On the sharpness of assumptions in the Federer theorem

The Federer theorem deals with the “massiveness” of the set of critical values for a t t -smooth map acting from R m \mathbb R^m to R n \mathbb R^n : it claims that the Hausdorff p p -measure of this set is zero for certain p p . If n ≥ m n\ge m , it has long been known that the assumption of that theorem relating the parameters m , n , t , p m,n,t,p is sharp. Here it is shown by an example that this assumption is also sharp for n > m n>m .

Normal form of equivariant maps in infinite dimensions

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.

Liftable vector fields, unfoldings and augmentations

We study liftable vector fields of smooth map-germs. We show how to obtain the module of liftable vector fields of any map-germ of finite singularity type from the module of liftable vector fields of a stable unfolding of it. As an application, we obtain the liftable vector fields for the family $$H_k$$ H k in Mond’s list. We then show the relation between the liftable vector fields of a stable germ and its augmentations.

Transformations of Closed Invariant Curves and Closed-Invariant-Curve-Like Chaotic Attractors in Piecewise Smooth Systems

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor.

Extreme snow loads in Austria

<p>In the framework of European standards for structural design, acceptable snow loads on constructions and buildings are based on maps for s<sub>k,</sub> the “characteristic snow load on the ground” with an average reoccurrence time of 50 years. The Austrian snow load standard is built on a very detailed zoning map from 2006, but underlying snow data is from the 1980s.</p><p>An updated snow load map for Austria is presented. It is based on 870 snow depth records with at least 30 years of regular daily observations between 1960 and 2019. ΔSNOW, a novel snow model, was used to simulate respective snow loads. Extreme value theory and generalized additive models led to a smooth map of extreme snow loads at 50x50m resolution. The methods are transparently published, reproducible and, thus, applicable in other regions as well.</p><p>The map can reasonably assign s<sub>k</sub> values up to 2000m altitude, a significant advantage compared to actual standards which are only valid up to 1500m. New insights in the spatial picture of extreme snow loads are provided and the quadratic altitude-s<sub>k</sub>-relation, which is widely used in snow load standards, is evaluated. Validation with station data reveals a higher accuracy for the presented map than for the currently used snow load map. The number of outliers, i.d. stations with significantly higher or lower s<sub>k</sub> values than the snow load maps would suggest, could be decreased in comparison with the actual standard. However, some problematic places remain, mostly in topographically and climatologically highly complex areas. In case the presented map will become a new base for future Austrian standards, those places will have to be treated in a special way.</p>

Bach and Einstein’s equations in presence of a field

The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four-dimensional smooth manifold [Formula: see text] in the presence of field [Formula: see text], given by a smooth map with source [Formula: see text] and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called [Formula: see text]-Bach tensor, that in absence of the field [Formula: see text] reduces to the classic Bach tensor, and by the vanishing another tensor related to the bi-energy of [Formula: see text]. Since solutions of the Einstein-massless scalar field equations, or more generally, of the Einstein field equations with source the wave map [Formula: see text] solves those generalized Bach’s equations, we include the latter in our analysis providing a systematic study for them, relying on the recent concept of [Formula: see text]-curvatures. We take the opportunity to discuss the related topic of warped product solutions.

Lagrangian Curve Flows on Symplectic Spaces

A smooth map γ in the symplectic space R2n is Lagrangian if γ,γx,…, γx(2n−1) are linearly independent and the span of γ,γx,…,γx(n−1) is a Lagrangian subspace of R2n. In this paper, we (i) construct a complete set of differential invariants for Lagrangian curves in R2n with respect to the symplectic group Sp(2n), (ii) construct two hierarchies of commuting Hamiltonian Lagrangian curve flows of C-type and A-type, (iii) show that the differential invariants of solutions of Lagrangian curve flows of C-type and A-type are solutions of the Drinfeld-Sokolov’s C^n(1)-KdV flows and A^2n−1(2)-KdV flows respectively, (iv) construct Darboux transforms, Permutability formulas, and scaling transforms, and give an algorithm to construct explicit soliton solutions, (v) give bi-Hamiltonian structures and commuting conservation laws for these curve flows.

Smooth functorial field theories from B-fields and D-branes

In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have boundaries; these are the endpoints of open strings stretched between D-branes. The values of our FFT are obtained from the B-field and its D-branes via transgression. Our construction generalises work of Bunke–Turner–Willerton to include open strings. At the same time, it generalises work of Moore–Segal about open-closed TQFTs to include target spaces. We provide a number of further features of our FFT: we show that it depends functorially on the B-field and the D-branes, we show that it is thin homotopy invariant, and we show that it comes equipped with a positive reflection structure in the sense of Freed–Hopkins. Finally, we describe how our construction is related to the classification of open-closed TQFTs obtained by Lauda–Pfeiffer.

Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.