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Register a new userAn Analytical and Numerical Study of Optimal Channel Networks
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Added by
Alessandro Flammini
Publication date
1996
fields
Physics
and research's language is
English
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We analyze the Optimal Channel Network model for river networks using both analytical and numerical approaches. This is a lattice model in which a functional describing the dissipated energy is introduced and minimized in order to find the optimal configurations. The fractal character of river networks is reflected in the power law behaviour of various quantities characterising the morphology of the basin. In the context of a finite size scaling Ansatz, the exponents describing the power law behaviour are calculated exactly and show mean field behaviour, except for two limiting values of a parameter characterizing the dissipated energy, for which the system belongs to different universality classes. Two modifi
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The mutual information of a single-layer perceptron with $N$ Gaussian inputs and $P$ deterministic binary outputs is studied by numerical simulations. The relevant parameters of the problem are the ratio between the number of output and input units, $alpha = P/N$, and those describing the two-point correlations between inputs. The main motivation of this work refers to the comparison between the replica computation of the mutual information and an analytical solution valid up to $alpha sim O(1)$. The most relevant results are: (1) the simulation supports the validity of the analytical prediction, and (2) it also verifies a previously proposed conjecture that the replica solution interpolates well between large and small values of $alpha$.
We present a numerical study based on Monte Carlo algorithm of the magnetic properties of a mixed Ising ferrimagnetic model on a cubic lattice where spins $sigma =pm 1/2$ and spins $S=0,pm 1$ are in alternating sites on the lattice. We carried out exact ground state calculations and employ a Monte Carlo simulation to obtain the finite-temperature phase diagram of the model. A compensation point appears when the next-nearest-neighbor interaction between the spins $sigma =pm 1/2$ exceeds a minimum value. We found a strong dependence of the compensation temperature with the interactions in the Hamiltonian, particulary the crystal field and the external field. An applied field can change the range of values of the compensation temperature from zero up to a maximum value that depends on the field.
There exists a variety of theories of the glass transition and many more numerical models. But because the models need built-in complexity to prevent crystallization, comparisons with theory can be difficult. We study the dynamics of a deeply supersaturated emph{monodisperse} four-dimensional (4D) hard-sphere fluid, which has no such complexity, but whose strong intrinsic geometrical frustration inhibits crystallization, even when deeply supersaturated. As an application, we compare its behavior to the mode-coupling theory (MCT) of glass formation. We find MCT to describe this system better than any other structural glass formers in lower dimensions. The reduction in dynamical heterogeneity in 4D suggested by a milder violation of the Stokes-Einstein relation could explain the agreement. These results are consistent with a mean-field scenario of the glass transition.
The exact formula for the average path length of Apollonian networks is found. With the help of recursion relations derived from the self-similar structure, we obtain the exact solution of average path length, $bar{d}_t$, for Apollonian networks. In contrast to the well-known numerical result $bar{d}_t propto (ln N_t)^{3/4}$ [Phys. Rev. Lett. textbf{94}, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmically as $bar{d}_t propto ln N_t$ in the infinite limit of network size $N_t$. The extensive numerical calculations completely agree with our closed-form solution.
Erosion by flowing water is one of the major forces shaping the surface of Earth. Studies in the last decade have shown, in particular, that the drainage region of rivers, where water is collected, exhibits scale invariant features characterized by exponents that are the same for rivers around the world. Here we show that from the data obtained by the MOLA altimeter of the Mars Global Surveyor one can perform the same analysis for mountain sides on Mars. We then show that in some regions fluid erosion might have played a role in the present martian landscape.
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