On the Square Peg Problem and its relatives

Toeplitz's Square Peg Problem asks whether every continuous simple closed curve in the plane contains the four vertices of a square. It has been proved for various classes of sufficiently smooth curves, some of which are dense, none of which are open. In this paper we prove it for several open classes of curves, one of which is also dense. This can be interpreted in saying that the Square Peg Problem is solved for generic curves. The latter class contains all previously known classes for which the Square Peg Problem has been proved in the affirmative. [footnote] We also prove results about rectangles inscribed in immersed curves. Finally, we show that the problem of finding a regular octahedron on metric 2-spheres has a "topological counter-example", that is, a certain test map with boundary condition exists

Computing integral points on X_ns^+(p)

We describe an algorithm for computing integral points on the modular curve of prime level p associated to the normalizer of a non-split Cartan subgroup of GL_2(F_p). Using our method, we show that for 7<p<101 the only integral points on this curve are the CM-points

Prodsimplicial-Neighborly Polytopes

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.Comment: 21 pages, 10 figure

Benefits of Exercise in Rheumatoid Arthritis

This paper aims to highlight the importance of exercise in patients with rheumatoid arthritis (RA) and to demonstrate the multitude of beneficial effects that properly designed exercise training has in this population. RA is a chronic, systemic, autoimmune disease characterised by decrements to joint health including joint pain and inflammation, fatigue, increased incidence and progression of cardiovascular disease, and accelerated loss of muscle mass, that is, “rheumatoid cachexia”. These factors contribute to functional limitation, disability, comorbidities, and reduced quality of life. Exercise training for RA patients has been shown to be efficacious in reversing cachexia and substantially improving function without exacerbating disease activity and is likely to reduce cardiovascular risk. Thus, all RA patients should be encouraged to include aerobic and resistance exercise training as part of routine care. Understanding the perceptions of RA patients and health professionals to exercise is key to patients initiating and adhering to effective exercise training

Enhanced firing of locus coeruleus neurons and SK channel dysfunction are conserved in distinct models of prodromal Parkinson's disease

Parkinson’s disease (PD) is clinically defined by the presence of the cardinal motor symptoms, which are associated with a loss of dopaminergic nigrostriatal neurons in the substantia nigra pars compacta (SNpc). While SNpc neurons serve as the prototypical cell-type to study cellular vulnerability in PD, there is an unmet need to extent our efforts to other neurons at risk. The noradrenergic locus coeruleus (LC) represents one of the first brain structures affected in Parkinson’s disease (PD) and plays not only a crucial role for the evolving non-motor symptomatology, but it is also believed to contribute to disease progression by efferent noradrenergic deficiency. Therefore, we sought to characterize the electrophysiological properties of LC neurons in two distinct PD models: (1) in an in vivo mouse model of focal α-synuclein overexpression; and (2) in an in vitro rotenone-induced PD model. Despite the fundamental differences of these two PD models, α-synuclein overexpression as well as rotenone exposure led to an accelerated autonomous pacemaker frequency of LC neurons, accompanied by severe alterations of the afterhyperpolarization amplitude. On the mechanistic side, we suggest that Ca(2+)-activated K(+) (SK) channels are mediators of the increased LC neuronal excitability, as pharmacological activation of these channels is sufficient to prevent increased LC pacemaking and subsequent neuronal loss in the LC following in vitro rotenone exposure. These findings suggest a role of SK channels in PD by linking α-synuclein- and rotenone-induced changes in LC firing rate to SK channel dysfunction