Given a measure space 〈X,μ〉, we define its measure algebra Aμ as the quotient of the algebra of all measurable subsets of X modulo the relation X μ ∼ Y if μ(X4Y ) = 0. If further X is endowed with a topology T , we can define an interior operator on Aμ analogous to the interior operator on P(X). Formulas of S4u (the modal logic S4 with a universal modality ∀ added) can then be assigned elements...

The comprehensive generalization of summation-by-parts of Del Rey Fernández et al. (J. Comput. Phys., 266, 2014) is extended to approximations of second derivatives with variable coefficients. This enables the construction of second-derivative operators with one or more of the following characteristics: i) non-repeating interior stencil, ii) nonuniform nodal distributions, and iii) exclusion of...

In this paper, we introduce a notion of generalized states from an EQ-algebra E1 to another EQ-algebra E2, which is a generalization of internal states (or state operators) on an EQ-algebra E. Also we give a type of special generalized state from an EQ-algebra E1 to E1, called generalized internal states (or GI-state). Then we give some examples and basic properties of generalized (internal) st...

We consider a different L-Minkowski combination of compact sets in R than the one introduced by Firey and we prove an L-BrunnMinkowski inequality, p ∈ [0, 1], for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function t 7→ μ(t 1 pA), p ∈ (0, 1], for unconditional con...

We consider a different L-Minkowski combination of compact sets in R than the one introduced by Firey and we prove an L-BrunnMinkowski inequality, p ∈ [0, 1], for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function t 7→ μ(t 1 pA), p ∈ (0, 1], for unconditional con...

Beam stiffeners have frequently been used for raising natural frequencies of base structures. In stiffener layout optimization problems, most of the previous researches considering the position and/or the length of the stiffener as design variables dealt with structures having just simple convex shapes such as a square or rectangle. The reason was because concave shape structures have difficult...

We construct a Banach space operator T ∈ B(X) such that the set JT (0) has a nonempty interior but JT (0) 6= X. This gives a negative answer to a problem raised by G. Costakis and A. Manoussos.