11 research outputs found
Pseudofractal Scale-free Web
We find that scale-free random networks are excellently modeled by a
deterministic graph. This graph has a discrete degree distribution (degree is
the number of connections of a vertex) which is characterized by a power-law
with exponent . Properties of this simple structure are
surprisingly close to those of growing random scale-free networks with
in the most interesting region, between 2 and 3. We succeed to find exactly and
numerically with high precision all main characteristics of the graph. In
particular, we obtain the exact shortest-path-length distribution. For the
large network () the distribution tends to a Gaussian of width
centered at . We show that the
eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail
with exponent .Comment: 5 pages, 3 figure
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
Non Linear Current Response of a Many-Level Tunneling System: Higher Harmonics Generation
The fully nonlinear response of a many-level tunneling system to a strong
alternating field of high frequency is studied in terms of the
Schwinger-Keldysh nonequilibrium Green functions. The nonlinear time dependent
tunneling current is calculated exactly and its resonance structure is
elucidated. In particular, it is shown that under certain reasonable conditions
on the physical parameters, the Fourier component is sharply peaked at
, where is the spacing between
two levels. This frequency multiplication results from the highly nonlinear
process of photon absorption (or emission) by the tunneling system. It is
also conjectured that this effect (which so far is studied mainly in the
context of nonlinear optics) might be experimentally feasible.Comment: 28 pages, LaTex, 7 figures are available upon request from
[email protected], submitted to Phys.Rev.