In this paper, we present a general framework for the study of interval type-2 rough fuzzy sets by using both constructive and axiomatic approaches. First, several concepts and properties of interval type-2 fuzzy sets are introduced. Then, a pair of lower and upper interval type-2 rough fuzzy approximation operators with respect to a crisp binary relation is proposed. Classical representations ...

We show global and interior higher-order log-Hölder regularity estimates for solutions of Dirichlet integral equations where the operator has a nonintegrable kernel with singularity at origin that is weaker than any fractional Laplacian. As consequence, under mild assumptions on right hand side, we existence classical problems involving logarithmic Laplacian Schrödinger operator.

We study the Extended Proximal Auxiliary Problem Principle-method (EPAPP) by Kaplan and Tichatschke [17, 20] for solving variational inequalities whose operator is the sum of a maximal monotone and a continuous operator. As in comparable methods using Bregman distances the authors required that the operator of the considered variational inequality (here called main operator) is paramonotone (se...

The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar’s constraint qualification— that is, whether or not “the sum theorem” is true— is themost famous open problem inMonotone Operator Theory. In his 2008monograph “From Hahn-Banach to Monotonicity”, Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal mo...

The Poincaré-Steklov operator provides a direct relation between the tangential electric and magnetic field at the boundary of a simply connected domain, and a discrete equivalent of the operator can be constructed from the sparse finite element (FE) matrix of that domain by forming the Schur complement to eliminate the interior unknowns. Identifying the FE system matrix as a discretized versio...

We define strict closure space algebras as Boolean algebras augmented with a unary operation Int that satisfies conditions analogous to those of the interior operation from strict closure spaces. We also consider a first order modal logic SCS in which the necessity operator satisfies analogous conditions. Sikorski proved that every interior algebra can be represented as an algebra of subsets of...

We introduce a new topological semantics for belief logics in which the belief modality is interpreted as the interior of the closure of the interior operator. We show that the system wKD45, a weakened version of KD45, is sound and complete w.r.t. the class of all topological spaces. Moreover, we point out a problem regarding updates on extremally disconnected spaces that appears in the setting...

We study the simplicial quantum gravity in three dimensions. Motivated by the Boulatov’s model which generates a sum over simplicial complexes weighted with the Turaev-Viro invariant, we introduce boundary operators in the simplicial gravity associated to compact orientable surfaces. An amplitude of the boundary operator is given by a sum over triangulations in the interior of the boundary surf...

A linear singularly perturbed time dependent convection–diffusion problem is examined. The initial condition is designed to have steep gradients in the vicinity of the inflow point, which are transported in time, thus creating a moving interior shock layer. The location of this interior layer is tracked by the characteristics of the reduced first order problem. A numerical method is designed an...