Some Theorems on Leibniz n-Algebras from the Category Un(Lb)

We study the Leibniz n-algebra Un(∑),whose multiplication is defined viathe bracket of a Leibniz algebra ∑ as[x1,...,xn]=[x1,[...,[xn-2,[xn-1,xn]]...]].Weshow that Un(∑) is simple if and only if ∑ is a simple Lie algebra.An analog of Levi'stheorem for Leibniz algebras in Un(Lb) is established and it is proven that the Leibnizn-kernel of Un(Σ) for any semisimple Leibniz algebra Σ is the n-algebra Un(Σ).     

Seeing the Unseen: Experimental Observation of Magnetic Anapole State Inside a High-Index Dielectric Particle

The existence of non-radiating electromagnetic sources attracts much attention in photonic community and gives rise to extensive discussions of various applications in lasing, medical imaging, sensing, and nonlinear optics. In this article, the existence of magnetic anapole states (or magnetic-type non-radiating sources) characterized by a suppressed magnetic dipole radiation in a dielectric cylindrical particle is theoretically predicted and experimentally demonstrated. The specific features of the magnetic anapole state under ideal conditions are identified, followed by a demonstration of how their existence can be detected in practical structures. The concept is valid in various frequency bands from visible range for nanoparticles to microwave range for millimeter size objects. The experimental study is performed in microwave frequency range which allows not only to measure the far-field (scattered field) characteristics, but also to probe the peculiar field profile directly inside the dielectric particle. The experimental results agree well with the analytical ones and pave the way to detect and identify nontrivial different-type anapole states.     

Cycloelimination of Imidazolidin‐2‐ylidene N‐Heterocyclic Carbenes: Mechanism and Insights into the Synthesis of Stable “NHC‐CDI” Amidinates

We report the discovery that 1,3‐bis(aryl)imidazolidin‐2‐ylidenes, one of the most widely studied classes of N‐heterocyclic carbenes (NHCs), undergo quantitative conversion to zwitterionic “NHC‐CDI” amidinates upon heating to ≈100 °C in solution. The mechanism of this novel NHC decomposition process is studied in detail. These studies enabled the rational synthesis of a new class of bench stable amidinates from a panel of NHCs and carbodiimides. We expect these constructs to find utility in a variety of applications.     

Gallai’s path decomposition conjecture for graphs of small maximum degree

Gallai’s path decomposition conjecture states that the edges of any connected graph on $n$ vertices can be decomposed into at most $\frac{n+1}{2}$ paths. We confirm that conjecture for all graphs with maximum degree at most five.