Average density of states of amorphous Hamiltonians: role of phonon mediated coupling of nano-clusters

Based on a description of an amorphous solid as a collection of coupled nanosize molecular clusters referred as basic blocks, we analyse the statistical properties of its Hamiltonian. The information is then used to derive the ensemble averaged density of the vibrational states (nonphonon) which turns out to be a Gaussian in the bulk of the spectrum and an Airy function in the low frequency regime. A comparison with experimental data for five glasses confirms validity of our theoretical predictions.

Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius.

Correlation Functions for a Chain of Short Range Oscillators

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate $$t^{-\frac{1}{3}}$$ t - 1 3 for position and momentum correlations and as $$t^{-\frac{2}{3}}$$ t - 2 3 for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate $$t^{-\frac{1}{4}}$$ t - 1 4 for position and momentum correlators and with rate $$t^{-\frac{1}{2}}$$ t - 1 2 for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly.

The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz.

The large-N partition function for non-parity-invariant Chern-Simons-matter theories

We extend the Fermi gas approach to a class of ABJM-like necklace quiver theories without parity invariance. The resulting partition function on S3 retains the form of an Airy function, but now includes a phase that scales as Nk in the large-N limit where k is an overall Chern-Simons level. We demonstrate the presence of this phase both analytically and numerically in the case of a three node quiver.

Implementation of Airy function using Graphics Processing Unit (GPU)

Special mathematical functions are an integral part of Fractional Calculus, one of them is the Airy function. But it’s a gruelling task for the processor as well as system that is constructed around the function when it comes to evaluating the special mathematical functions on an ordinary Central Processing Unit (CPU). The Parallel processing capabilities of a Graphics processing Unit (GPU) hence is used. In this paper GPU is used to get a speedup in time required, with respect to CPU time for evaluating the Airy function on its real domain. The objective of this paper is to provide a platform for computing the special functions which will accelerate the time required for obtaining the result and thus comparing the performance of numerical solution of Airy function using CPU and GPU.