Characterization of Rectifying Curves by Their Involutes and Evolutes

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

Upstream modes and antidots poison graphene quantum Hall effect

The quantum Hall effect is the seminal example of topological protection, as charge carriers are transmitted through one-dimensional edge channels where backscattering is prohibited. Graphene has made its marks as an exceptional platform to reveal new facets of this remarkable property. However, in conventional Hall bar geometries, topological protection of graphene edge channels is found regrettably less robust than in high mobility semi-conductors. Here, we explore graphene quantum Hall regime at the local scale, using a scanning gate microscope. We reveal the detrimental influence of antidots along the graphene edges, mediating backscattering towards upstream edge channels, hence triggering topological breakdown. Combined with simulations, our experimental results provide further insights into graphene quantum Hall channels vulnerability. In turn, this may ease future developments towards precise manipulation of topologically protected edge channels hosted in various types of two-dimensional crystals.

A remarkable property of cycloidal curves

The purpose of this theoretical work is to establish a connection between the most important properties of plane curves: cycloids and sinusoids. For this, a drawing mechanism is considered, which simultaneously draws a sinusoid and two cycloids. Based on the results obtained using this mechanical method of obtaining curves, the following important, previously unknown, theoretical facts are established. Firstly, new in theoretical terms is that the sinusoid is not represented as a graph of a trigonometric function, but as a locus of points equidistant from the current points of two cycloids: an ordinary and another cycloid congruent to the original one, inverted and shifted along the axis by half a period. Secondly, the line passing through the current points of these cycloids is nothing like a normal to the resulting sinusoid. This property greatly simplifies the graphical construction of such a normal. And, finally, a simple trigonometric relationship was established between the angle of rotation of the generating circle and the angle of deviation of the normal from the vertical.

Advanced Information Technologies under Manifold Coordinate Systems

In this paper, we regard information technologies under manifold coordinate systems, namely the concept of Ideal Ring Bundles (IRBs), which can be used for configure of the system with the minimized basis for vector data coding and processing designs. The concept involves establishing harmonious mutual penetration of symmetry and asymmetry law as the remarkable property of real space, which allows optimize information technologies based on the law for finding optimal solutions for wide classes of technological problems in informatics, using novel designs based on combinatorial configurations such as the IRBs and their manifold topological transformations. These design techniques make it possible to configure big vector data information systems with fewer numbers of code words than at present, while maintaining or improving on transformation content and the other operating characteristics of the system by means of combinatorial optimization.

On Münchhausen numbers

The remarkable property of the number 3435, namely 3435 = 3^3 + 4^4 + 3^3 + 5^5, was formalized to the notion of Münchhausen numbers. Properties of these numbers are studied and it is showed that although there are only finitely many Münchhausen numbers in a given base b, there are infinitely many Münchhausen numbers of the length 2 in all bases. A certain reversion in the definition gives the notion of so-called anti-Munchhausen numbers, search for them is computationally more effective.

How plants minimize somatic evolution

A remarkable property of plants is their ability to accumulate mutations at a very slow pace despite their potentially long lifespans, during which they continually form buds, each with the potential to become a new branch. Because replication errors in cell division represent an unavoidable source of mutations, minimizing mutation accumulation requires the minimization of cell divisions. Here we show that there exists a well defined theoretical minimum for the branching cost, defined as the number of cell divisions necessary for the creation of each branch. Most importantly, we also show that this theoretical minimum can be closely approached by a simple pattern of cell divisions in the meristematic tissue of apical buds during the generation of novel buds. Both the optimal pattern of cell divisions and the associated branching cost are consistent with recent experimental data, suggesting that plant evolution has led to the discovery of this mechanism.

Metallophthalocyanines as Catalysts in Aerobic Oxidation

The first remarkable property associated to metallophthalocyanines (MPcs) was their chemical “inertness”, which made and make them very attractive as stable and durable industrial dyes. Nevertheless, their rich redox chemistry was also explored in the last decades, making available a solid and detailed knowledge background for further studies on the suitability of MPcs as redox catalysts. An overlook of MPcs and their catalytic activity with dioxygen as oxidants will be discussed here with a special emphasis on the last decade. The mini-review begins with a short introduction to phthalocyanines, from their structure to their main features, going then through the redox chemistry of metallophthalocyanines and their catalytic activity in aerobic oxidation reactions. The most significant systems described in the literature comprise the oxidation of organosulfur compounds such as thiols and thiophenes, the functionalization of alkyl arenes, alcohols, olefins, among other substrates.

On local structure of one-dimensional basic sets of non-reversible A-endomorphisms of surfaces

Recently the authors of the article discovered a meaningful class of non-reversible endomorphisms on a two-dimensional torus. A remarkable property of these endomorphisms is that their non-wandering sets contain nontrivial one-dimensional strictly invariant hyperbolic basic sets (in the terminology of S. Smale and F. Pshetitsky) which have the uniqueness of an unstable one-dimensional bundle. It was proved that nontrivial (other than periodic isolated orbits) invariant sets can only be repellers. Note that this is not the case for reversible endomorphisms (diffeomorphisms). In the present paper, it is proved that one-dimensional expanding uniquely hyperbolic and strictly invariant one-dimensional expanding attractors and one-dimensional contracting repellers of non-reversible A-endomorphisms of closed orientable surfaces have the local structure of the product of an interval by a zero-dimensional closed set (finite or Cantor). This result contrasts with the existence of one-dimensional fractal repellers arising in complex dynamics on the Riemannian sphere and not possessing the properties of the existence of a single one-dimensional unstable bundle.

Redundancy in distributed proofs

Distributed proofs are mechanisms that enable the nodes of a network to collectively and efficiently check the correctness of Boolean predicates on the structure of the network (e.g., having a specific diameter), or on objects distributed over the nodes (e.g., a spanning tree). We consider well known mechanisms consisting of two components: a prover that assigns a certificate to each node, and a distributed algorithm called a verifier that is in charge of verifying the distributed proof formed by the collection of all certificates. We show that many network predicates have distributed proofs offering a high level of redundancy, explicitly or implicitly. We use this remarkable property of distributed proofs to establish perfect tradeoffs between the size of the certificate stored at every node, and the number of rounds of the verification protocol.

Five-Dimensional Contact CR-Submanifolds in S 7 ( 1 )

Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure (φ,ξ,η), we study its five-dimensional contact CR-submanifolds, which are the analogue of CR-submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field ξ is tangent to M, the tangent bundle of contact CR-submanifold M can be decomposed as T(M)=H(M)⊕E(M)⊕Rξ, where H(M) is invariant and E(M) is anti-invariant with respect to φ. On this occasion we obtain a complete classification of five-dimensional proper contact CR-submanifolds in S7(1) whose second fundamental form restricted to H(M) and E(M) vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres.