Power partitions and a generalized eta transformation property

In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number $p_s(n)$ of partitions of a positive integer~$n$ into $s$-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1-q^{n^s}\bigr)^{-1}$ near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case $s=1$. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for $p_s(n)$ of the same general type of the one that they gave for~$s=1$. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of $p(200)$ the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for $p_2(100000)$ these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying sizes

Bioinspired programmable wettability arrays for droplets manipulation

The manipulation of liquid droplets demonstrates great importance in various areas from laboratory research to our daily life. Here, inspired by the unique microstructure of plant stomata, we present a surface with programmable wettability arrays for droplets manipulation. The substrate film of this surface is constructed by using a coaxial capillary microfluidics to emulsify and pack graphene oxide (GO) hybridN-isopropylacrylamide (NIPAM) hydrogel solution into silica nanoparticles-dispersed ethoxylated trimethylolpropane triacrylate (ETPTA) phase. Because of the distribution of the silica nanoparticles on the ETPTA interface, the outer surface of the film could achieve favorable hydrophobic property under selective fluorosilane decoration. Owing to the outstanding photothermal energy transformation property of the GO, the encapsulated hydrophilic hydrogel arrays could shrink back into the holes to expose their hydrophobic surface with near-infrared (NIR) irradiation; this imparts the composite film with remotely switchable surface droplet adhesion status. Based on this phenomenon, we have demonstrated controllable droplet sliding on programmable wettability pathways, together with effective droplet transfer for printing with mask integration, which remains difficult to realize by existing techniques.

Institutions

This chapter offers a look at transformation processes from the perspective of the new institutional economics (NIE). It briefly describes the main pillars of this research area, including its assumptions, the definition of institutions, and their interplay. It is shown that the NIE can contribute to explaining the outcome of transformation processes by pointing at the different institutions relied upon during transition. In the section surveying the large literature on institutions and transition, special focus is laid on the role of constitutions for political transformation, property rights for economic transformation, and internal or informal institutions as institutions largely exempt from deliberate transformation which can, hence, constitute an important constraint in transformation processes. The chapter concludes by pointing out some research gaps.

The atomic anisotropic displacement tensor – completing the picture

A simplified approach for calculating the equivalent isotropic displacement parameter is presented and the transformation property of the tensor representationUto point-group operations is analysed. Complete tables have been compiled for the restrictions imposed upon the tensor owing to the site symmetry associated with all special positions as listed in Hahn [(2011),International Tables for Crystallography, Vol. A,Space-group Symmetry, 5th revised ed. Chichester: John Wiley and Sons, Ltd].

Green Preparation of an Environment-Responsive Polymer

Smart polymers are very important in many fields, such as biochemistry, fine chemicals and functional materials, etc. In this paper, poly (N-isopropyl acrylamide) (PNIPAm) was prepared by ultrasonics sonochemistry. In the reaction system, the monomer is N-isopropyl acrylamide (NIPAm), the crosslinker is N,N'-methylenebisacrylamide (BIS) and the solvent is tetrahydrofuran (THF). FTIR and UV-Vis were used to investigate the chemical constitution and phase transformation property of PNIPAm. The results show that it is successful to obtain PNIPAm. It has thermosensitivity. And its lower critical solution temperature is found to be 34°C. Therefore, a green preparation method for environment-responsive polymer is presented.

CHIRAL ANOMALY OF MASSLESS FERMION AT FINITE TEMPERATURE AND CHEMICAL POTENTIAL

In this paper, using the imaginary-time temperature field theory, we discuss the possible modification of chiral anomaly of a massless fermion in (3+1)-dimensional QED (QED4) when the temperature and chemical potential effects are included. It is found that the chiral anomaly is independent of the temperature and chemical potential. Meanwhile, we also introduce the chemical potential corresponding to chiral charge, and find that it will induce a Chern–Simons-like term, which has similar C transformation properties (and opposite P transformation property) to the corresponding Chern–Simons term in (2+1)-dimensional QED (QED3). More importantly, we find that when the chemical potential corresponding to the fermion number is space-dependent, it will induce an extra anomaly term. It is expected that this can yield new physical effects.