The geometry of the Wigner caustic and a decomposition of a curve into parallel arcs

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential.

High order curvature flows of plane curves with generalised Neumann boundary conditions

We consider the parabolic polyharmonic diffusion and the L 2 {L^{2}} -gradient flow for the square integral of the m-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L 2 {L^{2}} , then the evolving curve converges exponentially in the C ∞ {C^{\infty}} topology to a straight horizontal line segment. The same behaviour is shown for the L 2 {L^{2}} -gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.

Regularity in Topological Modules

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and m∗∈M∗ is a continuous linear functional on M which is open as a map between topological spaces, then m∗−1(int(B)) is regular open and m∗−1(B) is regular closed, for B any closed-unit zero-neighborhood in R.

The lattices of families of regular sets in topological spaces

The question as to the number of sets obtainable from a given subset of a topological space using the operators derived by composing members of the set {b, i, ∨, ∧}, where b, i, ∨ and ∧ denote the closure operator, the interior operator, the binary operators corresponding to union and intersection, respectively, is called the Kuratowski {b, i, ∨, ∧}-problem. This problem has been solved independently by Sherman [21] and, Gardner and Jackson [13], where the resulting 34 plus identity operators were depicted in the Hasse diagram. In this paper we investigate the sets of fixed points of these operators. We show that there are at most 23 such families of subsets. Twelve of them are the topology, the family of all closed subsets plus, well known generalizations of open sets, plus the families of their complements. Each of the other 11 families forms a complete complemented lattice under the operations of join, meet and negation defined according to a uniform procedure. Two of them are the well known Boolean algebras formed by the regular open sets and regular closed sets, any of the others in general need not be a Boolean algebras.

Isoperimetric equalities for rosettes

In this paper, we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with nonvanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented areas of the Wigner caustic, the Constant Width Measure Set and the Spherical Measure Set. We also study and find new results about the geometry of affine equidistants of rosettes and of the union of rosettes.

A Note On Neutrosophic Soft Menger Topological Spaces

In this paper, the concept of neutrosophic soft Mengerness, neutrosophic soft near Mengerness and neutrosophic soft almost Mengerness are introduced and studied. Some characterizations of neutrosophic soft almost Mengerness in terms of neutrosophic soft regular open or neutrosophic soft regular closed are given.

Natural extension of EF-spaces and EZ-spaces to the pointfree context

Two problems concerning EF-frames and EZ-frames are investigated. In [Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84–97], Tahirefar defines a Tychonoff space X to be an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). By extending these notions to locales, we give several characterizations of EF and EZ-frames, mostly in terms of certain ring-theoretic properties of 𝓡 L, the ring of real-valued continuous functions on L. We end by defining a qsz-frame which is a pointfree context of qsz-space and, give a characterization of these frames in terms of rings of real-valued continuous functions on L.