1,526 research outputs found
Control and Synchronization of Neuron Ensembles
Synchronization of oscillations is a phenomenon prevalent in natural, social,
and engineering systems. Controlling synchronization of oscillating systems is
motivated by a wide range of applications from neurological treatment of
Parkinson's disease to the design of neurocomputers. In this article, we study
the control of an ensemble of uncoupled neuron oscillators described by phase
models. We examine controllability of such a neuron ensemble for various phase
models and, furthermore, study the related optimal control problems. In
particular, by employing Pontryagin's maximum principle, we analytically derive
optimal controls for spiking single- and two-neuron systems, and analyze the
applicability of the latter to an ensemble system. Finally, we present a robust
computational method for optimal control of spiking neurons based on
pseudospectral approximations. The methodology developed here is universal to
the control of general nonlinear phase oscillators.Comment: 29 pages, 6 figure
Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation
We study nonlinear waves in Newton's cradle, a classical mechanical system
consisting of a chain of beads attached to linear pendula and interacting
nonlinearly via Hertz's contact forces. We formally derive a spatially discrete
modulation equation, for small amplitude nonlinear waves consisting of slow
modulations of time-periodic linear oscillations. The fully-nonlinear and
unilateral interactions between beads yield a nonstandard modulation equation
that we call the discrete p-Schroedinger (DpS) equation. It consists of a
spatial discretization of a generalized Schroedinger equation with p-Laplacian,
with fractional p>2 depending on the exponent of Hertz's contact force. We show
that the DpS equation admits explicit periodic travelling wave solutions, and
numerically find a plethora of standing wave solutions given by the orbits of a
discrete map, in particular spatially localized breather solutions. Using a
modified Lyapunov-Schmidt technique, we prove the existence of exact periodic
travelling waves in the chain of beads, close to the small amplitude modulated
waves given by the DpS equation. Using numerical simulations, we show that the
DpS equation captures several other important features of the dynamics in the
weakly nonlinear regime, namely modulational instabilities, the existence of
static and travelling breathers, and repulsive or attractive interactions of
these localized structures
Disturbing Implications of a Cosmological Constant
In this paper we consider the implications of a cosmological constant for the
evolution of the universe, under a set of assumptions motivated by the
holographic and horizon complementarity principles. We discuss the ``causal
patch" description of spacetime required by this framework, and present some
simple examples of cosmologies described this way. We argue that these
assumptions inevitably lead to very deep paradoxes, which seem to require major
revisions of our usual assumptions.Comment: 26 pages, 7 figures v2: references added v3: reference adde
Magnetic operations: a little fuzzy physics?
We examine the behaviour of charged particles in homogeneous, constant and/or
oscillating magnetic fields in the non-relativistic approximation. A special
role of the geometric center of the particle trajectory is elucidated. In
quantum case it becomes a 'fuzzy point' with non-commuting coordinates, an
element of non-commutative geometry which enters into the traditional control
problems. We show that its application extends beyond the usually considered
time independent magnetic fields of the quantum Hall effect. Some simple cases
of magnetic control by oscillating fields lead to the stability maps differing
from the traditional Strutt diagram.Comment: 28 pages, 8 figure
Integrable turbulence generated from modulational instability of cnoidal waves
We study numerically the nonlinear stage of modulational instability (MI) of
cnoidal waves, in the framework of the focusing one-dimensional Nonlinear
Schrodinger (NLS) equation. Cnoidal waves are the exact periodic solutions of
the NLS equation and can be represented as a lattice of overlapping solitons.
MI of these lattices lead to development of "integrable turbulence" [Zakharov
V.E., Stud. Appl. Math. 122, 219-234 (2009)]. We study the major
characteristics of the turbulence for dn-branch of cnoidal waves and
demonstrate how these characteristics depend on the degree of "overlapping"
between the solitons within the cnoidal wave.
Integrable turbulence, that develops from MI of dn-branch of cnoidal waves,
asymptotically approaches to it's stationary state in oscillatory way. During
this process kinetic and potential energies oscillate around their asymptotic
values. The amplitudes of these oscillations decay with time as t^{-a},
1<a<1.5, the phases contain nonlinear phase shift decaying as t^{-1/2}, and the
frequency of the oscillations is equal to the double maximal growth rate of the
MI, s=2g_{max}. In the asymptotic stationary state the ratio of potential to
kinetic energy is equal to -2. The asymptotic PDF of wave amplitudes is close
to Rayleigh distribution for cnoidal waves with strong overlapping, and is
significantly non-Rayleigh one for cnoidal waves with weak overlapping of
solitons. In the latter case the dynamics of the system reduces to two-soliton
collisions, which occur with exponentially small rate and provide up to
two-fold increase in amplitude compared with the original cnoidal wave.Comment: 36 pages, 25 figure
Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof
In this article, we provide a comprehensive historical survey of 183
different proofs of famous Euclid's theorem on the infinitude of prime numbers.
The author is trying to collect almost all the known proofs on infinitude of
primes, including some proofs that can be easily obtained as consequences of
some known problems or divisibility properties. Furthermore, here are listed
numerous elementary proofs of the infinitude of primes in different arithmetic
progressions.
All the references concerning the proofs of Euclid's theorem that use similar
methods and ideas are exposed subsequently. Namely, presented proofs are
divided into 8 subsections of Section 2 in dependence of the methods that are
used in them. {\bf Related new 14 proofs (2012-2017) are given in the last
subsection of Section 2.} In the next section, we survey mainly elementary
proofs of the infinitude of primes in different arithmetic progressions.
Presented proofs are special cases of Dirichlet's theorem. In Section 4, we
give a new simple "Euclidean's proof" of the infinitude of primes.Comment: 70 pages. In this extended third version of the article, 14 new
proofs of the infnitude of primes are added (2012-2017
A Mathematical Tumor Model with Immune Resistance and Drug Therapy: An Optimal Control Approach
We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four-population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug-free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor-drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy regimens. Using optimal control theory with constraints and numerical simulations, we obtain new therapy protocols that we then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations
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