Michał Jóźwikowski

Bundle-theoretic methods for higher-order variational calculus

We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.

A note on actions of some monoids

Smooth actions of the multiplicative monoid (R,⋅) ( R , ⋅ ) of real numbers on manifolds lead to an alternative, and for some reasons simpler, definitions of a vector bundle, a double vector bundle and related structures like a graded bundle (Grabowski and Rotkiewicz (2011) [10] ). For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid (R,⋅) ( R , ⋅ ) . Namely, we discuss geometric structures naturally related with: smooth and holomorphic actions of the monoid of multiplicative complex numbers, smooth actions of the monoid of second jets of punctured maps (R,0)→(R,0) ( R , 0 ) → ( R , 0 ) , smooth actions of the monoid of real 2 by 2 matrices and smooth actions of the multiplicative reals on a supermanifold. In particular cases we recover the notions of a holomorphic vector bundle, a complex vector bundle and a non-negatively graded manifold.

Coccidia (Apicomplexa: Eimeriidae) of the lowland European bison Bison bonasus bonasus (L.).

Coprological studies conducted between 2007 and 2011 in free-roaming and captive European bison Bison bonasus (Linnaeus, 1758) from Poland revealed 11 species of Eimeria infecting the host, i.e., Eimeria alabamensis, Eimeria auburnensis, Eimeria bovis, Eimeria brasiliensis, Eimeria bukidnonensis, Eimeria canadensis, Eimeria cylindrica, Eimeria ellipsoidalis, Eimeria pellita, Eimeria subspherica, and Eimeria zuernii. The typical host for all isolated species is cattle. The most prevalent species was E. bovis (29.7%), while E. brasiliensis was the rarest (0.5%). Five of the species (E. bovis, E. bukidnonensis, E. canadensis, E. ellipsoidalis, E. zuernii) have been observed previously in bison by other authors, 3 species were noticed by us in bison previously (E. alabamensis, E. cylindrica, E. pellita), while for 3 species (E. auburnensis, E. brasiliensis, and E. subspherica) these are new host and locality records. Oocysts of two species (E. brasiliensis, E. bukidnonensis) were noted only in the feces of bison kept in captivity. Moreover, the prevalence of positive samples was higher in the group of captive animals (55.4%) in comparison with the free-roaming herds (29.5%); although, oocysts per gram (OPG), counted with the conventional McMaster technique, was comparable in both groups, reaching maximally 6550 and 6400 in free-roaming and captive individuals, respectively. Overall, 142 fecal samples from 424 samples examined were positive for Eimeria (prevalence=33.5%). Age-related analysis revealed a higher percentage of Eimeria spp. positive samples and higher OPG values in bison under 1 year old as compared to older individuals (93.3% and 50-4050; 37.3% and 50-550, respectively). Additionally, greater eimerian species diversity was present among calves in comparison with older bison. In most cases single-species infections were observed (59.8%) with a predominance of E. bovis (85.9%). Multiple-species infections consisted of 2-7 species, usually including E. bovis. The observation was made that E. bovis infection appears conducive to the host acquiring more eimerian species. No symptoms of clinical coccidiosis occurred during the study.

New solutions of initial conditions in general relativity

We find new classes of exact solutions of the initial momentum constraint for vacuum Einsteins equations. Considered data are either invariant under a continuous symmetry or they are assumed to have the exterior curvature tensor of a simple form. In general the mean curvature H is non-constant and g is not conformally flat. In the generic case with the symmetry we obtain general solution in an explicit form. In other cases solutions are given up to quadrature. We also find a class of explicit solutions without symmetries which generalizes data induced by the Kerr metric or other metrics related to the Ernst equation. The conformal method of Lichnerowicz, Choquet-Bruhat and York is used to prove solvability of the Hamiltonian constraint if H vanishes. Existence of marginally outer trapped surfaces in initial manifold is discussed.

A contact covariant approach to optimal control with applications to sub-Riemannian geometry

We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of Ohsawa (Autom J IFAC 55:1–5, 2015), providing simple and elegant characterizations of normal and abnormal sub-Riemannian extremals.

Duality for Graded Manifolds

We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different category than the initial ones, namely graded polynomial (co)algebra bundles and free graded Weil (co)algebra bundles. Our results are then applied to obtain elegant characterizations of double vector bundles and graded bundles of degree 2. All these results have their supergeometric counterparts. For instance, we give a simple proof of a nice characterisation of

Why are normal sub-Riemannian extremals locally minimizing?

N

Prototypes of higher algebroids with applications to variational calculus

-manifolds of degree 2, announced in the literature.

Models for higher algebroids.

It is well-known that normal extremals in sub-Riemannian geometry are curves which locally minimize the energy functional. Most proofs of this fact do not make, however, an explicit use of relations between local optimality and the geometry of the problem. In this paper, we provide a new proof of that classical result, which gives insight into direct geometric reasons of local optimality. Also the relation of the regularity of normal extremals with their optimality becomes apparent in our approach.