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.

Expression of Smooth Muscle Myosin Heavy Chains and Unloaded Shortening in Single Smooth Muscle Cells

The functional significance of the variable expression of the smooth muscle myosin heavy chain (SM-MHC) tail isoforms, SM1 and SM2, was examined at the mRNA level (which correlates with the protein level) in individual permeabilized rabbit arterial smooth muscle cells (SMCs). The length of untethered single permeabilized SMCs was monitored during unloaded shortening in response to increased Ca2+ (pCa 6.0), histamine (1 μM), and phenylephrine (1 μM). Subsequent to contraction, the relative expression of SM1 and SM2 mRNAs from the same individual SMCs was determined by reverse transcription-polymerase chain reaction amplification and densitometric analysis. Correlational analyses between the SM2-to-SM1 ratio and unloaded shortening in saponin- and α-toxin-permeabilized SMCs (n = 28) reveal no significant relationship between the SM-MHC tail isoform ratio and unloaded shortening velocity. The best correlations between SM2/SM1 and the contraction characteristics of untethered vascular SMCs were with the minimum length attained following contraction (n = 20 andr = 0.72 for α-toxin,n = 8 andr = 0.78 for saponin). These results suggest that the primary effect of variable expression of the SM1 and SM2 SM-MHC tail isoforms is on the cell final length and not on shortening velocity

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

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