The abstruse meets the applicable: Some aspects of time-frequency analysis

The area of Fourier analysis connected to signal processing theory has undergone a rapid development in the last two decades. The aspect of this development that has received the most publicity is the theory of wavelets and their relatives, which involves expansions in terms of sets of functions generated from a single function by translations and dilations. However, there has also been much progress in the related area known astime-frequency analysis orGabor analysis, which involves expansions in terms of sets of functions generated from a single function by translations and modulations. In this area there are some questions of a concrete and practical nature whose study reveals connections with aspects of harmonic and functional analysis that were previously considered quite pure and perhaps rather exotic. In this expository paper, I give a survey of some of these interactions between the abstruse and the applicable. It is based on the thematic lectures which I gave at the Ninth Discussion Meeting on Harmonic Analysis at the Harish-Chandra Research Institute in Allahabad in October 2005.     

Value of capacity pooling in supply chains with heterogeneous customers

We investigate the value of pooling capacity in supply chains that serve product demands of different variabilities. We build and analyze models that integrate production queuing models with base stock inventory systems serving demands with different inter-arrival time distributions. The first model combines hyperexponential and exponential demand inter-arrival time distributions. Exact analysis of the model allows us to develop insights into the impact of the difference in demand variabilities on the value of pooling capacity. Simulation experiments allow us to validate these insights for more general settings. We then find one special case that combines exponential and deterministic demand arrivals with deterministic service, where pooling capacity results in increasing the total cost.     

On the limit cycle for the 1/r potential in momentum space

The renormalization of the attractive 1/r² potential has recently been studied using a variety of regulators. In particular, it was shown that renormalization with a square well in position space allows multiple solutions for the depth of the square well, including, but not requiring a renormalization group limit cycle. Here, we consider the renormalization of the 1/r² potential in momentum space. We regulate the problem with a momentum cutoff and absorb the cutoff dependence using a momentum-independent counterterm potential. The strength of this counterterm is uniquely determined and runs on a limit cycle. We also calculate the bound state spectrum and scattering observables, emphasizing the manifestation of the limit cycle in these observables.     

On stars and links of shellable polytopal complexes

The vertex stars of shellable polytopal complexes are shown to be shellable. The link of a vertex v of a shellable polytopal complex is also shown to be shellable, provided that all facets of the star of v are simple polytopes, or (more generally) if there exists a shelling F₁,…,F_n of the star of v such that, for every 1<j?n, the intersection of F_j with the previous facets is an initial segment of a line shelling of the boundary complex of F_j.     

Photon + hadron production in high energy deuteron (proton)–nucleus collisions

We apply the color glass condensate formalism to photon + hadron production cross section in high energy deuteron (proton)–gold collisions at RHIC. We investigate the dependence of the production cross section on the angle between the produced hadron and photon for various rapidities and transverse momenta. It is shown that the angular correlation between the produced hadron and photon is a sensitive probe of the saturation dynamics.     

The boundary rigidity problem in the presence of a magnetic field

For a compact Riemannian manifold with boundary, endowed with a magnetic potential α, we consider the problem of restoring the metric g and the magnetic potential α from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Mañé action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction. This problem can only be solved up to an isometry and a gauge transformation of α.For the linearized problem, we show injectivity, up to the natural obstruction, under explicit bounds on the curvature and on α. We also show injectivity and stability for g and α in a generic class G including real analytic ones.For the nonlinear problem, we show rigidity for real analytic simple (g,α), rigidity for metrics in a given conformal class, and locally, near any (g,α)∈G. We also show that simple magnetic systems on two-dimensional manifolds are always rigid.     

Dual coordinate ascent methods for non-strictly convex minimization

We consider a dual method for solving non-strictly convex programs possessing a certain separable structure. This method may be viewed as a dual version of a block coordinate ascent method studied by Auslender [1, Section 6]. We show that the decomposition methods of Han [6, 7] and the method of multipliers may be viewed as special cases of this method. We also prove a convergence result for this method which can be applied to sharpen the available convergence results for Han's methods.The main part of this research was conducted while the author was with the Laboratory for Information and Decision Systems, M.I.T., Cambridge, with support by the U.S. Army Research Office, Contract No. DAAL03-86-K-0171 (Center for Intelligent Control Systems) and by the National Science Foundation under Grant ECS-8519058.