Regioselective alkali metal reduction of dibenzocyclooctadiene lignan derivatives,demethoxylation followed by dehalogenation

The regioselective demethoxylation and dehalogenation of dihalogenated dibenzocyclooctadiene lignans derivatives were realized in a one-step reaction with excellent yields in the sodium and t-butanol reaction system.     

On the derived length of units in group algebra

Let G be a finite group G, K a field of characteristic p ≥ 17 and let U be the group of units in KG. We show that if the derived length of U does not exceed 4, then G must be abelian.     

On the proof of Erdős’ inequality

Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality \({\left\| {p'} \right\|_{\left[ { - 1,1} \right]}} \leqslant \frac{1}{2}{\left\| p \right\|_{\left[ { - 1,1} \right]}}\) for a constrained polynomial p of degree at most n, initially claimed by P. Erd?s, which is different from the one in the paper of T.Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval (?1, 1) and establish a new asymptotically sharp inequality.     

On 3-generated lattices with a completely modular element among generators

We consider a lattice generated by three elements, one of which is completely modular. The free lattice with this property is proved to be finite. It is not modular and contains exactly 39 elements. We have also found a finite set of defining relations for the generating elements of this lattice.     

Division closed partially ordered rings

Fuchs called a partially-ordered integral domain, say D, division closed if it has the property that whenever a > 0 and ab > 0, then b > 0. He showed that if D is a lattice-ordered division closed field, then D is totally ordered. In fact, it is known that for a lattice-ordered division ring, the following three conditions are equivalent: a) squares are positive, b) the order is total, and c) the ring is division closed. In the present article, our aim is to study \({\ell}\)-rings that possibly possess zerodivisors and focus on a natural generalization of the property of being division closed, what we call regular division closed. Our investigations lead us to the concept of a positive separating element in an \({\ell}\)-ring, which is related to the well-known concept of a positive d-element.     

On the covering radius of lattice zonotopes and its relation to view-obstructions and the lonely runner conjecture

The goal of this paper is twofold; first, we show the equivalence between certain problems in geometry, such as view-obstructions, billiard ball motions, and the estimation of covering radii of lattice zonotopes. Second, we utilize the latter interpretation and provide upper bounds of said radii by virtue of the Flatness Theorem. Our results allow us to specify how rational dependencies in the view-direction influence the obstruction parameter. These problems are similar in nature to the famous Lonely Runner Problem for which we draw analogous conclusions.     

O the volume growth of the hyperbolic regular $$\varvec{}$$-simplex

In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the n-dimensional volume of a regular simplex and the \((n-1)\)-dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for \(n=3\).     

J. L. Lions’ problem on maximal regularity

This is a survey on recent progress concerning maximal regularity of non-autonomous equations governed by time-dependent forms on a Hilbert space. It also contains two new results showing the limits of the theory.     

On second-order elliptic operators with complete first-order boundary degeneration and strong outward drift

Second order elliptic operators and their parabolic counterparts are studied in the case of complete boundary degeneration of the leading order coefficients in the presence of a strong outward pointing drift. It is shown that the problem generates a positive analytic \(C_0\)-semigroup with maximal \(L_p\)-regularity in \(L_q\)-spaces. This result is based on hard analysis estimates for integral operators in combination with modern functional analytic tools like \({\mathcal R}\)-boundedness and the operator-valued functional calculus.