Bispectral quantum Knizhnik-Zamolodchikov equations for arbitrary root systems

The bispectral quantum Knizhnik-Zamolodchikov (BqKZ) equation corresponding to the affine Hecke algebra $H$ of type $A_{N-1}$ is a consistent system of $q$-difference equations which in some sense contains two families of Cherednik's quantum affine Knizhnik-Zamolodchikov equations for meromorphic functions with values in principal series representations of $H$. In this paper we extend this construction of BqKZ to the case where $H$ is the affine Hecke algebra associated to an arbitrary irreducible reduced root system. We construct explicit solutions of BqKZ and describe its correspondence to a bispectral problem involving Macdonald's $q$-difference operators.Comment: 31 page

Computing Multi-Homogeneous Bezout Numbers is Hard

The multi-homogeneous Bezout number is a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one can ask for the optimal multi-homogenization that would minimize the Bezout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multi-homogeneous Bezout number is actually NP-hard. In terms of approximation theory for combinatorial optimization, the problem of computing the best multi-homogeneous structure does not belong to APX, unless P = NP. Moreover, polynomial time algorithms for estimating the minimal multi-homogeneous Bezout number up to a fixed factor cannot exist even in a randomized setting, unless BPP contains NP

Real Computational Universality: The Word Problem for a class of groups with infinite presentation

The word problem for discrete groups is well-known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real extension of the word problem for a certain class of groups which are presented as quotient groups of a free group and a normal subgroup. Most important, the free group will be generated by an uncountable set of generators with index running over certain sets of real numbers. This allows to include many mathematically important groups which are not captured in the framework of the classical word problem. Our contribution extends computational group theory from the discrete to the Blum-Shub-Smale (BSS) model of real number computation. We believe this to be an interesting step towards applying BSS theory, in addition to semi-algebraic geometry, also to further areas of mathematics. The main result establishes the word problem for such groups to be not only semi-decidable (and thus reducible FROM) but also reducible TO the Halting Problem for such machines. It thus provides the first non-trivial example of a problem COMPLETE, that is, computationally universal for this model.Comment: corrected Section 4.

WP 29 - Overcoming marginalisation? Gender and ethnic segregation in the Dutch construction, health, IT and printing industries

It is common knowledge that indigenous men generally have a better position in the labour market than women and ethnic minorities. This study deals with the question why this is the case in certain sectors of the Dutch economy. The text discusses the labour market attainment for women and ethnic minorities in economic sectors where they are underrepresented. In each of the sectors construction, IT and printing we have evaluated five hypotheses regarding the opportunities of access into and promotion within labour markets for the particular occupations of carpenters, software engineers and printers. We have selected the health sector and the occupation of nurses, as a contrasting sector where women outnumber men in absolute terms. Our hypotheses deal with the following issues: education and training; wage-setting; recruitment and selection; social benefits and active labour market policies. The study arrives at a conclusion about the differences in the factors explaining gender and ethnic segregation. The study is based on a literature overview, interviews with key informants and small case studies in 48 enterprises and organisations. This report includes the national overview for The Netherlands of the research project ‘Overcoming marginalisation’ that was funded under the fifth Framework by the European Commission. The research was executed simultaneously in Denmark, Germany, Italy, Spain and the UK. The international comparison and our comparative working paper on good practice examples will be published separately.

Local Variation as a Statistical Hypothesis Test

The goal of image oversegmentation is to divide an image into several pieces, each of which should ideally be part of an object. One of the simplest and yet most effective oversegmentation algorithms is known as local variation (LV) (Felzenszwalb and Huttenlocher 2004). In this work, we study this algorithm and show that algorithms similar to LV can be devised by applying different statistical models and decisions, thus providing further theoretical justification and a well-founded explanation for the unexpected high performance of the LV approach. Some of these algorithms are based on statistics of natural images and on a hypothesis testing decision; we denote these algorithms probabilistic local variation (pLV). The best pLV algorithm, which relies on censored estimation, presents state-of-the-art results while keeping the same computational complexity of the LV algorithm

Double affine Hecke algebras and bispectral quantum Knizhnik-Zamolodchikov equations

We use the double affine Hecke algebra of type GL_N to construct an explicit consistent system of q-difference equations, which we call the bispectral quantum Knizhnik-Zamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik's quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of q-difference equations acting on the central character of the principal series representations. We construct a meromorphic self-dual solution \Phi of BqKZ which, upon suitable specializations of the central character, reduces to symmetric self-dual Laurent polynomial solutions of quantum KZ equations. We give an explicit correspondence between solutions of BqKZ and solutions of a particular bispectral problem for the Ruijsenaars' commuting trigonometric q-difference operators. Under this correspondence \Phi becomes a self-dual Harish-Chandra series solution \Phi^+ of the bispectral problem. Specializing the central character as above, we recover from \Phi^+ the symmetric self-dual Macdonald polynomials.Comment: 52 page

Problems in the context evaluation of individualized courses

From 1970 to 1974 an Individualized Study System (ISS) for mathematics courses for first year engineering students was developed. Because of changes in the curriculum, new courses had to be developed from August 1974. The context evaluation of these new courses (ISS-calculus) consisted mainly of the evaluation of the mathematics courses developed during the preceding years. After a year the Department decided to suspend ISS as a teaching system for calculus partly because of dissatisfaction of the teachers with ISS-calculus.\ud This paper consists of two parts. Part one (sections 1,2) is a case study and summarizes the development of the system from 1970 to 1975. It examines in detail the problems encountered in this development with special attention to the role of the executive teacher. The organization of an ISS-course and the planning decisions to be taken become more complex according to the number of executive teachers. In part two (sections 3,4) we provide a classification of ISS courses to illustrate the complexity of the system and we offer some general advice on the management of individualized study systems

Prognostic significance of endogenous erythropoietin in long-term outcome of patients with acute decompensated heart failure

Aims Although previous reports suggest that an elevated endogenous erythropoietin (EPO) level is associated with worse clinical outcomes in chronic heart failure (HF) patients, the prognostic implication of EPO in patients with acute decompensated HF (ADHF) and underlying mechanisms of the high EPO level in severe HF patients who have a poor prognosis remain unclear. Methods and results We examined 539 consecutive ADHF patients with EPO measurement on admission from our registry. During a median follow-up period of 329 days, a higher EPO level on admission was independently associated with worse clinical outcomes [hazard ratio (HR) 1.25, 95% confidence interval (CI) 1.06–1.48, P = 0.008], and haemoglobin level was the strongest determinant of EPO level (P < 0.001), whereas estimated glomerular filtration rate (eGFR) was not significant in multivariate regression analysis. In the anaemic subgroup of 318 patients, a higher EPO level than expected on the basis of their haemoglobin level was related to increased adverse events (HR 1.63, 95% CI 1.05–2.49, P = 0.028). Moreover, estimated plasma volume excess rate was positively associated with EPO level (P = 0.003), and anaemic patients with a higher than expected EPO level tended to have a higher estimated plasma volume excess rate and plasma lactate level, and lower systemic oxygen saturation level with the preservation of the reticulocyte production index than those with a lower than expected EPO level. Conclusion A high EPO level predicts long-term worse clinical outcomes in ADHF patients, independent of anaemia and impaired renal function. Anaemia and hypoxia due to severe congestion may synergistically contribute to a high EPO level in high-risk HF patients

Granular Motor in the Non-Brownian Limit

In this work we experimentally study a granular rotor which is similar to the famous Smoluchowski-Feynman device and which consists of a rotor with four vanes immersed in a granular gas. Each side of the vanes can be composed of two different materials, creating a rotational asymmetry and turning the rotor into a ratchet. When the granular temperature is high, the rotor is in movement all the time, and its angular velocity distribution is well described by the Brownian Limit discussed in previous works. When the granular temperature is lowered considerably we enter the so-called Single Kick Limit, where collisions occur rarely and the unavoidable external friction causes the rotor to be at rest for most of the time. We find that the existing models are not capable of adequately describing the experimentally observed distribution in this limit. We trace back this discrepancy to the non-constancy of the deceleration due to external friction and show that incorporating this effect into the existing models leads to full agreement with our experiments. Subsequently, we extend this model to describe the angular velocity distribution of the rotor for any temperature of the gas, and obtain a very good agreement between the model and experimental data