Multiplicative perturbations of local C-cosine functions

"We establish some left and right multiplicative perturbations of a local C-cosine function C(.) on a complex Banach space X with non-densely defined generator, which can be applied to obtain some new additive perturbation results concerning C(.)."

The Drazin inverse matrix modification formulae with Peirce corners

Our motivation is to derive the Drazin inverse matrix modification formulae utilizing the Drazin inverses of adequate Peirce corners under some special cases, and the Drazin inverse of a special matrix with an additive perturbation. As applications, several new results for the expressions of the Drazin inverses of modified matrices A ?? CB and A ?? CDdB are obtained, and some well known results in the literature, as the Sherman-Morrison-Woodbury formula and Jacobson?s Lemma, are generalized.

Freeness over the diagonal and outliers detection in deformed random matrices with a variance profile

We study the eigenvalue distribution of a Gaussian unitary ensemble (GUE) matrix with a variance profile that is perturbed by an additive random matrix that may possess spikes. Our approach is guided by Voiculescu’s notion of freeness with amalgamation over the diagonal and by the notion of deterministic equivalent. This allows to derive a fixed point equation to approximate the spectral distribution of certain deformed GUE matrices with a variance profile and to characterize the location of potential outliers in such models in a non-asymptotic setting. We also consider the singular values distribution of a rectangular Gaussian random matrix with a variance profile in a similar setting of additive perturbation. We discuss the application of this approach to the study of low-rank matrix denoising models in the presence of heteroscedastic noise, that is when the amount of variance in the observed data matrix may change from entry to entry. Numerical experiments are used to illustrate our results. Deformed random matrix, Variance profile, Outlier detection, Free probability, Freeness with amalgamation, Operator-valued Stieltjes transform, Gaussian spiked model, Low-rank model. 2000 Math Subject Classification: 62G05, 62H12.

Fractional Order Sliding Mode Control of a Class of Second Order Perturbed Nonlinear Systems: Application to the Trajectory Tracking of a Quadrotor

A Fractional Order Sliding Mode Control (FOSMC) is proposed in this paper for an integer second order nonlinear system with an unknown additive perturbation term. A sufficient condition is given to assure the attractiveness to a given sliding surface where trajectory tracking is assured, despite the presence of the perturbation term. The control scheme is applied to the model of a quadrotor vehicle in order to have trajectory tracking in the space. Simulation results are presented to evaluate the performance of the control scheme.

Fluctuation of Matrix Entries and Application to Outliers of Elliptic Matrices

For any family of N ⨯ N randommatrices that is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type Tr(AkM), where the matrix M is deterministic (such random variables include, for example, the normalized matrix entries of Ak). A consequence is the asymptotic independence of the projection of the matrices Ak onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Formof the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other.These phenomena have already been observed with random matrices from the Single Ring Theorem.

Evaluating Re-Identification Risks of Data Protected by Additive Data Perturbation

Commercial organizations and government agencies that gather, store, share and disseminate data are facing increasing concerns over individual privacy and confidentiality. Confidential data is often masked in the database or prior to release to a third party, through methods such as data perturbation. In this study, re-identification risks of three major additive data perturbation techniques were compared using two different record linkage techniques. The results suggest that re-identification risk of Kim's multivariate noise addition method is similar to that of simple noise addition method. The general additive perturbation method (GADP) has the lowest re-identification risk and therefore provides the highest level of protection. The study also suggests that Fuller's method of assessing re-identification risk may be better suited than the probabilistic record-linkage method of Winkler, for numeric data. The results of this study should be help organizations and government agencies choose an appropriate additive perturbation technique.

LARGE DEVIATIONS FOR A NON-CENTERED WISHART MATRIX

We investigate an additive perturbation of a complex Wishart random matrix and prove that a large deviation principle holds for the spectral measures. The rate function is associated to a vector equilibrium problem coming from logarithmic potential theory, which in our case is a quadratic map involving the logarithmic energies, or Voiculescu's entropies, of two measures in the presence of an external field and an upper constraint. The proof is based on a two type particles Coulomb gas representation for the eigenvalue distribution, which gives a new insight on why such variational problems should describe the limiting spectral distribution. This representation is available because of a Nikishin structure satisfied by the weights of the multiple orthogonal polynomials hidden in the background.