Cheeger Sets for Rotationally Symmetric Planar Convex Bodies

In this note we obtain some properties of the Cheeger set $$C_\varOmega $$ C Ω associated to a k-rotationally symmetric planar convex body $$\varOmega $$ Ω . More precisely, we prove that $$C_\varOmega $$ C Ω is also k-rotationally symmetric and that the boundary of $$C_\varOmega $$ C Ω touches all the edges of $$\varOmega $$ Ω .

Equilibria of Plane Convex Bodies

We obtain a formula for the number of horizontal equilibria of a planar convex body K with respect to a center of mass O in terms of the winding number of the evolute of $$\partial K$$ ∂ K with respect to O. The formula extends to the case where O lies on the evolute of $$\partial K$$ ∂ K and a suitably modified version holds true for non-horizontal equilibria.

Surjective Identifications of Convex Noetherian Separations in Topological (C, R) Space

The interplay of symmetry of algebraic structures in a space and the corresponding topological properties of the space provides interesting insights. This paper proposes the formation of a predicate evaluated P-separation of the subspace of a topological (C, R) space, where the P-separations form countable and finite number of connected components. The Noetherian P-separated subspaces within the respective components admit triangulated planar convexes. The vertices of triangulated planar convexes in the topological (C, R) space are not in the interior of the Noetherian P-separated open subspaces. However, the P-separation points are interior to the respective locally dense planar triangulated convexes. The Noetherian P-separated subspaces are surjectively identified in another topological (C, R) space maintaining the corresponding local homeomorphism. The surjective identification of two triangulated planar convexes generates a quasiloop–quasigroupoid hybrid algebraic variety. However, the prime order of the two surjectively identified triangulated convexes allows the formation of a cyclic group structure in a countable discrete set under bijection. The surjectively identified topological subspace containing the quasiloop–quasigroupoid hybrid admits linear translation operation, where the right-identity element of the quasiloop–quasigroupoid hybrid structure preserves the symmetry of distribution of other elements. Interestingly, the vertices of a triangulated planar convex form the oriented multiplicative group structures. The surjectively identified planar triangulated convexes in a locally homeomorphic subspace maintain path-connection, where the right-identity element of the quasiloop–quasigroupoid hybrid behaves as a point of separation. Surjectively identified topological subspaces admitting multiple triangulated planar convexes preserve an alternative form of topological chained intersection property.

Banach–Mazur Distance from the Parallelogram to the Affine-Regular Hexagon and Other Affine-Regular Even-Gons

We show that the Banach–Mazur distance between the parallelogram and the affine-regular hexagon is $$\frac{3}{2}$$ 3 2 and we conclude that the diameter of the family of centrally-symmetric planar convex bodies is just $$\frac{3}{2}$$ 3 2 . A proof of this fact does not seem to be published earlier. Asplund announced this without a proof in his paper proving that the Banach–Mazur distance of any planar centrally-symmetric bodies is at most $$\frac{3}{2}$$ 3 2 . Analogously, we deal with the Banach–Mazur distances between the parallelogram and the remaining affine-regular even-gons.

Surfactant-Laden Janus Droplets with Tunable Morphologies and Enhanced Stability for Fabricating Lens-Shaped Polymeric Microparticles

Janus droplets can function as excellent templates for fabricating physically and chemically anisotropic particles. Here, we report new surfactant-laden Janus droplets with curvature controllability and enhanced stability against coalescence, suitable for fabricating shape-anisotropic polymer microparticles. Using a microfluidic flow-focusing device on a glass chip, nanoliter-sized biphasic droplets, comprising an acrylate monomer segment and a silicone-oil (SO) segment containing a surfactant, were produced in a co-flowing aqueous polyvinyl alcohol (PVA) solution. At equilibrium, the droplets formed a Janus geometry based on the minimization of interfacial energy, and each of the two Janus segments were uniform in size with coefficient-of-variation values below 3%. By varying the concentration of the surfactant in the SO phase, the curvature of the interface between the two lobes could be shifted among concave, planar, and convex shapes. In addition, the Janus droplets exhibited significantly improved stability against coalescence compared with previously reported Janus droplets carrying no surfactant that coalesced rapidly. Finally, via off-chip photopolymerization, concave-convex, planar-convex, and biconvex lens-shaped particles were fabricated.

Minirobots Moving at Different Partial Speeds

In this paper, we present the mathematical point of view of our research group regarding the multi-robot systems evolving in a multi-temporal way. We solve the minimum multi-time volume problem as optimal control problem for a group of planar micro-robots moving in the same direction at different partial speeds. We are motivated to solve this problem because a similar minimum-time optimal control problem is now in vogue for micro-scale and nano-scale robotic systems. Applying the (weak and strong) multi-time maximum principle, we obtain necessary conditions for optimality and that are used to guess a candidate control policy. The complexity of finding this policy for arbitrary initial conditions is dominated by the computation of a planar convex hull. We pointed this idea by applying the technique of multi-time Hamilton-Jacobi-Bellman PDE. Our results can be extended to consider obstacle avoidance by explicit parameterization of all possible optimal control policies.