Crossover Critical Behavior in the Three-Dimensional Ising Model

The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the three-dimensional Ising universality class. Recent numerical studies of Ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation-induced critical behavior and mean-field-type critical behavior. In this work, we compare these numerical results with a crossover Landau model based on renormalization-group matching. For this purpose we consider an application of the crossover Landau model to the three-dimensional Ising model without fitting to any adjustable parameters. The crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range (ten orders) of the scaled distance to the critical temperature. The dependence of the coupling constant on the interaction range, governing the crossover critical behavior, is discussed.     

Hydrophilic segmented block copolymers based on poly(ethylene oxide) and monodisperse amide segments

Segmented block copolymers based on poly(ethylene oxide) (PEO) flexible segments and monodisperse crystallizable bisester tetra‐amide segments were made via a polycondensation reaction. The molecular weight of the PEO segments varied from 600 to 4600 g/mol and a bisester tetra‐amide segment (T6T6T) based on dimethyl terephthalate (T) and hexamethylenediamine (6) was used. The resulting copolymers were melt‐processable and transparent. The crystallinity of the copolymers was investigated by differential scanning calorimetry (DSC) and Fourier Transform infrared (FTIR). The thermal properties were studied by DSC, temperature modulated synchrotron small angle X‐ray scattering (SAXS), and dynamic mechanical analysis (DMA). The elastic properties were evaluated by compression set (CS) test. The crystallinity of the T6T6T segments in the copolymers was high (>84%) and the crystallization fast due to the use of monodisperse tetra‐amide segments. DMA experiments showed that the materials had a low T_g, a broad and almost temperature independent rubbery plateau and a sharp flow temperature. With increasing PEO length both the PEO melting temperature and the PEO crystallinity increased. When the PEO segment length was longer than 2000 g/mol the PEO melting temperature was above room temperature and this resulted in a higher modulus and in higher compression set values at room temperature. The properties of PEO‐T6T6T copolymers were compared with similar poly(propylene oxide) and poly(tetramethylene oxide) copolymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 4522–4535, 2007     

Über eine Abbildung der Ebene auf eine gewisse Kummer'sche Fläche

Ohne Zusammenfassung     

Fluid-fluid phase separation in colloid-polymer mixtures studied with small angle light scattering and light microscopy

Mixtures of colloidal silica spheres and polydimethylsiloxane in cyclohexane with a colloid-polymer size ratio of about one were found to phase separate into two fluid phases, one which is colloid-rich and one which is colloid-poor. In this work the phase separation kinetics of this fluid-fluid phase separation is studied for different compositions of the colloid-polymer mixtures, and at several degrees of supersaturation, with small angle light scattering and with light microscopy. The small angle light scattering curve exhibits a peak that grows in intensity and that shifts to smaller wave vector with time. The characteristic length scale that is obtained from the scattering peak is of the order of a few μm, in agreement with observations by light microscopy. The domain size increases with time as , which might be an indication of coarsening by diffusion and coalescence, like in the case of binary liquid mixtures and polymer blends. For sufficiently low degrees of supersaturation the angular scattering intensity curves satisfy dynamical scaling behavior.     

An exponential lower bound to the size of bounded depth frege proofs of the pigeonhole principle

We prove lower bounds of the form exp(n^{ε d}), ε_d > 0, on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d, for any constant d. This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.     

The Module Structure Theorem for Sobolev Spaces on Open Manifolds

We prove a general embedding theorem for Sobolev spaces on open manifolds of bounded geometry and infer from this the module structure theorem. Thereafter we apply this to weighted Sobolev spaces.     

Interpolation by a Game

We introduce a notion of a real game (a generalisation of the Karchmer-Wigderson game (cf. [3]) and of real communication complexity, and relate this complexity to the size of monotone real formulas and circuits. We give an exponential lower bound for tree-like monotone protocols (defined in [4, Definition 2.2]) of small real communication complexity solving the monotone communication complexity problem associated with the bipartite perfect matching problem. This work is motivated by a research in interpolation theorems for prepositional logic (by a problem posed in [5, Section 8], in particular). Our main objective is to extend the communication complexity approach of [4, 5] to a wider class of proof systems. In this direction we obtain an effective interpolation in a form of a protocol of small real communication complexity. Together with the above mentioned lower bound for tree-like protocols this yields as a corollary a lower bound on the number of steps for particular semantic derivations of Hall's theorem (these include tree-like cutting planes proofs for which an exponential lower bound was demonstrated in [2]).