On low-dimensional projections of high-dimensional distributions

Let P be a probability distribution on q -dimensional space. The so-called Diaconis-Freedman effect means that for a fixed dimension d<sufficient conditions for this phenomenon in a suitable asymptotic framework with increasing dimension q . It turns out, that the conditions formulated by Diaconis and Freedman (1984) are not only sufficient but necessary as well. Moreover, letting P ^ be the empirical distribution of n independent random vectors with distribution P , we investigate the behavior of the empirical process n √ (P ^ −P) under random projections, conditional on P ^ .

Continued fractions built from convex sets and convex functions

In a partially ordered semigroup with the duality (or polarity) transform, it is pos- sible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

An Oka principle for equivariant isomorphisms

Let G be a reductive complex Lie group acting holomorphically on normal Stein spaces X and Y, which are locally G-biholomorphic over a common categorical quotient Q. When is there a global G-biholomorphism X → Y? If the actions of G on X and Y are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that X and Y are G-biholomorphic if X is K-contractible, where K is a maximal compact subgroup of G, or if X and Y are smooth and there is a G-diffeomorphism ψ : X → Y over Q, which is holomorphic when restricted to each fibre of the quotient map X → Q. We prove a similar theorem when ψ is only a G-homeomorphism, but with an assumption about its action on G-finite functions. When G is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of G-biholomorphisms from X to Y over Q. This sheaf can be badly singular, even for a low-dimensional representation of SL2(ℂ). Our work is in part motivated by the linearisation problem for actions on ℂn. It follows from one of our main results that a holomorphic G-action on ℂn, which is locally G-biholomorphic over a common quotient to a generic linear action, is linearisable.

Higher order elicitability and Osband’s principle

A statistical functional, such as the mean or the median, is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score. Such scoring functions are called strictly consistent for the functional. The elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores. In this paper, we explore the notion of elicitability for multi-dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions. We cover the case of functionals with elicitable components, but we also show that one-dimensional functionals that are not elicitable can be a component of a higher order elicitable functional. In the case of the variance, this is a known result. However, an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the “correct” quantiles. A direct consequence of applied interest is that the pair (Value at Risk, Expected Shortfall) is jointly elicitable under mild conditions that are usually fulfilled in risk management applications.

In vitro hemodynamic evaluation of ventricular suction conditions of the EVAHEART ventricular assist pump

Purpose: Mismatches between pump output and venous return in a continuous-flow ventricular assist device may elicit episodes of ventricular suction. This research describes a series of in vitro experiments to characterize the operating conditions under which the EVAHEART centrifugal blood pump (Sun Medical Technology Research Corp., Nagano, Japan) can be operated with minimal concern regarding left ventricular (LV) suction. Methods: The pump was interposed into a pneumatically driven pulsatile mock circulatory system (MCS) in the ventricular apex to aorta configuration. Under varying conditions of preload, afterload, and systolic pressure, the speed of the pump was increased step-wise until suction was observed. Identification of suction was based on pump inlet pressure. Results: In the case of reduced LV systolic pressure, reduced preload (=10 mmHg), and afterload (=60 mmHg), suction was observed for speeds =2,200 rpm. However, suction did not occur at any speed (up to a maximum speed of 2,400 rpm) when preload was kept within 10-14 mmHg and afterload =80 mmHg. Although in vitro experiments cannot replace in vivo models, the results indicated that ventricular suction can be avoided if sufficient preload and afterload are maintained. Conclusion: Conditions of hypovolemia and/or hypotension may increase the risk of suction at the highest speeds, irrespective of the native ventricular systolic pressure. However, in vitro guidelines are not directly transferrable to the clinical situation; therefore, patient-specific evaluation is recommended, which can be aided by ultrasonography at various points in the course of support.