Thermal Conversion of Methane to Formaldehyde Promoted by Gold in AuNbO3+ Cluster Cations

In addition to generation of a methyl radical, formation of a formaldehyde molecule was observed in the thermal reaction of methane with AuNbO₃⁺ heteronuclear oxide cluster cations. The clusters were prepared by laser ablation and mass‐selected to react with CH₄ in an ion‐trap reactor under thermal collision conditions. The reaction was studied by mass spectrometry and DFT calculations. The latter indicated that the gold atom promotes formaldehyde formation through transformation of an Au?O bond into an Au?Nb bond during the reaction.     

The Forgotten Pyrazines: Exploring the Dakin–West Reaction

Pyrazines are an underreported class of N-heterocycles available from nitrogen-rich biomass presenting an interesting functional alternative for current aromatics. In this work, access to pyrazines obtained from amino acids by using the 90 year old Dakin–West reaction was explored. After a qualitative screening several functional proteinogenic amino acids proved good substrates for this reaction, which were successfully scaled to multigram scale synthesis of the corresponding intermediate α-acetamido ketones. Subsequently, the conditions towards pyrazine formation using δ-amino-levulinic acid were optimized, and these were employed to synthesize a relevant set of five functional dimethylpyrazines in high purity. These pyrazines can be considered a versatile toolbox of aromatic building blocks for a wide range of applications, such as in the synthesis of polymers or metal–organic frameworks.     

关于不定方程x3-1=749y2

利用同余式、平方剩余、Pell方程的解的性质、递归序列证明了:不定方程x³-1=749y²仅有整数解(x,y)=(1,0).     

Errata and addendum to “On the prime geodesic theorem for hyperbolic ‐manifolds” Math. Nachr. 291 (2018), no. 14–15, 2160–2167

We correct the exponent in the error term of the prime geodesic theorem for hyperbolic 3‐manifolds 1 and in Park's theorem for higher dimensions [ 3 , 2 ].     

The first polymorph in the family of nucleobases: a second form of cytosine

A new polymorph of cytosine, C₄H₅N₃O, is reported half a century after the report of its first known crystal structure [Barker & Marsh (1964). Acta Cryst. 17 , 1581–1587]. Cytosine thus provides the first polymorphic example in the category of parent nucleobases. The new form, denoted (Ib), was observed unexpectedly during an attempt to cocrystallize cytosine with catechol. Form (Ib) crystallizes in the orthorhombic centrosymmetric space group Pccn with two molecules in the asymmetric unit. The previously known form, denoted (Ia), crystallizes in the orthorhombic noncentrosymmetric space group P2₁2₁2₁. The cytosine molecule is planar in both forms. Hydrogen‐bonding interactions are also similar for both forms. Infinite one‐dimensional ribbons composed of cytosine base‐pair dimers in R₂²(8) arrangements are observed in both (Ia) and (Ib). However, the way that the ribbons are packed differs in (Ia) and (Ib). This appears to guide the centrosymmetric versus noncentrosymmetric space‐group selection through the formation of an inversion‐related motif in polymorph (Ib) and a helical propagation in polymorph (Ia). A few selected polymorphic systems have been gathered from the Cambridge Structural Database to understand possible structural features responsible for achiral molecules adopting centro‐ and noncentrosymmetric space groups.     

Efficient remainder rule

Understanding the solution of a problem may require the reader to have background knowledge on the subject. For instance, finding an integer which, when divided by a nonzero integer leaves a remainder; but when divided by another nonzero integer may leave a different remainder. To find a smallest positive integer or a set of integers following the given conditions, one may need to understand the concept of modulo arithmetic in number theory. The Chinese Remainder Theorem is a known method to solve these types of problems using modulo arithmetic. In this paper, an efficient remainder rule has been proposed based on basic mathematical concepts. These core concepts are as follows: basic remainder rules of divisions, linear equation in slope intercept form, arithmetic progression and the use of a graphing calculator. These are easily understood by students who have taken prealgebra or intermediate algebra.     

On separating a fixed point from zero by invariants

Assume a fixed point v∈V^G can be separated from zero by a homogeneous invariant f∈𝕜[V]^G of degree p^rd, where p>0 is the characteristic of the ground field 𝕜 and p,d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r, which we obtain explicitly from f. It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power.     

Totally ordered sets and the prime spectra of rings

Let T be a totally ordered set and let D(T) denotes the set of all cuts of T. We prove the existence of a discrete valuation domain O_v such that T is order isomorphic to two special subsets of Spec(O_v). We prove that if A is a ring (not necessarily commutative), whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set U?Spec(A) such that the prime spectrum of A is order isomorphic to D(U). We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view.