4,541 research outputs found
Scaling and universality in nonlinear optical quantum graphs containing star motifs
Quantum graphs have recently emerged as models of nonlinear optical, quantum
confined systems with exquisite topological sensitivity and the potential for
predicting structures with an intrinsic, off-resonance response approaching the
fundamental limit. Loop topologies have modest responses, while bent wires have
larger responses, even when the bent wire and loop geometries are identical.
Topological enhancement of the nonlinear response of quantum graphs is even
greater for star graphs, for which the first hyperpolarizability can exceed
half the fundamental limit. In this paper, we investigate the nonlinear optical
properties of quantum graphs with the star vertex topology, introduce motifs
and develop new methods for computing the spectra of composite graphs. We show
that this class of graphs consistently produces intrinsic optical
nonlinearities near the limits predicted by potential optimization. All graphs
of this type have universal behavior for the scaling of their spectra and
transition moments as the nonlinearities approach the fundamental limit
Optimum topology of quasi-one dimensional nonlinear optical quantum systems
We determine the optimum topology of quasi-one dimensional nonlinear optical
structures using generalized quantum graph models. Quantum graphs are
relational graphs endowed with a metric and a multiparticle Hamiltonian acting
on the edges, and have a long application history in aromatic compounds,
mesoscopic and artificial materials, and quantum chaos. Quantum graphs have
recently emerged as models of quasi-one dimensional electron motion for
simulating quantum-confined nonlinear optical systems. This paper derives the
nonlinear optical properties of quantum graphs containing the basic star vertex
and compares their responses across topological and geometrical classes. We
show that such graphs have exactly the right topological properties to generate
energy spectra required to achieve large, intrinsic optical nonlinearities. The
graphs have the exquisite geometrical sensitivity required to tune wave
function overlap in a way that optimizes the transition moments. We show that
this class of graphs consistently produces intrinsic optical nonlinearities
near the fundamental limits. We discuss the application of the models to the
prediction and development of new nonlinear optical structures
Periodic behaviour of nonlinear second order discrete dynamical systems
In this work we provide conditions for the existence of periodic solutions to
nonlinear, second-order difference equations of the form \begin{equation*}
y(t+2)+by(t+1)+cy(t)=g(t,y(t)) \end{equation*} where , and
is continuous and periodic in
. Our analysis uses the Lyapunov-Schmidt reduction in combination with fixed
point methods and topological degree theory
Travelling Waves in Hamiltonian Systems on 2D Lattices with Nearest Neighbor Interactions
We study travelling waves on a two--dimensional lattice with linear and
nonlinear coupling between nearest particles and a periodic nonlinear substrate
potential. Such a discrete system can model molecules adsorbed on a substrate
crystal surface. We show the existence of both uniform sliding states and
periodic travelling waves as well in a two-dimensional sine-Gordon lattice
equation using topological and variational methods.Comment: 29 pages, 3 figures. to appear Nonlinearit
The k-core as a predictor of structural collapse in mutualistic ecosystems
Collapses of dynamical systems into irrecoverable states are observed in
ecosystems, human societies, financial systems and network infrastructures.
Despite their widespread occurrence and impact, these events remain largely
unpredictable. In searching for the causes for collapse and instability,
theoretical investigations have so far been unable to determine quantitatively
the influence of the structural features of the network formed by the
interacting species. Here, we derive the condition for the stability of a
mutualistic ecosystem as a constraint on the strength of the dynamical
interactions between species and a topological invariant of the network: the
k-core. Our solution predicts that when species located at the maximum k-core
of the network go extinct, as a consequence of sufficiently weak interaction
strengths, the system will reach the tipping point of its collapse. As a key
variable involved in collapse phenomena, monitoring the k-core of the network
may prove a powerful method to anticipate catastrophic events in the vast
context that stretches from ecological and biological networks to finance
Positive stationary solutions for p-Laplacian problems with nonpositive perturbation
The paper is devoted to the existence of positive solutions of nonlinear
elliptic equations with -Laplacian. We provide a general topological degree
that detects solutions of the problem where is a maximal monotone operator in a Banach
space and is a continuous mapping defined on a closed convex
cone . Next, we apply this general framework to a class of partial
differential equations with -Laplacian under Dirichlet boundary conditions
Wave patterns within the generalized convection-reaction-diffusion equation
A set of travelling wave solutions to a hyperbolic generalization of the
convection-reaction-diffusion is studied by the methods of local nonlinear
alnalysis and numerical simulation. Special attention is paid to displaying
appearance of the compactly supported soloutions, shock fronts, soliton-like
solutions and peakon
Observability/Identifiability of Rigid Motion under Perspective Projection
The "visual motion" problem consists of estimating the motion of an object viewed under projection. In this paper we address the feasibility of such a problem.
We will show that the model which defines the visual motion problem for feature points in the euclidean 3D space lacks of both linear and local (weak) observability. The locally observable manifold is covered with three levels of lie differentiations. Indeed, by imposing metric constraints on the state-space, it is possible to reduce the set of indistinguishable states.
We will then analyze a model for visual motion estimation in terms of identification of an Exterior Differential System, with the parameters living on a topological manifold, called the "essential manifold", which includes explicitly in its definition the forementioned metric constraints. We will show that rigid motion is globally observable/identifiable under perspective projection with zero level of lie differentiation under some general position conditions. Such conditions hold when the viewer does not move on a quadric surface containing all the visible points
Convergence acceleration of Kaczmarz's method
The method of alternation projections (MAP) is an iterative procedure for
finding the projection of a point on the intersection of closed subspaces of an
Hilbert space. The convergence of this method is usually slow, and several
methods for its acceleration have already been proposed. In this work, we
consider a special MAP, namely Kaczmarz' method for solving systems of linear
equations. The convergence of this method is discussed. After giving its matrix
formulation and its projection properties, we consider several procedures for
accelerating its convergence. They are based on sequence transformations whose
kernels contain sequences of the same form as the sequence of vectors generated
by Kaczmarz' method. Acceleration can be achieved either directly, that is
without modifying the sequence obtained by the method, or by restarting it from
the vector obtained by acceleration. Numerical examples show the effectiveness
of both procedures
Sparse Semidefinite Programs with Guaranteed Near-Linear Time Complexity via Dualized Clique Tree Conversion
Clique tree conversion solves large-scale semidefinite programs by splitting
an matrix variable into up to smaller matrix variables, each
representing a principal submatrix of up to . Its
fundamental weakness is the need to introduce overlap constraints that enforce
agreement between different matrix variables, because these can result in dense
coupling. In this paper, we show that by dualizing the clique tree conversion,
the coupling due to the overlap constraints is guaranteed to be sparse over
dense blocks, with a block sparsity pattern that coincides with the adjacency
matrix of a tree. We consider two classes of semidefinite programs with
favorable sparsity patterns that encompass the MAXCUT and MAX -CUT
relaxations, the Lovasz Theta problem, and the AC optimal power flow
relaxation. Assuming that , we prove that the per-iteration cost
of an interior-point method is linear time and memory, so an
-accurate and -feasible iterate is obtained after
iterations in near-linear
time. We confirm our theoretical insights with
numerical results on semidefinite programs as large as .
(Supporting code at https://github.com/ryz-codes/dual_ctc )Comment: [v1] appeared in IEEE CDC 2018; [v2+] To appear in Mathematical
Programmin
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