Lepage Research Institute

This paper is devoted to some selected topics of the theory of special orthogonal group SO(3). First, we discuss the trace of orthogonal matrices and its relation to the characteristic polynomial; on this basis, the partition of SO(3) formed by conjugation classes is described by trace mapping. Second, we show that every special orthogonal matrix can be expressed as the product of three elementary special orthogonal matrices. Explicit formulas for the decomposition as needed for applications in differential geometry and physics as symmetry transformations are given.

Discogenic back pain is one of the most diffused musculoskeletal pathologies and a hurdle to a good quality of life for millions of people. Existing therapeutic options are exclusively directed at reducing symptoms, not at targeting the underlying, still poorly understood, degenerative processes. Common intervertebral disc (IVD) disease models still do not fully replicate the course of degenerative IVD disease. Advanced disease models that incorporate mechanical loading are needed to investigate pathological causes and processes, as well as to identify therapeutic targets. Organs-on-chip (OoC) are microfluidic-based devices that aim at recapitulating tissue functions in vitro by introducing key features of the tissue microenvironment (e.g., 3D architecture, soluble signals and mechanical conditioning). In this review we analyze and depict existing OoC platforms used to investigate pathological alterations of IVD cells/tissues and discuss their benefits and limitations. Starting from the consideration that mechanobiology plays a pivotal role in both IVD homeostasis and degeneration, we then focus on OoC settings enabling to recapitulate physiological or aberrant mechanical loading, in conjunction with other relevant features (such as inflammation). Finally, we propose our view on design criteria for IVD-on-a-chip systems, offering a future perspective to model IVD mechanobiology.

We consider an inverse variational problem for the lines of constant curvature in (pseudo-)Euclidean two-, three-, and four-dimensional spaces. The accumulated results are physically meaningful in the case of relativistic mechanics of particles.