Palladium‐catalyzed decarboxylative coupling of α,β‐unsaturated carboxylic acids with aryl tosylates

We report a general method for selective cross‐coupling of α,β‐unsaturated carboxylic acids with aryl tosylates enabled by versatile Pd(II) complexes. This method features the general cross‐coupling of ubiquitous α,β‐unsaturated carboxylic acids by decarboxylation. The transformation is characterized by its operational simplicity, the use of inexpensive, air‐stable Pd(II) catalysts, scalability and wide substrate scope. The reaction proceeds with high trans selectivity to furnish valuable (E)‐1,2‐diarylethenes.     

Infinite family of transmission irregular trees of even order

Distance between two vertices is the number of edges in a shortest path connecting them in a connected graph $G$. The transmission of a vertex $v$ is the sum of distances from $v$ to all the other vertices of $G$. If transmissions of all vertices are mutually distinct, then $G$ is a transmission irregular graph. It is known that almost no graphs are transmission irregular. Infinite families of transmission irregular trees of odd order were presented in Alizadeh and Klav?ar (2018). The following problem was posed in Alizadeh and Klav?ar (2018): do there exist infinite families of transmission irregular trees of even order? In this article, such a family is constructed.     

Exploring mathematical connections of pre-university students through tasks involving rates of change

This paper reports the results of a research exploring the mathematical connections of pre-university students while they solving tasks which involving rates of change. We assume mathematical connections as a cognitive process through which a person finds real relationships between two or more ideas, concepts, definitions, theorems, procedures, representations or meanings or with other disciplines or the real-world. Four tasks were proposed to the 33 pre-university students that participated in this research; the central concept of the first task is the slope, the last three tasks contain concepts like velocity, speed and acceleration. Task-based interviews were conducted to collect data and later analysed with thematic analysis. Results showed most of the students made mathematical connections of the procedural type, the mathematical connections of the common features type are made in smaller quantities and the mathematical connection of the generalization type is scarcely made. Furthermore, students considered slope as a concept disconnected from velocity, speed and acceleration.     

The maximum multiplicity of the largest k-th eigenvalue in a matrix whose graph is acyclic or unicyclic

Given a graph $G$ we are interested in studying the symmetric matrices associated to $G$ with a fixed number of negative eigenvalues. For this class of matrices we focus on the maximum possible nullity. For trees this parameter has already been studied and plenty of applications are known. In this work we derive a formula for the maximum nullity and completely describe its behavior as a function of the number of negative eigenvalues. In addition, we also carefully describe the matrices associated with trees that attain this maximum nullity. The analysis is then extended to the more general class of unicyclic graphs. Further our work is applied to re-describing all possible partial inertias associated with trees, and is employed to study an instance of the inverse eigenvalue problem for certain trees.     

Interpretation of the Role of Composition on the Inclusion Efficiency of Monovalent Cations into Cobalt Hexacyanoferrate

Cobalt hexacyanoferrate of various compositions was prepared in flow mode and the role of the vacancy on the structure, thermogravimetric (TG) properties, and the adsorption efficiency was studied. The material, Na_yCo[Fe(CN)₆]_1−x ⋅ z H₂O, with a minimum vacancy of x=0.014 to the highest x=0.47, was obtained. The TG-differential scanning calorimetry (DSC) profile showed a distinct influence of the vacancy on the water release temperature. Materials with x>0.35 showed a smooth release of water at a relatively lower temperature. However, for the materials with x<0.35, water release took place in multiple steps, suggesting the existence of various forms of water. The FTIR profiles supported the existence of free and bonded water molecules. However, the materials with multiple water peaks in the FTIR spectra showed a shift of the major XRD peaks when heated at 285 °C in N₂ atmosphere. Regarding the effect of the vacancy on the adsorption behavior, for NH₄, the adsorption was found to be proportional to the number of Na atoms in the material, confirming the ion-exchange process. On the contrary, the materials with low vacancy and high Na content showed nominal Cs adsorption capacity. Interestingly, the K adsorption capacity was found to be in between that of the other two ions. This means the ionic size decides the rate of placement into the interstitial sites. For larger ions like Cs, the ease of percolation via the vacancy decides the overall adsorption efficiency.