Jensen measures and boundary values of plurisubharmonic functions

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions inB-regular domain. This theorem implies that the two classes of Jensen measures coincide inB-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above. The paper also contains a characterisation in terms of Jensen measures of those continuous functions that are boundary values of a continuous plurisubharmonic function.     

On one-sided versus two-sided classification

One-sided classifiers are computable devices which read the characteristic function of a set and output a sequence of guesses which converges to 1 iff the set on the input belongs to the gven class. Such a classifier istwo-sided if the sequence of its output in addition converges to 0 on setsnot belonging to the class. The present work obtains the below mentionedresults for one-sided classes (= Σ⁰ ₂ classes) with respect to four areas: Turing complexity, 1-reductions, index sets and measure. There are one-sided classes which are not two-sided. This can have two reasons: (1) the class has only high Turing complexity. Then there are some oracles which allow to construct noncomputale two-sided classifiers. (2) The class is difficult because of some topological constraints and then there are also no nonrecursive two-sided classifiers. For case (1), several results are obtainedto localize the Turing complexity of certain types of one-sided classes. The concepts of 1-reduction, 1-completeness and simple sets is transferred to one-sided classes: There are 1-complete classes and simple classes, but no class is at the same time 1-complete nd simple. The one-sided classes have a natural numbering. Most of the common index sets relative to this numbering have the high complexity Π¹ ₁: the index set of the class {0,1}^∞, the index set of the equality problem and the index set of all two-sided classes. On the other side the index set of the empty class has complexity Π⁰ ₂; Π⁰ ₂ and Σ⁰ ₂ are the least complexities any nontrivial index set can have. Lusin showed that any one-sided class is measurable. Concerning the effectiveness of this measure, it is shown that a one-sided class has recursive measure 0 if it has measure 0, but that thre are one-sided classes having measure 1 without having measure 1 effectively. The measure of a two-sided class can be computed in the limit. Received: 2 December 1999 / Revised version: 28 February 2000 / Published online: 15 June 2001     

The Elekes–Szabó Theorem in four dimensions

Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ? C be finite sets of size n. We show that F vanishes on at most O(n^8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.     

Dynamic reflection and refraction of neutrons at the boundaries of matter characterized by a variable magnetic induction

Optical phenomena that arise in the interaction of a neutron wave with matter characterized by a variable interaction potential are considered. The time dependence of the potential is assumed to be due to a change in the magnetization vector in matter with time. Since the interaction in question is time-dependent, the neutron energy is not conserved. If a neutron interacts with a sample that has a plane boundary, only the neutron-velocity component orthogonal to the matter boundary changes. Thus, reflected waves are characterized by a reflection angle that is different from the angle of incidence. Waves transmitted through a plane sample can also change direction. The changes in the neutron energy and in the neutron-velocity direction are closely related to the reversal of the neutron-spin projection. The question of whether a slab featuring a rotating magnetization vector can be used as a spin flipper or as a coherent wave splitter is considered.     

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.

     

Oscillator strengths for singly ionized oxygen

The electric dipole oscillator strengths for multiplet and individual lines between some doublet and quartet levels have been calculated using the weakest bound electron potential model theory (WBEPMT) in singly ionized oxygen. We employed both numerical Coulomb approximation (NCA) wave functions and numerical non-relativistic Hartree-Fock (NRHF) wave functions for expectation values of radii in determination of parameters. The calculated oscillator strengths have been compared with available theoretical and experimental results. A good agreement with results in literature has been obtained. Moreover, oscillator strengths not existing in the literature for some highly excited levels have been obtained using this method. Electronic supplementary material  Supplementary Online Material     

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.