7,302 research outputs found
A matrix product state based algorithm for determining dispersion relations of quantum spin chains with periodic boundary conditions
We study a matrix product state (MPS) algorithm to approximate excited states
of translationally invariant quantum spin systems with periodic boundary
conditions. By means of a momentum eigenstate ansatz generalizing the one of
\"Ostlund and Rommer [1], we separate the Hilbert space of the system into
subspaces with different momentum. This gives rise to a direct sum of effective
Hamiltonians, each one corresponding to a different momentum, and we determine
their spectrum by solving a generalized eigenvalue equation. Surprisingly, many
branches of the dispersion relation are approximated to a very good precision.
We benchmark the accuracy of the algorithm by comparison with the exact
solutions of the quantum Ising and the antiferromagnetic Heisenberg spin-1/2
model.Comment: 13 pages, 11 figures, 5 table
On Kostant's partial order on hyperbolic elements
We study Kostant's partial order on the elements of a semisimple Lie group in
relations with the finite dimensional representations. In particular, we prove
the converse statement of [3, Theorem 6.1] on hyperbolic elements.Comment: 7 page
Online Learning Models for Content Popularity Prediction In Wireless Edge Caching
Caching popular contents in advance is an important technique to achieve the
low latency requirement and to reduce the backhaul costs in future wireless
communications. Considering a network with base stations distributed as a
Poisson point process (PPP), optimal content placement caching probabilities
are derived for known popularity profile, which is unknown in practice. In this
paper, online prediction (OP) and online learning (OL) methods are presented
based on popularity prediction model (PPM) and Grassmannian prediction model
(GPM), to predict the content profile for future time slots for time-varying
popularities. In OP, the problem of finding the coefficients is modeled as a
constrained non-negative least squares (NNLS) problem which is solved with a
modified NNLS algorithm. In addition, these two models are compared with
log-request prediction model (RPM), information prediction model (IPM) and
average success probability (ASP) based model. Next, in OL methods for the
time-varying case, the cumulative mean squared error (MSE) is minimized and the
MSE regret is analyzed for each of the models. Moreover, for quasi-time varying
case where the popularity changes block-wise, KWIK (know what it knows)
learning method is modified for these models to improve the prediction MSE and
ASP performance. Simulation results show that for OP, PPM and GPM provides the
best ASP among these models, concluding that minimum mean squared error based
models do not necessarily result in optimal ASP. OL based models yield
approximately similar ASP and MSE, while for quasi-time varying case, KWIK
methods provide better performance, which has been verified with MovieLens
dataset.Comment: 9 figure, 29 page
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