we derive the formulas of the maximal andminimal ranks of four real matrices $x_{1},x_{2},x_{3}$ and $x_{4}$in common solution $x=x_{1}+x_{2}i+x_{3}j+x_{4}k$ to quaternionmatrix equations $a_{1}x=c_{1},xb_{2}=c_{2},a_{3}xb_{3}=c_{3}$. asapplications, we establish necessary and sufficient conditions forthe existence of the common real and complex solutions to the matrixequations. we give the exp...

Scale-free networks appear in many application domains such as social and biological networks [BA99, BB03, BO04]. Roughly speaking, scale-free networks have power-law degree distribution, ultra-short average path length and high clustering coefficient [BA99, BB03, BO04]. This paper presents new methods for visualising scale-free networks in three dimensions. To make effective use of the third d...

We show how to fix the renormalization scale for hard-scattering exclusive processes such as deeply virtual meson electroproduction by applying the BLM prescription to the imaginary part of the scattering amplitude and employing a fixed-t dispersion relation to obtain the scale-fixed real part. In this way we resolve the ambiguity in BLM renormalization scale-setting for complex scattering ampl...

Much of the recent research into imputation methodology has focused on developing optimal procedures for a single variable or set of variables, where the patterns of missingness and underlying distributions follow standard distributions. In contrast, it is frequently necessary to impute for many variables from a single survey, with an even larger set of potential covariates and complex covarian...

Optical phase contrast has for the first time been observed on a nanometer scale, with a near-field microscope of scattering type that maps the complete optical field of amplitude and phase. Backed by quasielectrostatic theory, we demonstrate the significance and experimental accessibility of even complex optical constants on a subwavelength scale. Further, our method can separate the near-fiel...

MOTIVATION There has recently been a notable shift in biomedical information extraction (IE) from relation models toward the more expressive event model, facilitated by the maturation of basic tools for biomedical text analysis and the availability of manually annotated resources. The event model allows detailed representation of complex natural language statements and can support a number of a...

After a long and turbulent history, the study of human cultural evolution is fi nally becoming comparable to the study of genetic evolution, with human history the counterpart of the biological fossil record. One of the most remarkable products of cultural evolution has been an increase in the scale of human societies by many orders of magnitude. Today, the great majority of humans live in comp...

Aberration correction in the Scanning Transmission Electron Microcope (STEM) is pushing the achievable spatial resolution for imaging and spectroscopy into the sub-angstrom regime, providing a new level of insight into the structure/property relations of complex materials. The combination of atomic-resolution Z-contrast scanning transmission electron microscopy and electron energy loss spectros...

A characteristic feature of complex systems is their deep structure, meaning that the definition of their states and observables depends on the level, or the scale, at which the system is considered. This scale dependence is reflected in the distinction of microand macro-states, referring to lower and higher levels of description. There are several conceptual and formal frameworks to address th...