Topological n-cells and Hilbert cubes in inverse limits

[EN] It has been shown by S. Mardešić that if a compact metrizable space X has dim X ≥ 1 and X is the inverse limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps, then X must contain an arc. We are going to prove that if X = (|Ka|,pba,(A,))is an inverse system in set theory of triangulated polyhedra|Ka|with simplicial bonding functions pba and X = lim X, then there exists a uniquely determined sub-inverse system XX= (|La|, pba|Lb|,(A,)) of X where for each a, La is a subcomplex of Ka, each pba|Lb|:|Lb| → |La| is surjective, and lim XX = X. We shall use this to generalize the Mardešić result by characterizing when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain a topological n-cell and do the same in the case of an inverse system of finite triangulated polyhedra with simplicial bonding maps. We shall also characterize when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain an embedded copy of the Hilbert cube. In each of the above settings, all the polyhedra have the weak topology or all have the metric topology(these topologies being identical when the polyhedra are finite).Rubin, LR. (2018). Topological n-cells and Hilbert cubes in inverse limits. Applied General Topology. 19(1):9-20. doi:10.4995/agt.2018.7061SWORD92019

An approximate inverse system approach to shape fibrations

The notion of shape fibration between compact metric spaces was introduced by S. Mardesic and T. B. Rushing. Mardesic extended the notion to arbitrary topological spaces. A shape fibration f : X → Y between topological spaces is defined by using the notion of resolution (p, q, f) of the map f, where p : X → X and q : Y → Y are polyhedral resolutions of X and Y, respectively, and the approximate homotopy lifting property for the system map f : X → Y. Although any map f : X → Y between topological spaces admits a resolution (p, q, f), if polyhedral resolutions p : X → X and q : Y → Y are chosen in advance, there may not exist a system map f : X → Y so that (p,q,f) is a resolution of f. To overcome this deficiency, T. Watanabe introduced the notion of approximate resolution. An approximate resolution of a map f : X → Y consists of approximate polyhedral resolutions p : X → X and q : Y → Y of X and Y, respectively, and an approximate map f : X → Y. In this paper we obtain the approximate homotopy lifting property for approximate maps and investigate its properties. Moreover, it is shown that the approximate homotopy lifting property is extended to the approximate pro-category and the approximate shape category in the sense of Watanabe. It is also shown that the approximate pro-category together with fibrations defined as morphisms having the approximate homotopy lifting property with respect to arbitrary spaces and weak equivalences defined as morphisms inducing isomorphisms in the pro-homotopy category satisfies the composition axiom for a fibration category in the sense of H. J. Baues. As an application it is shown that shape fibrations can be defined in terms of our approximate homotopy lifting property for approximate maps and that every homeomorphism is a shape fibration

Formal classification of unipotent parameterized diffeomorphisms

We provide a complete system of invariants for the formal classification of complex analytic unipotent germs of diffeomorphism at \cn{n} fixing the orbits of a regular vector field. We reduce the formal classification problem to solve a linear differential equation. Then we describe the formal invariants; their nature depends on the position of the fixed points set $Fix \phi$ with respect to the regular vector field preserved by $\phi$. We get invariants specifically attached to higher dimension ($n \geq 3$) although generically they are analogous to the one-dimensional ones.Comment: 35 page

Geotechnical Characteristics of Bauxite Residue Sand Mixed with Crumbed Rubber from Recycled Car Tires

The re-use of waste materials is an essential step in creating a sustainable future, and research into the re-use of different byproducts has often led to new materials that provide superior service or greater economy than those traditionally used. This paper aims to investigate the use of new mixtures of bauxite residue sand and crumbed rubber from recycled car tires, as fill material. In Australia alone, 20,000 tons of bauxite residue sand is produced daily as a by-product from the Aluminum industry and 20 million recycled car tires are annually disposed, leading to environmental health and hazard problems. In this paper, the geotechnical characteristics of the new mixtures are investigated under static and dynamic loading conditions. A series of laboratory experiments are conducted, including sieve analysis, compaction, permeability and static as well as repeated loading consolidateddrained triaxial tests. In addition, numerical modeling using the finite element method is used to examine the feasibility of the new materials in two geotechnical engineering applications, including slope stability and pavement design. The results indicate that the bauxite residue sand-tire crumb mixtures have a good potential for use as lightweight fill material in geotechnical engineering applications

Reply to the comment on the letter "Geometric Origin of the Tennis Racket Effect"

The author of the comment~[arXiv:2302.04190] criticizes our published results in Phys. Rev. Lett. \textbf{125}, 064301 (2020) about the Tennis Racket Effect (TRE). The TRE is a geometric effect which occurs in the free rotation of any asymmetric rigid body. We explain why the criticism of this comment is not valid.Comment: 3 page

Flags in zero dimensional complete intersections and indices of real vector fields

We introduce bilinear forms in a flag in a complete intersection local $\mathbb R$-algebra of dimension 0, related to the Eisenbud-Levine, Khimshiashvili bilinear form. We give a variational interpretation of these forms in terms of Jantzen's filtration and bilinear forms. We use the signatures of these forms to compute in the real case the constant relating the GSV-index with the signature function of vector fields tangent to an even dimensional hypersurface singularity, one being topologically defined and the other computable by finite dimensional commutative algebra methods.Comment: 17 pages. v2: Some changes in the introduction. A few typos corrected. To appear in Mathematische Zeitschrif