A note on Makeev's conjectures

A counterexample is given for the Knaster-like conjecture of Makeev for functions on $S^2$. Some particular cases of another conjecture of Makeev, on inscribing a quadrangle into a smooth simple closed curve, are solved positively

Dvoretzky type theorems for multivariate polynomials and sections of convex bodies

In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as the Birch theorem) and complex polynomials. We also consider a stronger conjecture on the homogeneous polynomial fields in the canonical bundle over real and complex Grassmannians. This conjecture is much stronger and false in general, but it is proved in the cases of $d=2$ (for $k$'s of certain type), odd $d$, and the complex Grassmannian (for odd and even $d$ and any $k$). Corollaries for the John ellipsoid of projections or sections of a convex body are deduced from the case $d=2$ of the polynomial field conjecture

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans

On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

Companions, codensity and causality

In the context of abstract coinduction in complete lattices, the notion of compatible function makes it possible to introduce enhancements of the coinduction proof principle. The largest compatible function, called the companion, subsumes most enhancements and has been proved to enjoy many good properties. Here we move to universal coalgebra, where the corresponding notion is that of a final distributive law. We show that when it exists the final distributive law is a monad, and that it coincides with the codensity monad of the final sequence of the given functor. On sets, we moreover characterise this codensity monad using a new abstract notion of causality. In particular, we recover the fact that on streams, the functions definable by a distributive law or GSOS specification are precisely the causal functions. Going back to enhancements of the coinductive proof principle, we finally obtain that any causal function gives rise to a valid up-to-context technique

Health-related quality of life change in patients treated at a multidisciplinary pain clinic

Background Multidisciplinary pain management (MPM) is a generally accepted method for treating chronic pain, but heterogeneous outcome measures provide only limited conclusions concerning its effectiveness. Therefore, further studies on the effectiveness of MPM are needed to identify subgroups of patients who benefit, or do not benefit, from these interventions. Our aim was to analyse health-related quality of life (HRQoL) changes after MPM and to identify factors associated with treatment outcomes. Methods We carried out a real world observational follow-up study of chronic pain patients referred to a tertiary multidisciplinary outpatient pain clinic to describe, using the validated HRQoL instrument 15D, the HRQoL change after MPM and to identify factors associated with this change. 1,043 patients responded to the 15D HRQoL questionnaire at baseline and 12 months after the start of treatment. Background data were collected from the pre-admission questionnaire of the pain clinic. Results Fifty-three percent of the patients reported a clinically important improvement and, of these, 81% had a major improvement. Thirty-five percent reported a clinically important deterioration, and 12% had no change in HRQoL. Binary logistic regression analysis revealed that major improvement was positively associated with shorter duration of pain (Peer reviewe

The genetics of neuropathic pain from model organisms to clinical application

Neuropathic pain (NeuP) arises due to injury of the somatosensory nervous system and is both common and disabling, rendering an urgent need for non-addictive, effective new therapies. Given the high evolutionary conservation of pain, investigative approaches from Drosophila mutagenesis to human Mendelian genetics have aided our understanding of the maladaptive plasticity underlying NeuP. Successes include the identification of ion channel variants causing hyper-excitability and the importance of neuro-immune signaling. Recent developments encompass improved sensory phenotyping in animal models and patients, brain imaging, and electrophysiology-based pain biomarkers, the collection of large well-phenotyped population cohorts, neurons derived from patient stem cells, and high-precision CRISPR generated genetic editing. We will discuss how to harness these resources to understand the pathophysiological drivers of NeuP, define its relationship with comorbidities such as anxiety, depression, and sleep disorders, and explore how to apply these findings to the prediction, diagnosis, and treatment of NeuP in the clinic