Data loci in algebraic optimization

We consider parametric optimization problems from an algebraic viewpoint. The idea is to find all of the critical points of an objective function thereby determining a global optimum. For generic parameters (data) in the objective function the number of critical points remains constant. This number is known as the algebraic degree of an optimization problem. In this article, we go further by considering the inverse problem of finding parameters of the objective function so it gives rise to critical points exhibiting a special structure. For example if the critical point is in the singular locus, has some symmetry, or satisfies some other algebraic property. Our main result is a theorem describing such parameters.     

Leray–Hopf solutions to a viscoelastoplastic fluid model with nonsmooth stress–strain relation

We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba–Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.     

Upper bound of Kähler angles on the β-symplectic critical surfaces

Let (M,g) be a Kähler surface and \mathrm{\Sigma } be a \beta-symplectic critical surface in M. If L_q\left(\mathrm{\Sigma }\right) is bounded for some q>3, then we give a uniform upper bound for the Kähler angle on \mathrm{\Sigma }. This bound only depends on M,q,\beta and the L_q functional of \mathrm{\Sigma }. For q>4, this estimate is known and we extend the scope of q.     

New results for diffusion in Lorentz lattice gas cellular automata

New calculations to over ten million time steps have revealed a more complex diffusive behavior than previously reported of a point particle on a square and triangular lattice randomly occupied by mirror or rotator scatterers. For the square lattice fully occupied by mirrors where extended closed particle orbits occur, anomalous diffusion was still found. However, for a not fully occupied lattice the superdiffusion, first noticed by Owczarek and Prellberg for a particular concentration, obtains for all concentrations. For the square lattice occupied by rotators and the triangular lattice occupied by mirrors or rotators, an absence of diffusion (trapping) was found for all concentrations, except on critical lines, where anomalous diffusion (extended closed orbits) occurs and hyperscaling holds for all closed orbits withuniversal exponentsd _f =7/4 and =15/7. Only one point on these critical lines can be related to a corresponding percolation problem. The questions arise therefore whether the other critical points can be mapped onto a new percolation-like problem and of the dynamical significance of hyperscaling.     

Critical exponents,hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2 ₄ – as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2 ₄ –. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R _g ² >/<R _e ² >=0.14026±0.00007, <R _m ² >/<R _e ² >=0.43961±0.00034, and ^*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponent ₁=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for ₁. Earlier Monte Carlo estimates ofv, which were 0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R _g ² >/<R _e ² >=0.1599±0.0002 and ^*=0.2471±0.0003; since ^*>0, hyperscaling holds. The approach to ^* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2 ₄ – for two-dimensional SAWs.     

Schrödinger invariance and strongly anisotropic critical systems

The extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated. For the special case of an anisotropy or dynamical exponent =z=2, the group of local scale transformation considered is the Schrödinger group, which can be obtained as the nonrelativistic limit of the conformal group. The requirement of Schrödinger invariance determines the two-point function in the bulk and reduces the three-point function to a scaling form of a single variable. Scaling forms are also derived for the two-point function close to a free surface which can be either spacelike or timelike. These results are reproduced in several exactly solvable statistical systems, namely the kinetic Ising model with Glauber dynamics, lattice diffusion, Lifshitz points in the spherical model, and critical dynamics of the spherical model with a nonconserved order parameter. For generic values of , evidence from higher-order Lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two-point function.     

Subtleties in data analysis related to the size of critical region

We comment on the analysis of the critical behavior of a layered driven diffusive system recently done by Achahbar and Marro. We discuss why we believe their method of taking the thermodynamic limit and determining the order-parameter exponent leads to unreliable estimates.     

Thermoreversible gelation of polyacrylonitrile/dimethylformamide-solution

The thermoreversible gelation of semi-diluted atactic polyacrylnitrile (PAN)/dimethylformamide (DMF)-solutions has been studied. The structural features of PAN/DMF-gels, formed by supercooling have been investigated by DSC-, x-ray- and swelling measurements. A new structural model has been introduced to describe the morphology of the junction zones of PAN/DMF-gels and to also explain the gelation behavior of PAN/DMF-solutions as the structural features of PAN/DMF-gels. The junction zones of a PAN/DMF-gel have been defined as ordered junction zones.A gelation enthalpy of about H=–6 kJ/mol supports the idea that an ordered junction zone is formed by intermolecularly neighboring stereoregular parts of atactic PAN chains due to a nucleation process in the solution. It can be defined as a strongly distributed fringed micelle.     

The theoretical basis of column chromatography in multicomponent separations. Part 2: Peak capacity. Column systems with good selectivity

The selectivity of a column system, S, defined by equation 28, includes a variation coefficient (β) of plate number with capacity ratio which has a large influence on the peak capacity, as shown in Fig. 18. Some typical chromatograms are given. In order to predict S for a column system from Kovat's Index, equations 40 and 41 are given to calculate the constant of the carbon number rule for squalane at different temperatures or for different stationary phases. The specific retention value of heptane on squalane at different temperatures can be calculated from equation 42. The nonpolarity index, defined by equation 43, was used to calculate the retention value of heptane on various stationary phases. In liquid chromatography, the order of elution may be reversed by changing the composition of the eluent on the same chemically bonded silica (manufactured in China). The linear relations between the log retention values of different kinds of solutes or of a single solute on the silicas of different surface areas when using the same eluent are given.     

Modifying the thermosensitivity of copolymers of sodium styrene sulfonate and<Emphasis Type="Italic"> N</Emphasis>-isopropylacrylamide with dodecyltrimethylammonium chloride

The interactions between copolymers of sodium styrene sulfonate (SSS) and N-isopropylacrylamide (NIPAM), anionic polyelectrolytes, and dodecyltrimethylammonium chloride (DTAC), a cationic surfactant, were studied in aqueous solutions of various ionic strengths. The copolymers were found to be thermoresponsive, showing a lower critical solution temperature (LCST). The influence of the polymer composition, the surfactant concentration, and the ionic strength on the LCST was studied. The surfactant was found to interact strongly with the polymer, forming mixed polymer-surfactant micelles. The critical aggregation concentration (cac) of the polymer-surfactant system was found from fluorescence spectroscopy using pyrene as a fluorescent probe. A strong dependence of the anionic polyelectrolyte-cationic surfactant interactions on the structure of the ionic comonomer was observed.