On Basis Constructions in Finite Element Exterior Calculus

We give a systematic and self-contained account of the construction of geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the families of finite element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between finite element spaces. These structural results show, for example, how to transfer linear dependencies between canonical spanning sets, or give a new derivation of the degrees of freedom

Symmetry and Invariant Bases in Finite Element Exterior Calculus

We study symmetries of bases and spanning sets in finite element exterior calculus using representation theory. The group of affine symmetries of a simplex is isomorphic to a permutation group and represented on simplicial finite element spaces by the pullback action. We want to know which vector-valued finite element spaces have bases that are invariant under permutation of vertex indices. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djokovic and Malzan's classification of monomial irreducible representations of the symmetric group and use symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Invariant bases are constructed in dimensions two and three for different spaces of finite element differential forms.Comment: 27 pages. Submitte

Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and bounded variation spaces

We describe higher-order chain rules for multivariate functions and tensor fields. We estimate Sobolev-Slobodeckij norms, Musielak-Orlicz norms, and the total variation seminorms of the higher derivatives of tensor fields after a change of variables and determine sufficient regularity conditions for the coordinate change. We also introduce a novel higher-order chain rule for composition chains of multivariate functions that is described via nested set partitions and generalized Bell polynomials; it is a natural extension of the Fa\`a di Bruno formula. Our discussion uses the coordinate-free language of tensor calculus and includes Fr\'echet-differentiable mappings between Banach spaces.Comment: Submitte

Smoothed projections over manifolds in finite element exterior calculus

We develop commuting finite element projections over smooth Riemannian manifolds. This extension of finite element exterior calculus establishes the stability and convergence of finite element methods for the Hodge-Laplace equation on manifolds. The commuting projections use localized mollification operators, building upon a classical construction by de Rham. These projections are uniformly bounded on Lebesgue spaces of differential forms and map onto intrinsic finite element spaces defined with respect to an intrinsic smooth triangulation of the manifold. We analyze the Galerkin approximation error. Since practical computations use extrinsic finite element methods over approximate computational manifolds, we also analyze the geometric error incurred.Comment: Submitted. 31 page

Local Finite Element Approximation of Sobolev Differential Forms

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Cl\'ement interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.Comment: 22 pages. Comments welcom

Clinical use of lithium salts: guide for users and prescribers

Lithium has been used clinically for 70 years, mainly to treat bipolar disorder. Competing treatments and exaggerated impressions about complexity and risks of lithium treatment have led to its declining use in some countries, encouraging this update about its safe clinical use. We conducted a nonsystematic review of recent research reports and developed consensus among international experts on the use of lithium to treat major mood disorders, aiming for a simple but authoritative guide for patients and prescribers

Lithium induced hypercalcemia:an expert opinion and management algorithm

Background: Lithium is the gold standard prophylactic treatment for bipolar disorder. Most clinical practice guidelines recommend regular calcium assessments as part of monitoring lithium treatment, but easy-to-implement specific management strategies in the event of abnormal calcium levels are lacking. Methods: Based on a narrative review of the effects of lithium on calcium and parathyroid hormone (PTH) homeostasis and its clinical implications, experts developed a step-by-step algorithm to guide the initial management of emergent hypercalcemia during lithium treatment. Results: In the event of albumin-corrected plasma calcium levels above the upper limit, PTH and calcium levels should be measured after two weeks. Measurement of PTH and calcium levels should preferably be repeated after one month in case of normal or high PTH level, and after one week in case of low PTH level, independently of calcium levels. Calcium levels above 2.8 mmol/l may require a more acute approach. If PTH and calcium levels are normalized, repeated measurements are suggested after six months. In case of persistent PTH and calcium abnormalities, referral to an endocrinologist is suggested since further examination may be needed. Conclusions: Standardized consensus driven management may diminish the potential risk of clinicians avoiding the use of lithium because of uncertainties about managing side-effects and consequently hindering some patients from receiving an optimal treatment

Aberrant intestinal microbiota in individuals with prediabetes

Aims/hypothesis: Individuals with type 2 diabetes have aberrant intestinal microbiota. However, recent studies suggest that metformin alters the composition and functional potential of gut microbiota, thereby interfering with the diabetes-related microbial signatures. We tested whether specific gut microbiota profiles are associated with prediabetes (defined as fasting plasma glucose of 6.1-7.0 mmol/l or HbA1c of 42-48 mmol/mol [6.0-6.5%]) and a range of clinical biomarkers of poor metabolic health. Methods: In the present case-control study, we analysed the gut microbiota of 134 Danish adults with prediabetes, overweight, insulin resistance, dyslipidaemia and low-grade inflammation and 134 age-and sex-matched individuals with normal glucose regulation. Results: We found that five bacterial genera and 36 operational taxonomic units (OTUs) were differentially abundant between individuals with prediabetes and those with normal glucose regulation. At the genus level, the abundance of Clostridium was decreased (mean log2 fold change -0.64 (SEM 0.23), p adj = 0.0497), whereas the abundances of Dorea, [Ruminococcus], Sutterella and Streptococcus were increased (mean log2 fold change 0.51 (SEM 0.12), p adj = 5 x 10-4; 0.51 (SEM 0.11), p adj = 1 x 10-4; 0.60 (SEM 0.21), p adj = 0.0497; and 0.92 (SEM0.21), p adj = 4 x 10-4, respectively). The two OTUs that differed the most were a member of the order Clostridiales (OTU 146564) and Akkermansia muciniphila, which both displayed lower abundance among individuals with prediabetes (mean log2 fold change -1.74 (SEM0.41), p adj = 2 x 10-3 and -1.65 (SEM0.34), p adj = 4 x 10-4, respectively). Faecal transfer from donors with prediabetes or screen-detected, drug-naive type 2 diabetes to germfree Swiss Webster or conventional C57BL/6 J mice did not induce impaired glucose regulation in recipient mice. Conclusions/interpretation: Collectively, our data show that individuals with prediabetes have aberrant intestinal microbiota characterised by a decreased abundance of the genus Clostridium and the mucin-degrading bacterium A. muciniphila. Our findings are comparable to observations in overt chronic diseases characterised by low-grade inflammation