Sylvester–Gallai Theorems for Complex Numbers and Quaternions

A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a deep inequality of Hirzebruch. We give an elementary proof of this result, and then extend it to show that an SG configuration in projective space over the quaternions must be contained in a three-dimensional flat.     

A Coaccelerated Observer

We analyze the situation of an observer coaccelerated relative to a linearly accelerated charge, in order to find whether he can observe the radiation emitted from the accelerated charge. It is found that the seemingly special situation of the coaccelerated observer relative to any other observer, is deduced from a wrong use of the retarded coordinate system, when such a system is inadmissible. It is also found that the coaccelerated observer has no special position other than any other observer, and hence, he can observe any physical events as any other observer.     

An analytical approximation for the number of states along the reaction coordinate

A variant of a new empirical method, enables one to express a collinear triatomic potential energy surface as a family of Morse curves along “natural” bond order coordinates orthogonal to the reaction coordinate. The procedure depends on a single adjustable parameter which is related to the barrier's height. Because an analytical expression for the number of vibrational states of a Morse oscillator is available, one has an analytical approximation for the number of states along the reaction coordinate. The extrema in the number of states are utilized in various versions of classical microcanonical variational transition state theory (among which is a new version, which is in better agreement with dynamical results), to estimate the probability of a collinear reactions, as a function of the total energy. The analytical expressions are also used to analyze the origins of the maximum and minima in the number of states.     

Fourth-order magnetic anisotropy and tunnel splittings in Mn(12) from spin-orbit-vibron interactions

From density-functional-theory based methods, we calculate the vibrational spectrum of the Mn(12)O(12)(COOH)(16)(H(2)O)(4) molecular magnet. Calculated infrared intensities are in accord with experimental studies. There have been no ab initio attempts at determining which interactions account for the fourth-order anisotropy. We show that vibrationally induced distortions of the molecule contribute to the fourth-order anisotropy Hamiltonian and that the magnitude and sign of the effect (-6.2 K) is in good agreement with all experiments. Vibrationally induced tunnel splittings in isotopically pure and natural samples are predicted.     

Simple physical model for chemomechanics

We present a simple model linking chemical interactions with macroscopic mechanical deformations for polymer‐coated microcantilever devices. The chemo‐mechanical process is treated in terms of Fickian diffusion followed by a structural relaxation of the polymer‐guest complex. The model is tested on a series of microcantilever bending experiments, in the presence of different isotopologues, and at different incident guest fluxes. In many cases, a nontrivial behavior, expressed by the appearance of an overshoot, is observed. The new model provides insight into the ways chemical interactions and reorganization processes manifest themselves in mechanical processes. © 2013 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys., 2013     

Absorption spectra and aqueous photochemistry of β-hydroxyalkyl nitrates of atmospheric interest

Molar absorption coefficients were measured for select alkyl nitrates and β-hydroxyalkyl nitrates in methanol. The presence of the β-hydroxyl group has a relatively minor effect on the absorption spectrum in the vicinity of the weak n → π* transition, which is responsible for photolysis of organic nitrates in the atmosphere. For both alkyl nitrates and β-hydroxyalkyl nitrates, there is an enhancement in the absorption coefficients in solution compared to the gas-phase values. The effect of the β-hydroxyl group on the spectra was modelled with molecular dynamics simulations using an OM2/GUGA-CI Hamiltonian for ethyl nitrate and β-hydroxyethyl nitrate. The simulation provided a qualitatively correct shape of the low energy tail of the absorption spectrum, which is important for atmospheric photochemistry. The role of direct aqueous photolysis in removal of β-hydroxyalkyl nitrates in cloud and fog water was modelled using a relative rate approach, and shown to be insignificant relative to gas-phase photochemical processes and aqueous OH oxidation under typical atmospheric conditions.     

Probabilistic Approach to an Image Reconstruction Problem

This aim of this note is to present a probabilistic approach to an algebraic problem in image reconstruction. The role of probability theory is to identify the relevant variables that characterize the solutions, and to provide alternative variational methods of reconstruction.     

Understanding and controlling the structure of polypropylene/layered silicate nanocomposites

The miscibility and structure in polypropylene/layered silicate nanocomposites is systematically investigated utilizing a maleic-anhydride grafted polypropylene with a low degree of functionalization acting as the compatibilizer. The morphology of the hybrids can be modified from phase separated to almost completely exfoliated in a controlled way by varying the ratio α of the compatibilizer to the organophilized clay; this ratio α is found to be the most important parameter in determining the final structure whereas exfoliated structures can be obtained for α values of 9 or higher. Furthermore, utilization of a “masterbatch” procedure can enhance the degree of exfoliation even for smaller values of α; in that case, polypropylene is essentially mixed with the already dispersed “hairy” platelets. Investigation of the thermal stability of the micro- and nanocomposites shows that high degree of exfoliation is vital in increasing the temperature that the polymer starts to degrade. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 2683–2695, 2008     

Concentration profiles near an activated enzyme

When a resting enzyme is activated, substrate concentration profile evolves in its vicinity, ultimately tending to steady state. We use modern theories for many-body effects on diffusion-influenced reactions to derive approximate analytical expressions for the steady-state profile and the Laplace transform of the transient concentration profiles. These show excellent agreement with accurate many-particle Brownian-dynamics simulations for the Michaelis-Menten kinetics. The steady-state profile has a hyperbolic dependence on the distance of the substrate from the enzyme, albeit with a prefactor containing the complexity of the many-body effects. These are most conspicuous for the substrate concentration at the surface of the enzyme. It shows an interesting transition as a function of the enzyme turnover rate. When it is high, the contact concentration decays monotonically to steady state. However, for slow turnover it is nonmonotonic, showing a minimum due to reversible substrate binding, then a maximum due to diffusion of new substrate toward the enzyme, and finally decay to steady state. Under certain conditions one can obtain a good estimate for the critical value of the turnover rate constant at the transition.     

On the sum of the L1 influences of bounded functions

Let f: {-1, 1}ⁿ → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L₁ influence of f can also be bounded as a function of d. Ba?kurs and Bavarian answered this question in the affirmative, providing a bound of O(d³) for general functions and O(d²) for homogeneous functions. We improve on their results by providing a bound of d² for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.