In addition its extension to a more general multifunction system is presented. Depending on the interaction between different functions of the system, two cases, namely reducible and irreducible, are discussed in detail. A multifunction electrical system is used for demonstration.The free energy of a model of uniformly weighted lattice knots of length n and knot type K confined to a lattice cube of side length L-1 is given by F_L(ϕ)=-1/Vlogp_n,L(K), where V=L^3 and where ϕ=n/V is the concentration of monomers of the lattice knot in the confining cube. The limiting free energy of the model is F_∞(ϕ)=lim_L→∞F_L(ϕ) and the limiting osmotic pressure of monomers leaving the lattice knot to become solvent molecules is defined by Π_∞(ϕ)=ϕ^2d/dϕ[F_∞(ϕ)/ϕ]. I show that, under very mild assumptions, the functions P_L(ϕ)=ϕ^2d/dϕ[F_L(ϕ)/ϕ]|_n and Π_L(ϕ)=ϕ^2d/dϕ[F_L(ϕ)/ϕ]|_L are finite-size approximations of Π_∞(ϕ).In this work, we model and simulate the shape evolution of critically charged droplets, from the initial spherical shape to the charge emission and back to the spherical shape. The shape deformation is described using the viscous correction for viscous potential flow model, which is a potential flow approximation of the Navier-Stokes equation for incompressible Newtonian fluids. The simulated shapes are compared to snapshots of experimentally observed drop deformations. We highlight the influence of the dimensionless viscosity and charge carrier mobility of the liquid on the shape evolution of droplets and discuss the observed trends. We give an explanation as to why the observed deformation pathways of positively and negatively charged pure water droplets differ and give a hint as to why negatively charged water droplets emit more charge during charge breakup than positively charged ones.An approach was developed to describe the first passage time (FPT) in multistep stochastic processes with discrete states governed by a master equation (ME). The approach is an extension of the totally absorbing boundary approach given for calculation of FPT in one-step processes [N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science Publishers, North Holland, Amsterdam, 2007)] to include multistep processes where jumps are not restricted to adjacent sites. In addition, a Fokker-Planck equation (FPE) was derived from the multistep ME, assuming the continuity of the state variable. The developed approach and an FPE based approach [C. W. Gardiner, Handbook of Stochastic Methods, 3rd ed. (Springer-Verlag, New York, 2004)] were used to find the mean first passage time (MFPT) of the transition between the negative and positive stable macrostates of dust grain charge when the charging process was bistable. The dust was in a plasma and charged by collecting ions and electrons, and emitting secondary electrons. The MFPTs for the transitioning of grain charge from one macrostate to the other were calculated by the two approaches for a range of grain sizes. Both approaches produced very similar results for the same grain except for when it was very small. The difference between MFPTs of two approaches for very small grains was attributed to the failure of the charge continuity assumption in the FPE description. For a given grain, the MFPT for a transition from the negative macrostate to the positive one was substantially larger than that for a transition in a reverse order. The normalized MFPT for a transition from the positive to the negative macrostate showed little sensitivity to the grain radius. For a reverse transition, with the increase of the grain radius, it dropped first and then increased. The probability density function of FPT was substantially wider for a transition from the positive to the negative macrostate, as compared to a reverse transition.As a unique subset of functional polymers, many biopolymers have a set of well-defined three-dimensional (3D) structural characteristics that can be described by spatial contacts between monomers. Statistical analysis of the contacts has been extremely productive in characterizing the biopolymer structural ensemble, such as for 3D chromosome structures. Often, native contacts and compartment structures are the focus of the studies, while the generic polymer aspect, such as the overall decaying of contacts with increasing sequence distance, is analyzed separately or preemptively removed. Here, we explore insights that can be gained by performing "compartment analysis" that keeps the distance decay, which we believe is particularly useful for characterizing the structure transformation of biopolymers. We tested contact analysis on several such transformations under physical perturbation or biological processes, including (1) unfolding of proteins induced by thermal denaturation, (2) chromosome conformation transition during the cell cycle, and (3) chromosome unpacking by physicochemical perturbations. Useful score functions were developed to further quantitatively characterize the transformation judging from the contact analysis. We also find that the sinusoidal undertone of eigenvector patterns (the "unwanted," low frequency signal, in contrast to the detailed A/B compartment) that had previously been attributed to biological effects of centromere proximal and distal interactions may in fact reflect a universal feature of polymers that have relatively weaker long-range contacts.Disordered packings of colloidal spheres show angle-independent structural color when the particles are on the scale of the wavelength of visible light.  https://www.selleckchem.com/products/piperaquine-phosphate.html  Previous work has shown that the positions of the peaks in the reflectance spectra can be predicted accurately from a single-scattering model that accounts for the effective refractive index of the material. This agreement shows that the main color peak arises from short-range correlations between particles. However, the single-scattering model does not quantitatively reproduce the observed color the main peak in the reflectance spectrum is much broader and the reflectance at low wavelengths is much larger than predicted by the model. We use a combination of experiment and theory to understand these features. We find that one significant contribution to the breadth of the main peak is light that is scattered, totally internally reflected from the boundary of the sample, and then scattered again. The high reflectance at low wavelengths also results from multiple scattering but can be traced to the increase in the scattering cross section of individual particles with decreasing wavelength.