21,492 research outputs found
Some Classes of third and Fourth-order iterative methods for solving nonlinear equations
The object of the present work is to present the new classes of third-order
and fourth-order iterative methods for solving nonlinear equations. Our
third-order method includes methods of Weerakoon \cite{Weerakoon}, Homeier
\cite{Homeier2}, Chun \cite{Chun} e.t.c. as particular cases. After that we
make this third-order method to fourth-order (optimal) by using a single weight
function rather than using two different weight functions in \cite{Soleymani}.
Finally some examples are given to illustrate the performance of the our method
by comparing with new existing third and fourth-order methods.Comment: arXiv admin note: substantial text overlap with arXiv:1307.733
Real-time Exponential Curve Fits Using Discrete Calculus
This paper presents an improved solution for curve fitting data to an exponential equation (Y = AeBt + C). This improvement is in four areas ? speed, stability, determinant processing time, and the removal of limits. The solution presented in this paper avoids iterative techniques and their stability errors by using three mathematical ideas ? discrete calculus, a special relationship (between exponential curves and the Mean Value Theorem for Derivatives), and a simple linear curve fit algorithm. This method can also be applied to fitting data to the general power law equation Y = AxB + C and the general geometric growth equation Y = AkBt + C
Nonlinear force-free modeling of the solar coronal magnetic field
The coronal magnetic field is an important quantity because the magnetic
field dominates the structure of the solar corona. Unfortunately direct
measurements of coronal magnetic fields are usually not available. The
photospheric magnetic field is measured routinely with vector magnetographs.
These photospheric measurements are extrapolated into the solar corona. The
extrapolated coronal magnetic field depends on assumptions regarding the
coronal plasma, e.g. force-freeness. Force-free means that all non-magnetic
forces like pressure gradients and gravity are neglected. This approach is well
justified in the solar corona due to the low plasma beta. One has to take care,
however, about ambiguities, noise and non-magnetic forces in the photosphere,
where the magnetic field vector is measured. Here we review different numerical
methods for a nonlinear force-free coronal magnetic field extrapolation:
Grad-Rubin codes, upward integration method, MHD-relaxation, optimization and
the boundary element approach. We briefly discuss the main features of the
different methods and concentrate mainly on recently developed new codes.Comment: 33 pages, 3 figures, Review articl
How single neuron properties shape chaotic dynamics and signal transmission in random neural networks
While most models of randomly connected networks assume nodes with simple
dynamics, nodes in realistic highly connected networks, such as neurons in the
brain, exhibit intrinsic dynamics over multiple timescales. We analyze how the
dynamical properties of nodes (such as single neurons) and recurrent
connections interact to shape the effective dynamics in large randomly
connected networks. A novel dynamical mean-field theory for strongly connected
networks of multi-dimensional rate units shows that the power spectrum of the
network activity in the chaotic phase emerges from a nonlinear sharpening of
the frequency response function of single units. For the case of
two-dimensional rate units with strong adaptation, we find that the network
exhibits a state of "resonant chaos", characterized by robust, narrow-band
stochastic oscillations. The coherence of stochastic oscillations is maximal at
the onset of chaos and their correlation time scales with the adaptation
timescale of single units. Surprisingly, the resonance frequency can be
predicted from the properties of isolated units, even in the presence of
heterogeneity in the adaptation parameters. In the presence of these
internally-generated chaotic fluctuations, the transmission of weak,
low-frequency signals is strongly enhanced by adaptation, whereas signal
transmission is not influenced by adaptation in the non-chaotic regime. Our
theoretical framework can be applied to other mechanisms at the level of single
nodes, such as synaptic filtering, refractoriness or spike synchronization.
These results advance our understanding of the interaction between the dynamics
of single units and recurrent connectivity, which is a fundamental step toward
the description of biologically realistic network models in the brain, or, more
generally, networks of other physical or man-made complex dynamical units
A Grassmann integral equation
The present study introduces and investigates a new type of equation which is
called Grassmann integral equation in analogy to integral equations studied in
real analysis. A Grassmann integral equation is an equation which involves
Grassmann integrations and which is to be obeyed by an unknown function over a
(finite-dimensional) Grassmann algebra G_m. A particular type of Grassmann
integral equations is explicitly studied for certain low-dimensional Grassmann
algebras. The choice of the equation under investigation is motivated by the
effective action formalism of (lattice) quantum field theory. In a very general
setting, for the Grassmann algebras G_2n, n = 2,3,4, the finite-dimensional
analogues of the generating functionals of the Green functions are worked out
explicitly by solving a coupled system of nonlinear matrix equations. Finally,
by imposing the condition G[{\bar\Psi},{\Psi}] = G_0[{\lambda\bar\Psi},
{\lambda\Psi}] + const., 0<\lambda\in R (\bar\Psi_k, \Psi_k, k=1,...,n, are the
generators of the Grassmann algebra G_2n), between the finite-dimensional
analogues G_0 and G of the (``classical'') action and effective action
functionals, respectively, a special Grassmann integral equation is being
established and solved which also is equivalent to a coupled system of
nonlinear matrix equations. If \lambda \not= 1, solutions to this Grassmann
integral equation exist for n=2 (and consequently, also for any even value of
n, specifically, for n=4) but not for n=3. If \lambda=1, the considered
Grassmann integral equation has always a solution which corresponds to a
Gaussian integral, but remarkably in the case n=4 a further solution is found
which corresponds to a non-Gaussian integral. The investigation sheds light on
the structures to be met for Grassmann algebras G_2n with arbitrarily chosen n.Comment: 58 pages LaTeX (v2: mainly, minor updates and corrections to the
reference section; v3: references [4], [17]-[21], [39], [46], [49]-[54],
[61], [64], [139] added
- …