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A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold -- which is viewed as the obstruction to the existence of a Q-invariant Berezin volume -- is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds., Comment: 11 pages. Comments are welcomed

Correct component placement is of significant importance to ensure optimal outcomes in total hip arthroplasty (THA). Traditionally, the Lewinnek plane has been referenced as an adequate "safe zone", formed between the anterior superior iliac spines and public tubercles to optimize acetabular orientation. However, recent evidence shows that the positioning of this plane may vary due to the biomechanical relationship between the lumbar spine and hip. Therefore, the plane acquired intraoperatively may not accurately recreate the actual functional plane and acetabular orientation encountered outside of the intraoperative environment. This review summarizes the hip-spine relationship and its implications on THA.

We present the notion of higher Kirillov brackets on the sections of an even line bundle over a supermanifold. When the line bundle is trivial we shall speak of higher Jacobi brackets. These brackets are understood furnishing the module of sections with an $L_{\infty}$-algebra, which we refer to as a homotopy Kirillov algebra. We are then to higher Kirillov algebroids as higher generalisations of Jacobi algebroids. Furthermore, we show how to associate a higher Kirillov algebroid and a homotopy BV-algebra with every higher Kirillov manifold. In short, we construct homotopy versions of some of the well-known theorems related to Kirillov's local Lie algebras., Comment: Change of title, improved introduction section and examples. Minor typos etc corrected. No major revision of the key results. 15 pages. Further comments welcomed

We prove that the category of -manifolds has all finite products. Further, we show that a -manifold (resp., a -morphism) can be reconstructed from its algebra of global -functions (resp., from its algebra morphism between global -function algebras). These results are of importance in the study of Lie groups. The investigation is all the more challenging, since the completed tensor product of the structure sheafs of two -manifolds is not a sheaf. We rely on a number of results on (pre)sheaves of topological algebras, which we establish in the appendix.

Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Z2n-manifold, i.e., a Z2n-manifold equipped with a Riemannian metric that may carry non-zero Z2n-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Z2n-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

A quantum mechanical model that realizes the $ \mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics., Comment: 15 pages, amsart, in v.2: minor changes, bibliography added

We extend the notion of super-Minkowski space-time to include Z 2 n -graded (Majorana) spinor coordinates. Our choice of the grading leads to spinor coordinates that are nilpotent but commute amongst themselves. The mathematical framework we employ is the recently developed category of Z 2 n -manifolds understood as locally ringed spaces. The formalism we present resembles N -extended superspace (in the presence of central charges), but with some subtle differences due to the exotic nature of the grading employed.

We show how the theory of $\mathbb{Z}_2^n$ -manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed., 9 pages. Several changes as compared with earlier versions. In particular, we have added "supersymmetries" of mixed symmetry tensors and have included a section on mixed symmetry tensors with values in vector bundles

Background: Understanding atrial fibrillation (AF) management in Scotland was identified as a key priority at the Cross Party Group (CPG) for Heart Disease and Stroke, which meets regularly at the Scottish Parliament. Therefore, a national inquiry was launched to provide clear recommendations to government to improve health for people living with AF. Methods: An advisory panel was convened comprising major third-sector representatives, academics, clinicians, Members of Scottish Parliament (MSPs), and people living with AF. Two questionnaires—one for clinicians, one for people living with AF—were circulated through multiple channels and four roundtable events were held. Results: In total, there were 262 responses to the questionnaires (n=203 clinician; n=59 people living with AF); and roundtables were attended by 35 people, including members of the advisory panel, MSPs, clinicians, academics and people living with AF. Ten key recommendations emerged relating to the detection and diagnosis of AF; initial treatment and subsequent management of AF; use of data to improve detection, diagnosis and treatment of AF in Scotland; and patient information and public awareness. Conclusions: It is hoped that the results of the inquiry discussed in the current article will provide the blueprint for change that will improve AF management across Scotland.

We show that the function sheaf of a $\mathbb{Z}_2^n$-manifold is a nuclear Fr\'echet sheaf of $\mathbb{Z}_2^n$-graded $\mathbb{Z}_2^n$-commutative associative unital algebras. Further, we prove that the components of the pullback sheaf morphism of a $\mathbb{Z}_2^n$-morphism are all continuous. These results are essential for the existence of categorical products in the category of $\mathbb{Z}_2^n$-manifolds. All proofs are self-contained and explicit., Comment: 27 pages. Comments welcomed

We present the notion of a filtered bundle as a generalisation of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalisations of double vector bundles and double affine bundles. Furthermore, we show that the linearisation of a filtered bundle - which can be seen as a partial polarisation of the admissible changes of local coordinates - is well defined., 23 pages