Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n−s=∏pprime11−p−s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended ζ(s) to the complex plane z and conjectured that all nontrivial zeros are in the R(z)=1/2 axis. The nonadditive entropy Sq=k∑ipilnq(1/pi)(q∈R;S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz≡z1−q−11−q(ln1z=lnz). It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as 〈x〉q≡elnqx, which recover the number x for q=1. The q-prime numbers are then defined as the q-natural numbers 〈n〉q≡elnqn(n=1,2,3,⋯), where n is a prime number p=2,3,5,7,⋯ We show that, for any value of q, infinitely many q-prime numbers exist; for q≤1 they diverge for increasing prime number, whereas they converge for q>1; the standard prime numbers are recovered for q=1. For q≤1, we generalize the ζ(s) function as follows: ζq(s)≡〈ζ(s)〉q (s∈R). We show that this function appears to diverge at s=1+0, ∀q. Also, we alternatively define, for q≤1, ζq∑(s)≡∑n=1∞1〈n〉qs=1+1〈2〉qs+⋯ and ζq∏(s)≡∏pprime11−〈p〉q−s=11−〈2〉q−s11−〈3〉q−s11−〈5〉q−s⋯, which, for q<1, generically satisfy ζq∑(s)<ζq∏(s), in variance with the q=1 case, where of course ζ1∑(s)=ζ1∏(s).     

Structural Insights into the Design of Synthetic Nanobody Libraries

Single domain antibodies from camelids, or nanobodies, are a unique class of antibody fragments with several advantageous characteristics: small monomeric size, high stability and solubility and easy tailoring for multiple applications. Nanobodies are gaining increasing acceptance as diagnostic tools and promising therapeutic agents in cancer and other diseases. While most nanobodies are obtained from immunized animals of the camelid family, a few synthetic nanobody libraries constructed in recent years have shown the capability of generating high quality nanobodies in terms of affinity and stability. Since this synthetic approach has important advantages over the use of animals, the recent advances are indeed encouraging. Here we review over a dozen synthetic nanobody libraries reported so far and discuss the different approaches followed in their construction and validation, with an emphasis on framework and hypervariable loop design as critical issues defining their potential as high-class nanobody sources.     

Relative effectiveness of pretreatments on performance of Rhizomucor miehei lipase in nonpolar reaction media

Enzymes can be used in nonpolar reaction media to modify waterinsoluble substrates. A variety of pretreatments, applied to the enzyme prior to introduction to the nonpolar media, can improve enzyme activity. However, the various pretreatments have not been studied using directly comparable conditions, nor have they been applied simultaneously to test for interactive effects. This work evaluates pretreatment of lipase with various classes of additives. The pretreated lipase is used to catalyze esterification between citronellol and acetic acid in a medium of n-hexane. The effectiveness of a particular pretreatment is presented in terms of relative performance (RP), which is equal to the number of times faster the pretreated lipase catalyzes the reaction relative to untreated lipase. The individual and interactive effects of the pretreatment factors were studied and compared. Buffer salts had a much stronger performance-enhancing effect than nonbuffer salts; pretreatment with 90% (w/w) sodium phosphate yielded lipase with an RP of approx 64. A strong interaction was found between the treatments with sodium phosphate and pH adjustment. These treatments may mitigate the inhibitory effect of acetic acid. Activating effects of phase interfaces and active-site protectants are shown to be complementary to other treatments, demonstrating that they likely act by distinct mechanisms.     

Magnetic nanotips: the realisation of an atomic-scale beam-splitter

We present the first realisation of ferromagnetic Fe nanotips. Studies of the field emitted beams of electrons and metallic ions above and under the Curie temperature T_c were made. We have observed a reversible splitting of the electron beams when the temperature crossed through T_c and multiple spot patterns for the emitted metallic ion beams. These phenomena were not observed with Fe microtips or non-magnetic nanotips and thus were correlated to a strong magnetic interaction at the atomic scale apex of the nanotips. The Fe nanotips constitute an atomic scale beam-splitter.     

Positron spectroscopy analysis in metallocene propylene/1‐octadecene copolymers: Parameters dependence on monoclinic and mesomorphic polymorphs

Copolymers of propylene and 1‐octadecene synthesized by a metallocene catalyst were characterized by PALS, WAXD, DSC, and density measurements. The change in the sizes and the number density of free‐volume holes as a function of the 1‐octadecene content were compared to the master laws published for copolymers of propylene with α‐olefins of shorter lengths. It was found that the samples were separated into two groups. The first set of samples obeyed the master laws while the second group deviated. This behavior was explained by the difference in the copolymer structure. The WAXD analysis confirmed the separation as well. The deviation was explained as due to the mesomorphic crystalline structure of the second group of samples, in contrast to the monoclinic one for the first group, which was a result of the relatively fast cooling from the molten state and from the high comonomer molar content. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1994–2002, 2010