Precision PEGylated Polymers Obtained by Sequence‐Controlled Copolymerization and Postpolymerization Modification

Copolymers containing water‐soluble poly(ethylene glycol) (PEG) side chains and precisely controlled functional microstructures were synthesized by sequence‐controlled copolymerization of donor and acceptor comonomers, that is, styrene derivatives and N‐substituted maleimides. Two routes were compared for the preparation of these structures: a) the direct use of a PEG–styrene macromonomer as a donor comonomer, and b) the use of an alkyne‐functionalized styrenic comonomer, which was PEGylated by copper‐catalyzed alkyne–azide cycloaddition after polymerization. The latter method was found to be the most versatile and enabled the synthesis of high‐precision copolymers. For example, PEGylated copolymers containing precisely positioned fluorescent (e.g. pyrene), switchable (e.g. azobenzene), and reactive functionalities (e.g. an activated ester) were prepared.     

On generalized square-full numbers in an arithmetic progression

Czechoslovak Mathematical Journal - Let a and $$b \in \mathbb{N}$$ . Denote by Ra,b the set of all integers n > 1 whose canonical prime representation $$n = p_1^{{\alpha _1}}p_2^{{\alpha...     

The Mean-value of Meromorphic Functions with Respect to a Generalized Boolean Transformation

A formula for the mean-value distribution of certain meromorphic functions on a vertical line s = σ +iR under a generalized Boolean transformation, called rational Boolean transformation from R into itself, is derived using Birkhoff 's ergodic theorem. This formula is represented as a computable integral. Using the Cauchy's integral theorem, values of this integral corresponding to various possible cases are explicitly computed.     

An algorithm for the number of path homomorphisms

A homomorphism of a graph G₁=(V₁,E₁) to a graph G₂=(V₂,E₂) is a mapping from the vertex set V₁ of G₁ to the vertex set V₂ of G₂ which preserves edges. In this paper we provide an algorithm to determine the number of homomorphisms from an arbitrary finite undirected path to another arbitrary finite undirected path.     

Characterizations of Complete Sublattices of a Given Complete Lattice

To characterize all complete sublattices of a given complete lattice is an important problem. In this paper we will give three different characterizations of complete sublattices of a complete lattice by using closure operators, kernel operators, and by using Galoisclosed subrelations.The research was supported by a grant of the faculty of Science ChiangMai University Thailand.     

Thermal characterization of longitudinal merging of turbulent spots

This study presents the thermal behavior of two young turbulent spots merging into a longitudinal direction on an isothermal flat plate for the local Reynolds number between 6.1 × 10⁴ and 1.3 × 10⁵ in a low freestream turbulence water tunnel having a turbulent intensity of 1.16%. The two turbulent spots are generated by water injection through a 1-mm-diameter hole in the perpendicular direction of the mainstream flow with a dimensionless separating time (Δτ) of 42.08, 84.16, and 126.24. Thermochromic liquid crystals are utilized mutually with an image processing technique to extract the spot characteristics qualitatively and quantitatively. The results demonstrate that the following turbulent spot directly causes an increase in the local Nusselt number and heat rate within the footprint of the merging spots. The relatively highest increase in this study occurs when Δτ = 84.16. The average Nusselt number and effectiveness characterize differently in the intersection zone, non-intersection zone of the leading spot, and non-intersection zone of the following spot. The results confirm that turbulent spots under the boundary layer transition augment the heat transfer rate to the level of full turbulence by not only their spot maturity but also the merging mechanism. Finally, the heat transfer mechanism is discussed and the predictive formulas for the Nusselt number and heat flux of the longitudinal merging of turbulent spots for Δτ from 0 to 126.24 are provided.     

On the distribution of (k,r)-integers in Piatetski-Shapiro sequences

Czechoslovak Mathematical Journal - A natural number n is said to be a (k, r)-integer if n = akb, where k > r > 1 and b is not divisible by the rth power of any prime. We study the...     

Precision polyelectrolytes

Charged macromolecules with controlled microstructures were prepared. Well-defined non-ionic precursors were first synthesized by sequence-controlled radical polymerization of tert-butyl 4-vinyl benzoate with various N-substituted maleimides. Afterwards, these macromolecules were hydrolyzed into polyanions.     

On the distribution of square-full and cube-full primitive roots

Periodica Mathematica Hungarica - A positive integer n is called an r-full integer if for all primes $$p\mid n$$ we have $$p^r\mid n.$$ Let p be an odd prime. For $$\gcd (n,p)=1$$, the smallest...