On the existence of flat orthogonal matrices

In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $\frac{1}{\sqrt{n}}$. Entries of $\pm \frac{1}{\sqrt{n}}$ correspond to Hadamard matrices, so the question of existence of flat orthogonal matrices can be viewed as a relaxation of the Hadamard problem.Comment: 10 page

Intrinsic aging and effective viscosity in the slow dynamics of a soft glass with tunable elasticity

We investigate by rheology and light scattering the influence of the elastic modulus, $G_0$, on the slow dynamics and the aging of a soft glass. We show that the slow dynamics and the aging can be entirely described by the evolution of an effective viscosity, $\eta_{eff}$, defined as the characteristic time measured in a stress relaxation experiment times $G_0$. At all time, $\eta_{eff}$ is found to be independent of $G_0$, of elastic perturbations, and of the rate at which the sample is quenched in the glassy phase. We propose a simple model that links $\eta_{eff}$ to the internal stress built up at the fluid-to-solid transition

Enumeration of three term arithmetic progressions in fixed density sets

Additive combinatorics is built around the famous theorem by Szemer\'edi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemer\'edi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a generalization of Szemer\'edi's theorem using methods from real algebraic geometry.Comment: 62 pages. Update v2: Corrected some references. Update v3: Incorporated feedbac

Microscopic Model for High-spin vs. Low-spin ground state in [Ni2M(CN)8][Ni_2{M(CN)_8]} (M=MoV,WV,NbIVM=Mo^V, W^V, Nb^{IV}) magnetic clusters

Conventional superexchange rules predict ferromagnetic exchange interaction between Ni(II) and M (M=Mo(V), W(V), Nb(IV)). Recent experiments show that in some systems this superexchange is antiferromagnetic. To understand this feature, in this paper we develop a microscopic model for Ni(II)-M systems and solve it exactly using a valence bond approach. We identify the direct exchange coupling, the splitting of the magnetic orbitals and the inter-orbital electron repulsions, on the M site as the parameters which control the ground state spin of various clusters of the Ni(II)-M system. We present quantum phase diagrams which delineate the high-spin and low-spin ground states in the parameter space. We fit the spin gap to a spin Hamiltonian and extract the effective exchange constant within the experimentally observed range, for reasonable parameter values. We also find a region in the parameter space where an intermediate spin state is the ground state. These results indicate that the spin spectrum of the microscopic model cannot be reproduced by a simple Heisenberg exchange Hamiltonian.Comment: 8 pages including 7 figure

Dispersions of ellipsoidal particles in a nematic liquid crystal

Colloidal particles dispersed in a partially ordered medium, such as a liquid crystal (LC) phase, disturb its alignment and are subject to elastic forces. These forces are long-ranged, anisotropic and tunable through temperature or external fields, making them a valuable asset to control colloidal assembly. The latter is very sensitive to the particle geometry since it alters the interactions between the colloids. We here present a detailed numerical analysis of the energetics of elongated objects, namely prolate ellipsoids, immersed in a nematic host. The results, complemented with qualitative experiments, reveal novel LC configurations with peculiar topological properties around the ellipsoids, depending on their aspect ratio and the boundary conditions imposed on the nematic order parameter. The latter also determine the preferred orientation of ellipsoids in the nematic field, because of elastic torques, as well as the morphology of particles aggregates.Comment: 31 pages, 11 figure