15,828 research outputs found
Degree counting and shadow system for Toda system: one bubbling
Here we initiate the program for computing the Leray-Schauder topological
degree for Toda system. This program still contains a lot of
challenging problems for analysts. The first step of our approach is to answer
whether concentration phenomena holds or not. In this paper, we prove the
concentration phenomena holds while crosses , and . However, for , the question whether
concentration holds or not still remains open up to now. The second step is to
study the corresponding shadow system and its degree counting formula. The last
step is to construct bubbling solution of Toda system via a
non-degenerate solution of the shadow system. Using this construction, we
succeed to calculate the degree for and
.Comment: 64 page
Solving Three Dimensional Maxwell Eigenvalue Problem with Fourteen Bravais Lattices
Calculation of band structure of three dimensional photonic crystals amounts
to solving large-scale Maxwell eigenvalue problems, which are notoriously
challenging due to high multiplicity of zero eigenvalue. In this paper, we try
to address this problem in such a broad context that band structure of three
dimensional isotropic photonic crystals with all 14 Bravais lattices can be
efficiently computed in a unified framework. We uncover the delicate machinery
behind several key results of our work and on the basis of this new
understanding we drastically simplify the derivations, proofs and arguments in
our framework. In this work particular effort is made on reformulating the
Bloch boundary condition for all 14 Bravais lattices in the redefined
orthogonal coordinate system, and establishing eigen-decomposition of discrete
partial derivative operators by systematic use of commutativity among them,
which has been overlooked previously, and reducing eigen-decomposition of
double-curl operator to the canonical form of a 3x3 complex skew-symmetric
matrix under unitary congruence. With the validity of the novel nullspace free
method in the broad context, we perform some calculations on one benchmark
system to demonstrate the accuracy and efficiency of our algorithm.Comment: 37 pages, 20 figure
A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible M-matrix
In this paper, based on the Noda iteration, we present inexact Noda
iterations (INI), to find the smallest eigenvalue and the associated positive
eigenvector of a large irreducible nonsingular M-matrix. The positivity of
approximations is critical in applications, and if the approximations lose the
positivity then they will be physically meaningless. We propose two different
inner tolerance strategies for solving the inner linear systems involved, and
prove that the resulting INI algorithms are globally linear and superlinear
with the convergence order , respectively. The proposed
INI algorithms are structure preserving and maintains the positivity of
approximate eigenvectors. We also revisit the exact Noda iteration and
establish a new quadratic convergence result. All the above is first done for
the problem of computing the Perron root and the positive Perron vector of an
irreducible nonnegative matrix and then adapted to that of computing the
smallest eigenpair of the irreducible nonsingular M-matrix. Numerical examples
illustrate that the proposed INI algorithms are practical, and they always
preserve the positivity of approximate eigenvectors. We compare them with the
positivity non-preserving Jacobi--Davidson method and implicitly restarted
Arnoldi method, which often compute physically meaningless eigenvectors, and
illustrate that the overall efficiency of the INI algorithms is competitive
with and can be considerably higher than the latter two methods.Comment: 28 pages, 8 figure
Intermittent behaviors in weakly coupled map lattices
In this paper, we study intermittent behaviors of coupled piecewise-expanding
map lattices with two nodes and a weak coupling. We show that the successive
phase transition between ordered and disordered phases occurs for almost every
orbit. That is, we prove and
, where correspond to the coordinates of two nodes at the iterative step .
We also prove the same conclusion for weakly coupled tent-map lattices with any
multi-nodes.Comment: 33 pages, 4 figure
Pion and Kaon Distribution Amplitudes in the Continuum Limit
We present a lattice-QCD calculation of the pion, kaon and
distribution amplitudes using large-momentum effective theory (LaMET). Our
calculation is carried out using three ensembles with 2+1+1 flavors of highly
improved staggered quarks (HISQ), generated by MILC collaboration, at 310 MeV
pion mass with 0.06, 0.09 and 0.12 fm lattice spacings. We use clover fermion
action for the valence quarks and tune the quark mass to match the lightest
light and strange masses in the sea. The resulting lattice matrix elements are
nonperturbatively renormalized in regularization-independent
momentum-subtraction (RI/MOM) scheme and extrapolated to the continuum. We use
two approaches to extract the -dependence of the meson distribution
amplitudes: 1) we fit the renormalized matrix elements in coordinate space to
an assumed distribution form through a one-loop matching kernel; 2) we use a
machine-learning algorithm trained on pseudo lattice-QCD data to make
predictions on the lattice data. We found the results are consistent between
these methods with the latter method giving a less smooth shape. Both
approaches suggest that as the quark mass increases, the distribution amplitude
becomes narrower. Our pion distribution amplitude has broader distribution than
predicted by light-front constituent-quark model, and the moments of our pion
distributions agree with previous lattice-QCD results using the operator
production expansion
iSIRA: Integrated Shift-Invert Residual Arnoldi Method for Graph Laplacian Matrices from Big Data
The eigenvalue problem of a graph Laplacian matrix arising from a simple,
connected and undirected graph has been given more attention due to its
extensive applications, such as spectral clustering, community detection,
complex network, image processing and so on. The associated graph Laplacian
matrix is symmetric, positive semi-definite, and is usually large and sparse.
Computing some smallest positive eigenvalues and corresponding eigenvectors is
often of interest.
However, the singularity of makes the classical eigensolvers inefficient
since we need to factorize for the purpose of solving large and sparse
linear systems exactly. The next difficulty is that it is usually time
consuming or even unavailable to factorize a large and sparse matrix arising
from real network problems from big data such as social media transactional
databases, and sensor systems because there is in general not only local
connections.
In this paper, we propose an eignsolver based on the inexact residual Arnoldi
method together with an implicit remedy of the singularity and an effective
deflation for convergent eigenvalues. Numerical experiments reveal that the
integrated eigensolver outperforms the classical Arnoldi/Lanczos method for
computing some smallest positive eigeninformation provided the LU factorization
is not available
Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem
The discretized Bethe-Salpeter eigenvalue problem arises in the Green's
function evaluation in many body physics and quantum chemistry. Discretization
leads to a matrix eigenvalue problem for with
a Hamiltonian-like structure. After an appropriate transformation of to a
standard symplectic form, the structure-preserving doubling algorithm,
originally for algebraic Riccati equations, is extended for the discretized
Bethe-Salpeter eigenvalue problem. Potential breakdowns of the algorithm, due
to the ill condition or singularity of certain matrices, can be avoided with a
double-Cayley transform or a three-recursion remedy. A detailed convergence
analysis is conducted for the proposed algorithm, especially on the benign
effects of the double-Cayley transform. Numerical results are presented to
demonstrate the efficiency and structure-preserving nature of the algorithm.Comment: 22 page
Singular Value Decompositions for Single-Curl Operators in Three-Dimensional Maxwell's Equations for Complex Media
This article focuses on solving the generalized eigenvalue problems (GEP)
arising in the source-free Maxwell equation with magnetoelectric coupling
effects that models three-dimensional complex media. The goal is to compute the
smallest positive eigenvalues, and the main challenge is that the coefficient
matrix in the discrete Maxwell equation is indefinite and degenerate. To
overcome this difficulty, we derive a singular value decomposition (SVD) of the
discrete single-curl operator and then explicitly express the basis of the
invariant subspace corresponding to the nonzero eigenvalues of the GEP.
Consequently, we reduce the GEP to a null space free standard eigenvalue
problem (NFSEP) that contains only the nonzero (complex) eigenvalues of the GEP
and can be solved by the shift-and-invert Arnoldi method without being
disturbed by the null space. Furthermore, the basis of the eigendecomposition
is chosen carefully so that we can apply fast Fourier transformation
(FFT)-based matrix vector multiplication to solve the embedded linear systems
efficiently by an iterative method. For chiral and pseudochiral complex media,
which are of great interest in magnetoelectric applications, the NFSEP can be
further transformed to a null space free generalized eigenvalue problem whose
coefficient matrices are Hermitian and Hermitian positive definite
(HHPD-NFGEP). This HHPD-NFGEP can be solved by using the invert Lanczos method
without shifting. Furthermore, the embedded linear system can be solved
efficiently by using the conjugate gradient method without preconditioning and
the FFT-based matrix vector multiplications. Numerical results are presented to
demonstrate the efficiency of the proposed methods
Flavor Structure of the Nucleon Sea from Lattice QCD
We present the first direct lattice calculation of the isovector sea-quark
parton distributions using the formalism developed recently by one of the
authors. We use HISQ lattice gauge ensembles (generated by MILC
Collaboration) and clover valence fermions with pion mass 310 MeV. We are able
to obtain the qualitative features of the nucleon sea flavor structure even at
this large pion mass: We observe violation of the Gottfried sum rule,
indicating ; the helicity distribution obeys
, which is consistent with the
STAR data at large and small leptonic pseudorapidity.Comment: 5 pages, 3 figures, version to appear at PR
p-wave holographic superconductor in scalar hairy black holes
We study the properties of the p-wave holographic superconductor for the
scalar hairy black holes in the probe limit. The black hole solutions in
question possess planar topology, which are derived from the Einstein gravity
theory minimally coupled to a scalar field with a generic scalar potential.
These solutions can be viewed as characterized by two independent parameters,
namely, and , where AdS vacuum is manifestly restored when
. Consequently, the p-wave holographic superconductor is
investigated by employing the above static planar black hole spacetime as the
background metric, where a Maxwell field is introduced to the model by
nonminimally coupling it to a complex vector field. The latter is shown to
condensate and furnish the superconducting phase when the temperature is below
a critical value. By numerical calculations, we examine in detail how the
scalar field in the background affects the properties of the superconductivity.
It is found that the critical temperature depends crucially on the parameters
and , which subsequently affects the condensation process. By
employing the Kubo formula, the real, as well as imaginary parts of the
conductivity, are calculated and presented as functions of frequency. The
results are discussed regarding the poles of the Green function, and the
typical values of the BCS theory.Comment: 15 pages, 6 figure
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