ascribed to various events charged by uncertainty, are comparable only quantitatively by the relatio...

In this paper, -subgradients for convex set-valued maps are defined. We prove an existence theorem for -subgradients of convex set-valued maps. Also, we give necessary optimality conditions for an -solution of a convex set-valued optimization problem (CSP). Moreover, using the single-valued function induced from the set-valued map, we obtain theorems describing the -subgradient sum formula for ...

This paper deals with the concept of complete residuated lattice-valued (referred to as L-valued) finite tree automata. In this regard, we first define an L-valued regular tree language, and then we prove a necessary and sufficient condition for the regularity of an L-valued tree language. Furthermore, we generalize the pumping lemma for L-valued finite tree automata (L-FTA). Afterwards, the be...