Statistical Self-Similar Properties of Complex Networks

It has been shown that many complex networks shared distinctive features, which differ in many ways from the random and the regular networks. Although these features capture important characteristics of complex networks, their applicability depends on the type of networks. To unravel ubiquitous characteristics that complex networks may have in common, we adopt the clustering coefficient as the probability measure, and present a systematic analysis of various types of complex networks from the perspective of statistical self-similarity. We find that the probability distribution of the clustering coefficient is best characterized by the multifractal; moreover, the support of the measure had a fractal dimension. These two features enable us to describe complex networks in a unified way; at the same time, offer unforeseen possibilities to comprehend complex networks.Comment: 11 pages, 4 figure

A non-hyponormal operator generating Stieltjes moment sequences

A linear operator $S$ in a complex Hilbert space \hh for which the set \dzn{S} of its $C^\infty$-vectors is dense in \hh and $\{\|S^n f\|^2\}_{n=0}^\infty$ is a Stieltjes moment sequence for every f \in \dzn{S} is said to generate Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal operator $S$ which generates Stieltjes moment sequences. What is more, \dzn{S} is a core of any power $S^n$ of $S$. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of indeterminate Stieltjes moment sequences. As a consequence, it is shown that there exists a non-hyponormal composition operator in an $L^2$-space (over a $\sigma$-finite measure space) which is injective, paranormal and which generates Stieltjes moment sequences. In contrast to the case of abstract Hilbert space operators, composition operators which are formally normal and which generate Stieltjes moment sequences are always subnormal (in fact normal). The independence assertion of Barry Simon's theorem which parameterizes von Neumann extensions of a closed real symmetric operator with deficiency indices $(1,1)$ is shown to be false

Stability boundaries of roll and square convection in binary fluid mixtures with positive separation ratio

Rayleigh-B\'{e}nard convection in horizontal layers of binary fluid mixtures heated from below with realistic horizontal boundary conditions is studied theoretically using multi-mode Galerkin expansions. For positive separation ratios the main difference between the mixtures and pure fluids lies in the existence of stable three dimensional patterns near onset in a wide range of the parameter space. We evaluated the stationary solutions of roll, crossroll, and square convection and we determined the location of the stability boundaries for many parameter combinations thereby obtaining the Busse balloon for roll and square patterns.Comment: 19 pages + 15 figures, accepted by Journal of Fluid Mechanic

Track clustering with a quantum annealer for primary vertex reconstruction at hadron colliders

Clustering of charged particle tracks along the beam axis is the first step in reconstructing the positions of hadronic interactions, also known as primary vertices, at hadron collider experiments. We use a 2036 qubit D-Wave quantum annealer to perform track clustering in a limited capacity on artificial events where the positions of primary vertices and tracks resemble those measured by the Compact Muon Solenoid experiment at the Large Hadron Collider. The algorithm, which is not a classical-quantum hybrid but relies entirely on quantum annealing, is tested on a variety of event topologies from 2 primary vertices and 10 tracks up to 5 primary vertices and 15 tracks. It is benchmarked against simulated annealing executed on a commercial CPU constrained to the same processor time per anneal as time in the physical annealer, and performance is found to be comparable for small numbers of vertices with an intriguing advantage noted for 2 vertices and 16 tracks