A Theorem on Analytic Strong Multiplicity One

Let $K$ be an algebraic number field, and $\pi=\otimes\pi_{v}$ an irreducible, automorphic, cuspidal representation of \GL_{m}(\mathbb{A}_{K}) with analytic conductor $C(\pi)$. The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant $c$ depending on $\varepsilon>0, m,$ and $K$ only, such that $\pi$ can be decided completely by its local components $\pi_{v}$ with norm $N(v)<c\cdot C(\pi)^{2m+\varepsilon}.$Comment: accepted by J. Number Theor

Sensitivity analysis in linear models

AbstractIn this work, we consider the general linear model or its variants with the ordinary least squares, generalised least squares or restricted least squares estimators of the regression coefficients and variance. We propose a newly unified set of definitions for local sensitivity for both situations, one for the estimators of the regression coefficients, and the other for the estimators of the variance. Based on these definitions, we present the estimators’ sensitivity results.We include brief remarks on possible links of these definitions and sensitivity results to local influence and other existing results.</jats:p

Remote State Estimation with Smart Sensors over Markov Fading Channels

We consider a fundamental remote state estimation problem of discrete-time linear time-invariant (LTI) systems. A smart sensor forwards its local state estimate to a remote estimator over a time-correlated $M$-state Markov fading channel, where the packet drop probability is time-varying and depends on the current fading channel state. We establish a necessary and sufficient condition for mean-square stability of the remote estimation error covariance as $\rho^2(\mathbf{A})\rho(\mathbf{DM})<1$, where $\rho(\cdot)$ denotes the spectral radius, $\mathbf{A}$ is the state transition matrix of the LTI system, $\mathbf{D}$ is a diagonal matrix containing the packet drop probabilities in different channel states, and $\mathbf{M}$ is the transition probability matrix of the Markov channel states. To derive this result, we propose a novel estimation-cycle based approach, and provide new element-wise bounds of matrix powers. The stability condition is verified by numerical results, and is shown more effective than existing sufficient conditions in the literature. We observe that the stability region in terms of the packet drop probabilities in different channel states can either be convex or concave depending on the transition probability matrix $\mathbf{M}$. Our numerical results suggest that the stability conditions for remote estimation may coincide for setups with a smart sensor and with a conventional one (which sends raw measurements to the remote estimator), though the smart sensor setup achieves a better estimation performance.Comment: The paper has been accepted by IEEE Transactions on Automatic Control. Copyright may be transferred without notice, after which this version may no longer be accessibl