Bismuth (III) Acetate: A New Catalyst for Preparation of Azlactones via the Erlenmeyer Synthesis

Bismuth(III) acetate catalyzes the synthesis of azlactones from aromatic aldehydes in moderate to good yields via the Erlenmeyer synthesis. The relatively low toxicity and low cost of bismuth(III) acetate make this procedure particularly attractive.     

Some matrix identities on colored Motzkin paths

Merlini and Sprugnoli (2017) give both an algebraic and a combinatorial proof for an identity proposed by Louis Shapiro by using Riordan arrays and a particular model of lattice paths. In this paper, we revisit the identity and emphasize the use of colored partial Motzkin paths as appropriate tool. By using colored Motzkin paths with weight defined according to the height of its last point, we can generalize the identity in several ways. These identities allow us to move from Fibonacci polynomials, Lucas polynomials, and Chebyshev polynomials, to the polynomials of the form ${(z+b)}^n$.     

递推数列的非线性表达式的构造

A type of nonlinear expressions of Lucas sequences are established inspired by Hsu [A nonlinear expression for Fibonacci numbers and its consequences.J.Math.Res.Appl.,2012,32(6):654–658].Using the relationships between the Lucas sequence and other linear recurring sequences satisfying the same recurrence relation of order 2,i.e.,the Horadam sequences,we may transfer the identities of Lucas sequences to the latter.     

Eulerian polynomials and B-splines

An interrelationship between Eulerian polynomials, Eulerian fractions and Euler–Frobe nius polynomials, Euler–Frobenius fractions, and B-splines is presented. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed.     

Elementary operators and invariant subalgebras

We provide an example of an elementary operator which leaves invariant a nest algebra but which cannot be written as a finite sum of multiplications each of which leaves the nest algebra invariant. We also prove that the given operator lies in the completely bounded norm closure of the linear span of the multiplications which leave the nest algebra invariant.

     

A Phase Space Approach to Gravitational Entropy

We examine the definition S=ln as a candidate function for gravitational entropy. We calculate its behavior for gravitational and density perturbations in closed, open and flat cosmological models and find that in all cases it increases monotonically. We are also able to calculate the entropy density of gravitational radiation produced by inflation. We compare the results with the behavior of the Weyl-tensor squared. Applying the formalism to black holes has proved more problematical.     

Solubility of gases in liquids. XVI. Henry's law coefficients for nitrogen in water at 5 to 50°C

The solubility of nitrogen in pure liquid water was measured in the pressure range 45 to 115 kPa and in the temperature range 5 to 50°C. These data are used to obtain Henry coefficients H _2,1 (T,P _s,1 ) at the vapor pressure P _s,1 of water. The temperature dependence of H _2,1 (T,P _s,1 ) is accounted for by both a Clarke-Glew (CG) type fitting equation, and a power series in T^–1, as suggested by Benson and Krause (BK). The imprecision of our measurements is characterized by an average deviation of ±0.038% from a four-term CG equation, and by an average deviation of ±0.042% from a three-term BK equation. From the temperature variation of H _2,1 (T,P _s,1 ) partial molar quantities referring to the solution process, such as enthalpies and heat capacities of solution, are obtained. They are given in tabular form, together with H _2,1 (T,P _s,1 ) and derived Ostwald coefficients L, at rounded temperatures. Finally, experimental results are compared with values calculated via scaled particle theory.     

<Emphasis Type="Italic">Rhodobacter capsulatus</Emphasis> porphobilinogen synthase,a high activity metal ion independent hexamer

Background  

The enzyme porphobilinogen synthase (PBGS), which is central to the biosynthesis of heme, chlorophyll and cobalamins, has long been known to use a variety of metal ions and has recently been shown able to exist in two very different quaternary forms that are related to metal ion usage. This paper reports new information on the metal ion independence and quaternary structure of PBGS from the photosynthetic bacterium Rhodobacter capsulatus.