The significance of the geometry of differential equations was well understood already in the nineteenth century, and in the last few decades such aspects have played an increasing role in numerical methods for differential equations. Nowadays, there is a rich selection of integrators which preserve properties like symplecticity, reversibility, phase volume and first integrals, either exactly o...

Linear stationary subdivision rules take a sequence of input data and produce ever denser sequences of subdivided data from it. They are employed in multiresolution modeling and have intimate connections with wavelet and more general pyramid transforms. Data which naturally do not live in a vector space, but in a nonlinear geometry like a surface, symmetric space, or a Lie group (e.g. motion ca...

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent then its upper (or lower) Lie nilpotency index is at most |G| + 1, where |G| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal have already been determined. Here we determine G for which upper (or lower) L...

We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus T, n ≥ 2, namely codimension one foliations, flows, and the so-called nonquadratic foliations. We show in particular that non-quadratic foliations are rigid, in the sense that they do not admit transverse diffeomorphisms other than ±id and translations. The computation is an app...

Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the foliation. The case in which the foliation is given by the orbits of an action of a Lie group has a particularly nice structure, which we study in detail, g...

The present paper aims at introducing the concepts and mathematical details of unsupervised neural learning with orthonormality constrains. The neural structures considered are single non-linear layers and the learnable parameters are organized in matrices, as usual, which gives the parameters spaces the geometrical structure of the Euclidean manifold. The constraint of orthonormality for the c...

The φ 4 field model is generalized to the case when the field φ(x) is defined on a Lie group: S[φ] = x∈G L[φ(x)]dµ(x), dµ(x) is the left-invariant measure on a locally compact group G. For the particular case of the affine group G : x ′ = ax +b, a ∈ R+, x, b ∈ R n the Feynman perturbation expansion for the Green functions is shown to have no ultraviolet divergences for certain choice of λ(a) ∼ ...

This project continues a strong record of research in the Department of Bioethics that is aimed at facilitating fair and effective allocation of resources to promote health. The research involves a variety of methodologies including conceptual analysis and empirical methods including survey research, focus groups to promote democratic deliberation, secondary analysis of large datasets, and meta...

This paper will provide some simple explanation about matrix Lie groups. I intend for the reader to have a background in calculus, point-set topology, group theory, and linear algebra, particularly the Spectral Theorem. I focus on SO(3), the group of rotations in three-dimensional space.