Bounded Point Evaluations and Local Spectral Theory

We study in this paper the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams {\it Dynamic Systems and Apllications} 3(1994) 103-112. Furthermore, we generalize some results of Williams and give a simple proof of theorem 2.5 of L.R. Williams (The Local Spectra of Pure Quasinormal Operators J. Math. anal. Appl. 187(1994) 842-850) that non normal hyponormal weighted shifts have fat local spectra.Comment: 44pp, diploma thesi

Cholesky-factorized sparse Kernel in support vector machines

Support Vector Machine (SVM) is one of the most powerful machine learning algorithms due to its convex optimization formulation and handling non-linear classification. However, one of its main drawbacks is the long time it takes to train large data sets. This limitation is often aroused when applying non-linear kernels (e.g. RBF Kernel) which are usually required to obtain better separation for linearly inseparable data sets. In this thesis, we study an approach that aims to speed-up the training time by combining both the better performance of RBF kernels and fast training by a linear solver, LIBLINEAR. The approach uses an RBF kernel with a sparse matrix which is factorized using Cholesky decomposition. The method is tested on large artificial and real data sets and compared to the standard RBF and linear kernels where both the accuracy and training time are reported. For most data sets, the result shows a huge training time reduction, over 90\%, whilst maintaining the accuracy

Additive maps preserving the reduced minimum modulus of Banach space operators

Let ${\mathcal B}(X)$ be the algebra of all bounded linear operators on an infinite dimensional complex Banach space $X$. We prove that an additive surjective map $\phi$ on ${\mathcal B}(X)$ preserves the reduced minimum modulus if and only if either there are bijective isometries $U:X\to X$ and $V:X\to X$ both linear or both conjugate linear such that $\phi(T)=UTV$ for all $T\in{\mathcal B}(X)$, or $X$ is reflexive and there are bijective isometries $U:X^*\to X$ and $V:X\to X^*$ both linear or both conjugate linear such that $\phi(T)=UT^*V$ for all $T\in{\mathcal B}(X)$. As immediate consequences of the ingredients used in the proof of this result, we get the complete description of surjective additive maps preserving the minimum, the surjectivity and the maximum moduli of Banach space operators.Comment: The abstract of this paper was posted on May 2009 in the web page of the analysis group of Laval University (http://newton.mat.ulaval.ca/analyse/abstracts/2009-06.pdf

Achievable Regions for Interference Channels with Generalized and Intermittent Feedback

In this paper, we first study a two-user interference channel with generalized feedback. We establish an inner bound on its capacity region. The coding scheme that we employ for the inner bound is based on an appropriate combination of Han-Kobayash rate splitting and compress-and-forward at the senders. Each sender compresses the channel output that is observes using a compression scheme that is \`a-la Lim et al. noisy network coding and Avestimeher et al. quantize-map-and-forward. Next, we study an injective deterministic model in which the senders obtain output feedback only intermittently. Specializing the coding scheme of the model with generalized feedback to this scenario, we obtain useful insights onto effective ways of combining noisy network coding with interference alignment techniques. We also apply our results to linear deterministic interference channels with intermittent feedback.Comment: To appear in Proc. of the 2014 IEEE International Symposium on Information Theory, 6 pages, 2 figure

On Cooperative Multiple Access Channels with Delayed CSI at Transmitters

We consider a cooperative two-user multiaccess channel in which the transmission is controlled by a random state. Both encoders transmit a common message and, one of the encoders also transmits an individual message. We study the capacity region of this communication model for different degrees of availability of the states at the encoders, causally or strictly causally. In the case in which the states are revealed causally to both encoders but not to the decoder we find an explicit characterization of the capacity region in the discrete memoryless case. In the case in which the states are revealed only strictly causally to both encoders, we establish inner and outer bounds on the capacity region. The outer bound is non-trivial, and has a relatively simple form. It has the advantage of incorporating only one auxiliary random variable. We then introduce a class of cooperative multiaccess channels with states known strictly causally at both encoders for which the inner and outer bounds agree; and so we characterize the capacity region for this class. In this class of channels, the state can be obtained as a deterministic function of the channel inputs and output. We also study the model in which the states are revealed, strictly causally, in an asymmetric manner, to only one encoder. Throughout the paper, we discuss a number of examples; and compute the capacity region of some of these examples. The results shed more light on the utility of delayed channel state information for increasing the capacity region of state-dependent cooperative multiaccess channels; and tie with recent progress in this framework.Comment: 54 pages. To appear in IEEE Transactions on Information Theory. arXiv admin note: substantial text overlap with arXiv:1201.327