6,331 research outputs found
Dunkl translations, Dunkl-type space and Riesz transforms for Dunkl transform on
In this paper, we will give some results on the support of Dunkl translations
on compactly supported functions. Then we will define Dunkl-type space
and Riesz transforms for Dunkl transform on , and prove the
boundedness of Riesz transforms from to Dunkl-type space under
the uniform boundedness assumption of Dunkl translations. The proof and the
definition in Dunkl setting will be harder than in the classical case for the
lack of some similar properties of Dunkl translations to that of classical
translations. We will also extend the preciseness of the description of support
of Dunkl translations on characteristic functions by Gallardo and Rejeb to that
on all nonnegative radial functions in .Comment: 12 pages;accepted for publication in Functional Analysis and its
Applications after some minor revision
A strategy for a university cafe during holidays
With no existing strategy for a cafe business and a highly competitive market, the organisation requires a planned strategy. This research proposes to research a café to determine the best strategy for the organisation. A questionnaire will collect quantitative and qualitative data and the organisation will be observed to determine business strategies
Generic Secure Repair for Distributed Storage
This paper studies the problem of repairing secret sharing schemes, i.e.,
schemes that encode a message into shares, assigned to nodes, so that
any nodes can decode the message but any colluding nodes cannot infer
any information about the message. In the event of node failures so that shares
held by the failed nodes are lost, the system needs to be repaired by
reconstructing and reassigning the lost shares to the failed (or replacement)
nodes. This can be achieved trivially by a trustworthy third-party that
receives the shares of the available nodes, recompute and reassign the lost
shares. The interesting question, studied in the paper, is how to repair
without a trustworthy third-party. The main issue that arises is repair
security: how to maintain the requirement that any colluding nodes,
including the failed nodes, cannot learn any information about the message,
during and after the repair process? We solve this secure repair problem from
the perspective of secure multi-party computation. Specifically, we design
generic repair schemes that can securely repair any (scalar or vector) linear
secret sharing schemes. We prove a lower bound on the repair bandwidth of
secure repair schemes and show that the proposed secure repair schemes achieve
the optimal repair bandwidth up to a small constant factor when dominates
, or when the secret sharing scheme being repaired has optimal rate. We
adopt a formal information-theoretic approach in our analysis and bounds. A
main idea in our schemes is to allow a more flexible repair model than the
straightforward one-round repair model implicitly assumed by existing secure
regenerating codes. Particularly, the proposed secure repair schemes are simple
and efficient two-round protocols
Approximations of Shannon Mutual Information for Discrete Variables with Applications to Neural Population Coding
Although Shannon mutual information has been widely used, its effective
calculation is often difficult for many practical problems, including those in
neural population coding. Asymptotic formulas based on Fisher information
sometimes provide accurate approximations to the mutual information but this
approach is restricted to continuous variables because the calculation of
Fisher information requires derivatives with respect to the encoded variables.
In this paper, we consider information-theoretic bounds and approximations of
the mutual information based on Kullback--Leibler divergence and R\'{e}nyi
divergence. We propose several information metrics to approximate Shannon
mutual information in the context of neural population coding. While our
asymptotic formulas all work for discrete variables, one of them has consistent
performance and high accuracy regardless of whether the encoded variables are
discrete or continuous. We performed numerical simulations and confirmed that
our approximation formulas were highly accurate for approximating the mutual
information between the stimuli and the responses of a large neural population.
These approximation formulas may potentially bring convenience to the
applications of information theory to many practical and theoretical problems.Comment: 31 pages, 6 figure
On the Asymptotic Efficiency of Approximate Bayesian Computation Estimators
Many statistical applications involve models for which it is difficult to
evaluate the likelihood, but from which it is relatively easy to sample.
Approximate Bayesian computation is a likelihood-free method for implementing
Bayesian inference in such cases. We present results on the asymptotic variance
of estimators obtained using approximate Bayesian computation in a large-data
limit. Our key assumption is that the data are summarized by a
fixed-dimensional summary statistic that obeys a central limit theorem. We
prove asymptotic normality of the mean of the approximate Bayesian computation
posterior. This result also shows that, in terms of asymptotic variance, we
should use a summary statistic that is the same dimension as the parameter
vector, p; and that any summary statistic of higher dimension can be reduced,
through a linear transformation, to dimension p in a way that can only reduce
the asymptotic variance of the posterior mean. We look at how the Monte Carlo
error of an importance sampling algorithm that samples from the approximate
Bayesian computation posterior affects the accuracy of estimators. We give
conditions on the importance sampling proposal distribution such that the
variance of the estimator will be the same order as that of the maximum
likelihood estimator based on the summary statistics used. This suggests an
iterative importance sampling algorithm, which we evaluate empirically on a
stochastic volatility model.Comment: Main text shortened and proof revised. To appear in Biometrik
Connecting Multiple-unicast and Network Error Correction: Reduction and Unachievability
We show that solving a multiple-unicast network coding problem can be reduced
to solving a single-unicast network error correction problem, where an
adversary may jam at most a single edge in the network. Specifically, we
present an efficient reduction that maps a multiple-unicast network coding
instance to a network error correction instance while preserving feasibility.
The reduction holds for both the zero probability of error model and the
vanishing probability of error model. Previous reductions are restricted to the
zero-error case. As an application of the reduction, we present a constructive
example showing that the single-unicast network error correction capacity may
not be achievable, a result of separate interest.Comment: ISIT 2015. arXiv admin note: text overlap with arXiv:1410.190
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