Recollections of Fred Almgren

While visiting Princeton for the 1997–98 year I have had the privilege of using Fred’s office and helping Jean to go through his papers. It has been moving to see the magnitude of Fred’s helpful correspondence with students and colleagues, some of which Jean and I returned to them with an invitation to contribute short recollections. This article gives some selections from their responses. For more recollections see Mackenzie’s article on “Fred Almgren: Lover of Mathematics, Family, and Life’s Adventures” (Notices AMS 44, 1997, 1102–1106) and the memorial issue ofExperimental Mathematics (Vol. 6, 1997, 1–12).     

Sensitivity of portfolio VaR and CVaR to portfolio return characteristics

Risk management through marginal rebalancing is important for institutional investors due to the size of their portfolios. We consider the problem of improving marginally portfolio VaR and CVaR through a marginal change in the portfolio return characteristics. We study the relative significance of standard deviation, mean, tail thickness, and skewness in a parametric setting assuming a Student’s t or a stable distribution for portfolio returns. We also carry out an empirical study with the constituents of DAX30, CAC40, and SMI. Our analysis leads to practical implications for institutional investors and regulators.     

A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s

In this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients. We aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by Brownian motion. An important and rather surprising consequence of our method which is based on Malliavin calculus is that the solutions derived by Veretennikov (Theory Probab Appl 24:354–366, 1979) for Brownian motion with bounded and measurable drift in $\mathbb{R }^{d}$ are Malliavin differentiable. Further, a strength of our approach, which does not rely on a pathwise uniqueness argument, is that it can be transferred and applied to the analysis of various other types of (stochastic) equations: We obtain a Bismut–Elworthy–Li formula (Elworthy and Li, J Funct Anal 125:252–286, 1994) for spatial derivatives of solutions to the Kolmogorov equation with bounded and measurable drift coefficients. To derive the formula, we use that our approach can be applied to obtain Sobolev differentiability in the initial condition in addition to Malliavin differentiability of the associated stochastic differential equations. Another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields. Moreover, our approach is also applicable to the construction of solutions of stochastic evolution equations on Hilbert spaces.     

An old conjecture of Erdos-Turán on additive bases

There is a 1941 conjecture of Erdos and Turán on what is now called additive basis that we restate:

Conjecture 0.1(Erdos and Turán). Suppose that is an increasing sequence of integers and

Suppose that

If 0$"> for all , then is unbounded.

Our main purpose is to show that the sequence cannot be bounded by . There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.

     

Dynamic reflection and refraction of neutrons

Optical phenomena that arise in the interaction of a neutron wave with matter characterized by a variable magnetic-induction vector are considered. Since the interaction is time-dependent, the neutron energy is not conserved. Only the neutron velocity component, which is orthogonal to the matter boundary, changes. The changes in the neutron energy and in direction of propagation are closely related to the reversal of the neutron-spin projection. Possibilities of application of the thin slab with rotating magnetization are considered.     

Shape tailoring of hexagonally ordered triangular gold nanoparticles with nanosecond-pulsed laser light

In this contribution recent results on selective and precise tailoring of triangular gold nanoparticles (NPs) using ns-pulsed laser light are presented. The NPs were prepared by nanosphere lithography and subsequently tailored with ns-pulsed laser light using different fluences and wavelengths. The method is based on the size and shape dependent localized surface plasmon polariton resonance (SPR) of the NPs. We will demonstrate that the gap size between triangular NPs can be tuned from approximately 102±14 nm to 122±11 nm, due to a shape change of the NP from triangular to oblate. These morphological changes are accompanied by a significant shift of the surface plasmon resonance from λ_SPR=730 nm to λ_SPR=680 nm. Most importantly if the laser wavelength is chosen such that the dipolar SPR is excited, the hexagonal order of the NPs remains intact after irradiation, in contrast to excitation via the quadrupole SPR or within the interband transition. A tuneable gap size and the conservation of the hexagonal order of the NP array is the precondition for applications, where the NPs should serve as anchor points, e.g. for functional molecular nanowires, which can be used to utilize molecular devices.     

On realistic terrains

We study worst-case complexities of visibility and distance structures on terrains under realistic assumptions on edge length ratios and the angles of the triangles, and a more general low-density assumption. We show that the visibility map of a point for a realistic terrain with n triangles has complexity . We also prove that the shortest path between two points p and q on a realistic terrain passes through triangles, and that the bisector of p and q has complexity . We use these results to show that the shortest path map for any point on a realistic terrain has complexity , and that the Voronoi diagram for any set of m points on a realistic terrain has complexity and . Our results immediately imply more efficient algorithms for computing the various structures on realistic terrains.     

Phase transitions of nonlinear waves in quadratic waveguide arrays

We study two-color parametric nonlinear modes in waveguide arrays with a quadratic nonlinear response. We predict theoretically and observe experimentally a new type of phase transition manifested in an abrupt power-controlled change of the mode structure from unstaggered to staggered, due to the interplay of localization and synchronization in parametrically driven discrete systems.     

Embedding some Riemann surfaces into $${\mathbb {C}^2}$$ with interpolation

We prove that several types of open Riemann surfaces, including the finitely connected planar domains, embed properly into such that the values on any given discrete sequence can be arbitrarily prescribed. Kutzschebauch supported by Schweizerische Nationalfonds grant 200021-107477/1.     

Cohomology over Fiber products of local rings

Let S and T be local rings with common residue field k, let R be the fiber product $S{\times }_{k}T$, and let M be an S-module. The Poincaré series $P_M^R$ of M has been expressed in terms of $P_M^S$, $P_k^S$ and $P_k^T$ by Kostrikin and Shafarevich, and by Dress and Krämer. Here, an explicit minimal resolution, as well as theorems on the structure of ${\mathrm{Ext}}_R(k,k)$ and ${\mathrm{Ext}}_R(M,k)$ are given that illuminate these equalities. Structure theorems for the cohomology modules of fiber products of modules are also given. As an application of these results, we compute the depth of cohomology modules over a fiber product.