Regularity for solutions of quasilinear elliptic equations under minimal assumptions

We review some recent results in the regularity theory for elliptic second order P.D.E. obtained under assumptions which can be shown in some instances to be necessary.     

Magnetic monopoles in SU(N) gauge theories

The potential $\text{A}\text{(}\text{r}\text{) ≡}\text{M}\text{(}\text{r}\text{?}\text{×}\text{n}\text{?}\text{)(r?}\text{r}\text{·}\text{n}\text{?}\text{)}{}^{\mathrm{?1}}$ is a static solution to the classical theory of non-abelian gauge fields coupled to a point magnetic source, for any matrix $\text{M}$ in the Lie algebra of the gauge group G. This solution is rotationally invariant if the eigenvalues of $\text{M}$ in the adjoint representation of G are quantized in half-integer units, but is stable to small perturbations only if all non-vanishing eigenvalues are $\text{±}\text{1}\text{2}$. In this paper, for the gauge groups G = SU(N), it is shown which sets of eigenvalues of $\text{M}$ are consistent with the group structure, which consistent sets are gauge inequivalent, and which consistent gauge inequivalent sets correspond to stable monopoles. It is found that there are N inequivalent stable monopoles, including the trivial case $\text{M}\text{=}\text{0}$. Equivalence here is with respect to non-singular gauge transformations—the symmetry transformations of the classical theory. Singular gauge transformations are, in contrast, not symmetries but they are nevertheless useful for classifying solutions and for relating the above concept of local stability to the global, or topological, stability associated with the Dirac strings. In this context, it is shown that there are N distinct topological classes of monopoles, with the group structure of the center $\text{Z}{}_{\mathrm{N}}\text{?Π}{}_1\text{(}\text{SU}\text{(N)/}\text{Z}{}_{\mathrm{N}}\text{)}$ of SU(N), that each class contains exactly one stable monopole, and that any other monopole in the same class has a strictly larger value of the magnetic charge magnitude $\text{tr}\text{M}{}^2$. This leads to an interesting physical picture of local stability as a consequence of the minimization of magnetic energy. The paper concludes with some comments on related topics: the empirical absence of magnetic charge, `t Hooft's calculation of magnetic energy, magnetic confinement, and spontaneously broken theories.     

Weak coupling analysis of the finite-temperature twisted Eguchi-Kawai model

We show that with a suitable twist the finite-temperature twisted Eguchi-Kawai model reproduces the planar perturbation expansion of the full Wilson theory in the weak coupling sector.     

Phase Transition between Synchronous and Asynchronous Updating Algorithms

We update a one-dimensional chain of Ising spins of length L with algorithms which are parameterized by the probability p for a certain site to get updated in one time step. The result of the update event itself is determined by the energy change due to the local change in the configuration. In this way we interpolate between the Metropolis algorithm at zero temperature when p is of the order of 1/L and L is large, and a synchronous deterministic updating procedure for p=1. As a function of p we observe a phase transition between the stationary states to which the algorithm drives the system. These are non-absorbing stationary states with antiferromagnetic domains for p>p _c , and absorbing states with ferromagnetic domains for p≤p _c . This means that above this transition the stationary states have lost any remnants of the ferromagnetic Ising interaction. A measurement of the critical exponents shows that this transition belongs to the universality class of parity conservation.     

Sensitivity analysis of full field methods for residual stress measurement

The hole drilling technique is a well known experimental method for residual stress investigation. This technique is usually used in combination with electrical strain gauges but there is no reason to enforce this choice and other approaches, in particular some full-field optical techniques, can be advantageously used. Since all these techniques give full field data, it becomes important to properly use this redundant information content to increase the robustness and reliability of the analysis.In this work, various well known approaches to the hole drilling/full-field data analysis will be investigated using a two-step approach. In the first one, a sensitivity analysis will be performed on the simpler algorithms and then the reliability of the methods will be estimated by Montecarlo analysis using a known displacement field as a reference.     

Modelling the Interaction between Carboxylic Acids and Zinc Oxide: Insight into Degradation of ZnO Pigments

Computational modelling applied to cultural heritage can assist the characterization of painting materials and help to understand their intrinsic and external degradation processes. The degradation of the widely employed zinc oxide (ZnO)—a white pigment mostly used in oil paints—leads to the formation of metal soaps, complexes of Zn ions and long-chain fatty acids coming from the degradation of the oil binder. Being a serious problem affecting the appearance and the structural integrity of many oil paintings, it is relevant to characterize the structure of these complexes and to understand the reaction pathways associated with this degradation process. Density functional theory (DFT) calculations were performed to investigate the adsorption of the acetate and acetic acid on relatively large ZnO clusters and the formation of Zn–acetate complexes. Carboxylic acids with longer alkyl chains were then investigated as more realistic models of the fatty acids present in the oil medium. In addition, DFT calculations using a periodic ZnO slab were performed in order to compare the obtained results at different levels of theory. Optimization calculations as well as the formation energies of the ZnO@carboxylate coupled systems and the thermodynamics leading to possible degradation products were computed. Our results highlight the potential for DFT calculations to provide a better understanding of oil paint degradation, with the aim of contributing to the development of strengthening and conservation strategies of paintings.     

Toda flows and real Hessenberg manifolds

We relate the Toda flow on the “p-part” of a semi-simple Lie algebra to the topology of real Hessenberg manifolds, and we obtain their mod2 Betti numbers by reversing Morse inequalities using a theorem of Floyd and a result on the Weyl group.     

Cyclohexenylphenyldiazene: a simple surrogate of the azobenzene photochromic unit

We have carried out an experimental and computational study on the ground- and excited-state photochemical and photophysical properties of (1-cyclohexenyl)phenyldiazene (CPD), a species formally derived from azobenzene in which one of the phenyl rings is replaced by a 1-cyclohexene substituent. The results show that CPD does substantially behave like azobenzene, but with a higher (approximately 70%) Phi(Z-->E) (npi*) photoisomerization quantum yield, calling for CPD as an effective alternative of azobenzene itself with new functionalization possibilities. By use of state-of-the-art ab initio CASPT2//CASSCF minimum energy path computations, we have identified the most efficient decay and isomerization routes of the absorbing singlet (pipi*), S1 (npi*), T1, and S0 states of CPD. The resulting mechanistic scheme agrees with experimental findings and provides a rationale of the observed photoisomerization quantum yields. Furthermore, this study provides a deep insight on the photophysical and photochemical properties of compounds based on the -N=N- double bond which supplies a general model for the photoreactivity of azobenzene-type compounds in general. This is expected to be a useful guideline for the design of novel photoreactive azo compounds.     

Controlled sound field with a dual layer loudspeaker array

Controlled sound interference has been extensively investigated using a prototype dual layer loudspeaker array comprised of 16 loudspeakers. Results are presented for measures of array performance such as input signal power, directivity of sound radiation and accuracy of sound reproduction resulting from the application of conventional control methods such as minimization of error in mean squared pressure, maximization of energy difference and minimization of weighted pressure error and energy. Procedures for selecting the tuning parameters have also been introduced. With these conventional concepts aimed at the production of acoustically bright and dark zones, all the control methods used require a trade-off between radiation directivity and reproduction accuracy in the bright zone. An alternative solution is proposed which can achieve better performance based on the measures presented simultaneously by inserting a low priority zone named as the “gray” zone. This involves the weighted minimization of mean-squared errors in both bright and dark zones together with the gray zone in which the minimization error is given less importance. This results in the production of directional bright zone in which the accuracy of sound reproduction is maintained with less required input power. The results of simulations and experiments are shown to be in excellent agreement.