2,190 research outputs found
Manifolds with non-negative Ricci curvature and Nash inequalities
We prove that for any complete n-dimensional Riemannian manifold with
nonnegative Ricci curvature, if the Nash inequality is satisfied, then it is
diffeomorphic to l.Comment: five pages, latex fil
General Sobolev Inequality on Riemannian Manifold
Let M be a complete n-dimensional Riemannian manifold, if the sobolev
inqualities hold on M, then the geodesic ball has maximal volume growth; if the
Ricci curvature of M is nonnegative, and one of the general Sobolev
inequalities holds on M, then M is diffeomorphic to .Comment: 4 page
On the -error linear complexity of binary sequences derived from polynomial quotients
We investigate the -error linear complexity of -periodic binary
sequences defined from the polynomial quotients (including the well-studied
Fermat quotients), which is defined by where is an
odd prime and . Indeed, first for all integers , we determine
exact values of the -error linear complexity over the finite field \F_2
for these binary sequences under the assumption of f2 being a primitive root
modulo , and then we determine their -error linear complexity over the
finite field \F_p for either when or when . Theoretical results obtained indicate that such sequences possess `good'
error linear complexity.Comment: 2 figure
Quantum steerability based on joint measurability
Occupying a position between entanglement and Bell nonlocality,
Einstein-Podolsky-Rosen (EPR) steering has attracted increasing attention in
recent years. Many criteria have been proposed and experimentally implemented
to characterize EPR-steering. Nevertheless, only a few results are available to
quantify steerability using analytical results. In this work, we propose a
method for quantifying the steerability in two-qubit quantum states in the
two-setting EPR-steering scenario, using the connection between joint
measurability and steerability. We derive an analytical formula for the
steerability of a class of X-states. The sufficient and necessary conditions
for two-setting EPRsteering are presented. Based on these results, a class of
asymmetric states, namely, one-way steerable states, are obtained.Comment: 8 pages, 5 figure
Linear complexity problems of level sequences of Euler quotients and their related binary sequences
The Euler quotient modulo an odd-prime power can be uniquely
decomposed as a -adic number of the form where for and we set all
if . We firstly study certain arithmetic properties of
the level sequences over via introducing a
new quotient. Then we determine the exact values of linear complexity of
and values of -error linear complexity for binary
sequences defined by .Comment: 16 page
Regret vs. Communication: Distributed Stochastic Multi-Armed Bandits and Beyond
In this paper, we consider the distributed stochastic multi-armed bandit
problem, where a global arm set can be accessed by multiple players
independently. The players are allowed to exchange their history of
observations with each other at specific points in time. We study the
relationship between regret and communication. When the time horizon is known,
we propose the Over-Exploration strategy, which only requires one-round
communication and whose regret does not scale with the number of players. When
the time horizon is unknown, we measure the frequency of communication through
a new notion called the density of the communication set, and give an exact
characterization of the interplay between regret and communication.
Specifically, a lower bound is established and stable strategies that match the
lower bound are developed. The results and analyses in this paper are specific
but can be translated into more general settings
High-order Green Operators on the Disk and the Polydisc
In this paper, we give the explicit expressions of high-order Green operators
on the disk and the polydisc, and hence the kernel functions of high-order
Green operators are also presented. As applications, we present the explicit
integral expressions of all the solutions for linear high-order partial
differential equations in the disk
Conservation law for Uncertainty relations and quantum correlations
Uncertainty principle, a fundamental principle in quantum physics, has been
studied intensively via various uncertainty inequalities. Here we derive an
uncertainty equality in terms of linear entropy, and show that the sum of
uncertainty in complementary local bases is equal to a fixed quantity. We also
introduce a measure of correlation in a bipartite state, and show that the sum
of correlations revealed in a full set of complementary bases is equal to the
total correlation in the bipartite state. The surprising simple equality
relations we obtain imply that the study on uncertainty principle and
correlations can rely on the use of linear entropy, a simple quantity that is
very convenient for calculation
Spectral density of mixtures of random density matrices for qubits
We derive the spectral density of the equiprobable mixture of two random
density matrices of a two-level quantum system. We also work out the spectral
density of mixture under the so-called quantum addition rule. We use the
spectral densities to calculate the average entropy of mixtures of random
density matrices, and show that the average entropy of the
arithmetic-mean-state of qubit density matrices randomly chosen from the
Hilbert-Schmidt ensemble is never decreasing with the number . We also get
the exact value of the average squared fidelity. Some conjectures and open
problems related to von Neumann entropy are also proposed.Comment: 21 pages, LaTex, 6 figure
Genuine Multipartite Entanglement of Superpositions
We investigate how the genuine multipartite entanglement is distributed among
the components of superposed states. Analytical lower and upper bounds for the
usual multipartite negativity and the genuine multipartite entanglement
negativity are derived. These bounds are shown to be tight by detailed
examples.Comment: 5 page
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