Functional Inequalities for Two-Level Concentration

Probability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

On Singular Liouville Equations and Systems

We consider some singular Liouville equations and systems motivated by uniformization problems in a non-smooth setting, as well as from models in mathematical physics. We will study the existence of solutions from a variational point of view, using suitable improvements of the Moser–Trudinger inequality. These reduce the problem to a topological one by studying the concentration property of conformal volume, which will be constrained by the functional inequalities of geometric flavour. We will mainly describe some common strategies from the papers [

Semi-Definite Programming Based Waveform Design for Spectrum Sensing

In order to solve the problem of spectrum shortages, dynamic spectrum access based cognitive radio technology was proposed, and the technology of spectrum sensing is the foundation of cognitive radio. In this paper, a semi-definite programming based waveform design algorithm is proposed for spectrum sensing. Considering the energy concentration property of Chirp signal, the Chirp signal is adopted as the basic function, which makes the composite waveform perform like impulse function in fractional Fourier transform domain. Simulation results show that the NESP (Normalized Effective Signal Power) of the designed waveform using the proposed algorithm can be over 70%, and the designed waveform also retains good energy concentration property in fractional domain.

A Subspace Embedding Method inL2Norm via Fast Cauchy Transform

We propose a subspace embedding method via Fast Cauchy Transform (FCT) inL2norm. It is motivated by and complements the work of the subspace embedding method inLpnorm, for allp∈[1,∞]exceptp=2, by K. L. Clarkson (ACM-SIAM, 2013). Unlike the traditionally used orthogonal basis in Johnson-Lindenstrauss (JL) embedding, we employ the well-conditioned basis inL2norm to obtain concentration property of FCT inL2norm.