Characterization of Nanofluids Using Multifractal Analysis of a Liquid Droplet Trace

This paper presents a novel approach to the analysis of nanofluids by using a nonlinear multifractal algorithm. Multifractal analysis allows to present detailed local descriptions of complex scaling behavior using a spectrum of singularity exponents. Nanoliquids prepared from nanoparticles of SiO2 (~0,01g) suspended in 100 ml of demineralized water and in 100 ml of 99,5% isopropanol were subjected to classical methods of analysis: determination of the contact angle, determination of the zeta potential, pH, and examination with a particle size analyzer. The obtained results show that the obtained nanofluid is stable and well prepared, while further nonlinear analyzes show that the usage of multifractal analysis for nanofluids can significantly improve both the process of analyzing this issue as well as its preparation, based on the multifractional spectrum.

Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

We consider Galerkin approximations of eigenvalue problems for holomorphic Fredholm operator functions for which the operators do not have the structure “coercive+compact”. In this case the regularity (in the vocabulary of discrete approximation schemes) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. The technique is based on the knowledge of a suitable Test function operator. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework can be successfully applied to analyze e.g. complex scaling/perfectly matched layer methods, problems involving sign-changing coefficients due to meta-materials and also (boundary element) approximations of Maxwell-type equations. We demonstrate the application of our framework to the Maxwell eigenvalue problem for a conductive material.

Transparent boundary conditions for discretized non-Hermitian eigenvalue problems

A method is presented for transparent, energy-dependent boundary conditions for open, non-Hermitian systems, and is illustrated on an example of Stark resonances in a single-particle quantum system. The approach provides an alternative to external complex scaling, and is applicable when asymptotic solutions can be characterized at large distances from the origin. Its main benefit consists in a drastic reduction of the dimesnionality of the underlying eigenvalue problem. Besides application to quantum mechanics, the method can be used in other contexts such as in systems involving unstable optical cavities and lossy waveguides.

Complex scaling: Physics of unbound light nuclei and perspective

The complex scaling method (CSM) is one of the most powerful methods of describing the resonances with complex energy eigenstates based on non-Hermitian quantum mechanics. We present the basic application of CSM to the properties of the unbound phenomena of light nuclei. In particular, we focus on many-body resonant and non-resonant continuum states observed in unstable nuclei. We also investigate the continuum level density (CLD) in the scattering problem in terms of the Green’s function with CSM. We discuss the explicit effects of resonant and non-resonant contributions in CLD and transition strength functions.