Homomorphisms of matrix algebras and constructions of Butson–Hadamard matrices

An $n\times n$ matrix $H$ is Butson–Hadamard if its entries are $k$th roots of unity and it satisfies $H{H}^1=n{I}_n$. Write $\mathrm{BH}(n,k)$ for the set of such matrices.Suppose that $k=p^{\alpha }{q}^{\beta }$ where $p$ and $q$ are primes and $\alpha \ge 1$. A recent result of Östergård and Paavola uses a matrix $H\in \mathrm{BH}(n,pk)$ to construct $H^{\prime }\in \mathrm{BH}(pn,k)$. We simplify the proof of this result and remove the restriction on the number of prime divisors of $k$. More precisely, we prove that if $k=mt$, and each prime divisor of $k$ divides $t$, then we can construct a matrix $H^{\prime }\in \mathrm{BH}(mn,t)$ from any $H\in \mathrm{BH}(n,k)$.     

An Aldehyde Responsive,Cleavable Linker for Glucose Responsive Insulins

A glucose responsive insulin (GRI) that responds to changes in blood glucose concentrations has remained an elusive goal. Here we describe the development of glucose cleavable linkers based on hydrazone and thiazolidine structures. We developed linkers with low levels of spontaneous hydrolysis but increased level of hydrolysis with rising concentrations of glucose, which demonstrated their glucose responsiveness in vitro. Lipidated hydrazones and thiazolidines were conjugated to the LysB29 side-chain of HI by pH-controlled acylations providing GRIs with glucose responsiveness confirmed in vitro for thiazolidines. Clamp studies showed increased glucose infusion at hyperglycemic conditions for one GRI indicative of a true glucose response. The glucose responsive cleavable linker in these GRIs allow changes in glucose levels to drive the release of active insulin from a circulating depot. We have demonstrated an unprecedented, chemically responsive linker concept for biopharmaceuticals.     

A Frustrated Phosphane–Borane Lewis Pair and Hydrogen: A Kinetics Study

The energy profile of a frustrated Lewis pair (FLP) dihydrogen splitting system was determined by a combined experimental kinetic and DFT study. A trimethylene‐bridged phosphane–borane FLP was converted into its endothermic H₂‐cleavage product by sequential H⁺/H^? addition. The system could be handled at low temperature, and the kinetics of the H₂ elimination were determined to give a rate constant of k_HH,exp(299 K)=(2.87±0.1)×10^?4 s^?1 in solution. The primary kinetic isotope effects were determined; for example, (k_HH/k_DD)_exp=3.19. The system was accurately analyzed by DFT calculations.     

Green’s Functional for Higher-Order Ordinary Differential Equations with General Nonlocal Conditions and Variable Principal Coefficient

Ukrainian Mathematical Journal - The method of Green’s functional is a little-known technique for the construction of fundamental solutions to linear ordinary differential equations (ODE)...     

The Radical Formula for Noncommutative Rings

Ukrainian Mathematical Journal - We determine some classes of left modules satisfying the radical formula in a noncommutative ring. We also show that, under a certain condition, a finitely...     

Correction to: Characterization of essential oil from Matricaria sevanensis by microwave-assisted distillation

Journal of Thermal Analysis and Calorimetry - In the original publication of the article, peak numbers were not given in figure 4. The corrected figure 4 is given below.