Deformation Retraction Research Articles

Deformation retraction of the group of strict contactomorphisms of the three-sphere to the unitary group

We prove that the group of strict contactomorphisms of the standard tight contact structure on the three-sphere deformation retracts to its unitary subgroup U(2).

Framework for Numerical 6DOF Simulation with Focus on a Wing Deforming UAV in Perch Landing

The perch landing maneuver of a wing-deforming unmanned aerial vehicle (UAV) was investigated through a framework that uses the free, open-source OpenFOAM with 6 degrees of freedom (6DOF) simulations. The framework uses a moving grid to follow the trajectory of the UAV, reducing computational resources. Together with the ability to allow internal grid deformation, sliding mesh, and algorithm addition, it can accurately mimic the entire landing process. Different wing deformation speeds, additional elevator rotation and emulated propeller lift were added to the 6DOF simulations to investigate their effects on the landing maneuver. The results showed that the wing deformation retraction speed has a considerable effect on the trajectory and velocity of the UAV. The wing deformation reduced the forward velocity of the UAV by 32%, from 13.89 to 9 m/s. With the elevator control, the velocity was reduced to 5 m/s. Lastly, and an activation time of 1 s for the emulated propeller lift can further decrease the velocity to around 4.2 m/s. A better algorithm for the emulated propeller lift may be able to give a superior performance. This framework allows us to understand the underlying perch landing maneuver aerodynamics. It can also be used on problems like fast-turning agile and flapping wing flight.

Direct 4D printing of ceramics driven by hydrogel dehydration

4D printing technology combines 3D printing and stimulus-responsive materials, enabling construction of complex 3D objects efficiently. However, unlike smart soft materials, 4D printing of ceramics is a great challenge due to the extremely weak deformability of ceramics. Here, we report a feasible and efficient manufacturing and design approach to realize direct 4D printing of ceramics. Photocurable ceramic elastomer slurry and hydrogel precursor are developed for the fabrication of hydrogel-ceramic laminates via multimaterial digital light processing 3D printing. Flat patterned laminates evolve into complex 3D structures driven by hydrogel dehydration, and then turn into pure ceramics after sintering. Considering the dehydration-induced deformation and sintering-induced shape retraction, we develop a theoretical model to calculate the curvatures of bent laminate and sintered ceramic part. Then, we build a design flow for direct 4D printing of various complex ceramic objects. This approach opens a new avenue for the development of ceramic 4D printing technology.

Lipschitz metric isometries between outer spaces of virtually free groups

Dowdall and Taylor observed that, given a finite-index subgroup of a free group, taking covers induces an embedding from the outer space of the free group to the outer space of the subgroup, this embedding is an isometry with respect to the (asymmetric) Lipschitz metric and that the embedding sends folding paths to folding paths. The purpose of this note is to extend this result to virtually free groups. We further extend a result of Francaviglia and Martino, proving the existence of “candidates” for the Lipschitz distance between points in the outer space of the virtually free group. Additionally, we identify a deformation retraction of the spine of the outer space for the virtually free group with the space considered by Krstić and Vogtmann.

Spherical tropicalization and Berkovich analytification

Let X be a spherical variety. We show that Tevelev and Vogiannou’s tropicalization map from X to its tropicalization factors through the Berkovich analytification $$X^{ \text {an} }$$ , as in the case for toric varieties. Furthermore we show that the tropicalization is a strong deformation retraction of $$X^{ \text {an} }$$ . We also give a strong deformation retraction of Thuillier’s analytification $$X^{\beth }$$ onto a subspace described using the colored fan of X.

Regular points of extremal subsets in Alexandrov spaces

We define regular points of an extremal subset in an Alexandrov space and study their basic properties. We show that a neighborhood of a regular point in an extremal subset is almost isometric to an open subset in Euclidean space and that the set of regular points in an extremal subset has full measure and is dense in it. These results actually hold for strained points in an extremal subset. Applications include the volume convergence of extremal subsets under a noncollapsing convergence of Alexandrov spaces, and the existence of a cone fibration structure of a metric neighborhood of the regular part of an extremal subset. In an appendix, a deformation retraction of a metric neighborhood of a general extremal subset is constructed.

Constructions and Properties of Quasi Sigma-Algebra in Topological Measure Space

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in a Hausdorff topological space based on the locally compactable neighborhood systems without considering strictly (metrized) Borel variety. The null extension gives rise to a quasi sigma-semiring based on sigma-neighborhoods, which are rectifiable in view of Dieudonné measure in n-space. The concepts of symmetric signed measure, uniformly pushforward measure, and its interval-valued Lebesgue variety within a topological measure space are introduced. The symmetric signed measure preserves the total ordering on the real line; however, the collapse of symmetry admits Dieudonné measure within the topological space. The locally constant measures in compact supports in sigma-neighborhood systems are invariant under topological deformation retraction in a simply connected space where the sequence of deformation retractions induces a strongly convergent sequence of measures. Moreover, the extended sigma-structures in an automorphic and isomorphic topological space preserve the properties of uniformly pushforward measure. The Haar-measurable group algebraic structures equivalent to additive integer groups arise under the locally constant and signed measures as long as the topological space is non-compact and the null-extended sigma-neighborhood system admits compact groups. The comparative analyses of the proposed concepts with respect to existing results are outlined.

Cohomology of algebraic varieties over non-archimedean fields

Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma _{\infty }$ , where $\Gamma $ denotes the value group of K. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma _{\infty }$ . In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

New metric structures on [formula omitted]-foliations

In the paper we introduce new metric structures on g-foliations that are less rigid than the well-known structures: almost contact and 3-Sasakian structures as well as f-structures with parallelizable kernel and almost para-ϕ-structures with complemented frames. We discuss the properties of the new structures in order to demonstrate similarities with the corresponding classical structures. Then using the flow of metrics on a g-foliation, we build deformation retraction of our structures with positive partial Ricci curvature onto the subspace of the aforementioned classical structures.

Fundamental Group of LB-Valued General Fuzzy Automata

This study aims to investigate [Formula: see text]-valued GFA from algebraic and topological perspectives, where [Formula: see text] stands for residuated lattice and B is a set of propositions about the general fuzzy automata, in which its underlying structure is a complete infinitely distributive lattice. Further, the concepts of [Formula: see text]-valued general fuzzy automata (or simply [Formula: see text]-valued GFA) contractible spaces, [Formula: see text]-valued GFA path homotopy, [Formula: see text]-valued GFA retraction, [Formula: see text]-valued GFA deformation retraction, [Formula: see text]-valued GFA path connected space and [Formula: see text]-valued GFA homotopy equivalent space are introduced and explicated. In addition, [Formula: see text]-valued GFA fundamental groups are proposed and studied. Regarding these issues, some properties are also established and explained.

On Fuzzy Fundamental Groups and Fuzzy Folding of Fuzzy Minkowski Space

In this paper, we studied the relations between new types of fuzzy retractions, fuzzy foldings, and fuzzy deformation retractions, on fuzzy fundamental groups of the fuzzy Minkowski space M ˜ 4 . These geometrical transformations are used to give a combinatorial characterization of the fundamental groups of fuzzy submanifolds on M ˜ 4 . Then, the fuzzy fundamental groups of the fuzzy geodesics and the limit fuzzy foldings of M ˜ 4 are presented and obtained. Finally, we proved a sequence of theorems concerning the isomorphism between the fuzzy fundamental group and the fuzzy identity group.

A complex Euclidean reflection group with a non-positively curved complement complex

The complement of a hyperplane arrangement in \mathbb{C}^n deformation retracts onto an n -dimensional cell complex, but the known procedures only apply to complexifications of real arrangements (Salvetti) or the cell complex produced depends on an initial choice of coordinates (Björner–Ziegler). In this article we consider the unique complex Euclidean reflection group acting cocompactly by isometries on \mathbb{C}^2 whose linear part is the finite complex reflection group known as G_4 in the Shephard-Todd classification and we construct a choice-free deformation retraction from its hyperplane complement onto a 2 -dimensional complex K where every 2 -cell is a Euclidean equilateral triangle and every vertex link is a Möbius–Kantor graph. The hyperplane complement contains non-regular points, the action of the reflection group on K is not free, and the braid group is not torsion-free. Despite all of this, since K is non-positively curved, the corresponding braid group is a \operatorname{CAT}(0) group.

Investigating compatibility between TPU and bitumen SARA fractions by means of Hansen solubility parameters and interfacial tension measurements

Hansen solubility parameters (HSP) and model oils were used to predict the affinity of each block of the TPU (thermoplastic polyurethanes) with the SARA fractions (Saturates, Aromatics, Resins, Asphaltenes) of bitumen, revealing the preferential compatibility of the bitumen with the soft block of the TPU. More specifically, aromatics are highly compatible with the soft block while resins are compatible with soft and hard blocks of the TPU. Companion SAXS analysis highlighted that the microstructure of TPU is however maintained with semicrystalline hard nanodomains dispersed in a soft matrix with its amorphous part being swollen by resins. Interfacial tension measured by drop deformation retraction method (DDR) highlighted the compatibilizing role of resins as a lower interfacial tension value was measured at high resin content. In turn, different mechanical properties of TPU-modified bitumens were obtained when changing their composition. With similar morphologies for the blends (i.e. continuous TPU phase swollen with the same ratio), a decrease in rubbery storage modulus and flowing temperature was observed as the amount of resins increases.

Topological Analysis of Fibrations in Multidimensional (C, R) Space

A holomorphically fibred space generates locally trivial bundles with positive dimensional fibers. This paper proposes two varieties of fibrations (compact and non-compact) in the non-uniformly scalable quasinormed topological (C, R) space admitting cylindrically symmetric continuous functions. The projective base space is dense, containing a complex plane, and the corresponding surjective fiber projection on the base space can be fixed at any point on real subspace. The contact category fibers support multiple oriented singularities of piecewise continuous functions within the topological space. A composite algebraic operation comprised of continuous linear translation and arithmetic addition generates an associative magma in the non-compact fiber space. The finite translation is continuous on complex planar subspace under non-compact projection. Interestingly, the associative magma resists transforming into a monoid due to the non-commutativity of composite algebraic operation. However, an additive group algebraic structure can be admitted in the fiber space if the fibration is a non-compact variety. Moreover, the projection on base space supports additive group structure, if and only if the planar base space passes through the real origin of the topological (C, R) space. The topological analysis shows that outward deformation retraction is not admissible within the dense topological fiber space. The comparative analysis of the proposed fiber space with respect to Minkowski space and Seifert fiber space illustrates that the group algebraic structures in each fiber spaces are of different varieties. The proposed topological fiber bundles are rigid, preserving sigma-sections as compared to the fiber bundles on manifolds.

Theoretical Study on Prestress Loss in Cross-Tensioned Concrete Pavement with BFRP Tendons

Cross-tensioned concrete pavement can reduce transverse joints and cracks and improve the durability of the pavement, and the decrease in slab thickness can be achieved without damaging the performance of the pavement. However, the corrosion of the steel can cause serious damage to the pavement structure, resulting in higher maintenance costs and shorter service life. Basalt fiber-reinforced polymer (BFRP) has been proven to be an effective alternative in both jointed plain concrete pavement (JPCP) and continuously reinforced concrete pavement (CRCP) due to its lightweight and corrosion-resistant properties. In this paper, a systematic theoretical method for determining the prestress loss of BFRP tendons in cross-tensioned concrete pavement was proposed, with the impact of the slab width and distribution angle of the prestressed tendon on the prestress loss being studied and compared to the results of traditional steel strands. Results showed that the proportion of the prestress loss due to anchorage deformation and prestress retraction in the prestressing stage rose with the increase in distribution angle and the decrease in slab width, while the prestress loss during the in-service stage was a constant value for both BFRP tendons and steel strands. The prestress loss of BFRP tendons was far lower than that of steel strands in both prestressing stage and in-service stage for a given slab width (3 m, 4.5 m, 9.0 m, 12.75 m) and distribution angle (20°, 25°, 30°, 35°, 40°, 45°), and the difference ranged from 6.4% to 16%, signifying the feasibility of BFRP tendons in cross-tensioned concrete pavement. Overall, the smaller the slab width, the greater the difference of the prestress loss between BFRP tendons and steel strands.

On the Deformation Retractions of Frenet Curves in Minkowski 4 - Space

In this paper, the position vector equation of   the Frenet curves with constant curvatures in Minkowski 4 -space has been presented. New types for retractions and deformation retracts of Frenet curves in  are deduced. The relations between the Frenet apparatus of the Frenet curves before and after the deformation retracts are obtained.

Fixed poin sets in digital topology, 1

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

Orbit configuration spaces of small covers and quasi-toric manifolds

In this article, we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology, such as small covers, quasi-toric manifolds and (real) moment-angle manifolds; especially for the cases of small covers and quasi-toric manifolds. These kinds of orbit configuration spaces have non-free group actions, and they are all noncompact, but still built via simple convex polytopes. We obtain an explicit formula of the Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope. As a by-product of our method, we also obtain a formula of the Euler characteristic for the classical configuration space, which generalizes the Felix-Thomas formula. In addition, we also study the homotopy type of such orbit configuration spaces. In particular, we determine an equivariant strong deformation retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold, which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.

A Riemann–Hurwitz formula for skeleta in non-Archimedean geometry

Let k be an algebraically closed non-Archimedean non-trivially real valued field which is complete with respect to its valuation. Let \(\phi :C' \rightarrow C\) be a finite morphism between smooth projective irreducible k-curves. The morphism \(\phi \) induces a morphism \(\phi ^{\mathrm {an}} :C'^{\mathrm {an}} \rightarrow C^{\mathrm {an}}\) between the Berkovich analytifications of the curves. We construct a pair of deformation retractions of \(C'^{\mathrm {an}}\) and \(C^{\mathrm {an}}\) which are compatible with the morphism \(\phi ^{\mathrm {an}}\) and whose images \(\Sigma _{C'^{\mathrm {\mathrm {an}}}},\Sigma _{C^{\mathrm {\mathrm {an}}}}\) are closed subspaces of that are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta. In addition, the subspaces \(\Sigma _{C'^{\mathrm {\mathrm {an}}}}\) and \(\Sigma _{C^{\mathrm {\mathrm {an}}}}\) are such that their complements in their respective analytifications decompose into the disjoint union of isomorphic copies of Berkovich open disks. The pair of compatible deformation retractions forces the morphism \(\phi ^{\mathrm {an}}\) to restrict to a map \(\Sigma _{C'^{\mathrm {\mathrm {an}}}} \rightarrow \Sigma _{C^{\mathrm {\mathrm {an}}}}\). The first Betti number of the skeleton C is well defined and an invariant of the curve which we call \(g^{\mathrm {an}}(C)\). We study how the genus of \(\Sigma _{C'^{\mathrm {\mathrm {an}}}}\) can be calculated using the morphism \(\phi ^{\mathrm {an}}_{|\Sigma _{C'^{\mathrm {\mathrm {an}}}}}\) and invariants defined on \(\Sigma _{C^{\mathrm {an}}}\) via the deformation retractions.

The topology of Baumslag–Solitar representations

Let [Formula: see text] be a Baumslag–Solitar group and let [Formula: see text] be a complex reductive algebraic group with maximal compact subgroup [Formula: see text]. We show that, when [Formula: see text] and [Formula: see text] are relatively prime with distinct absolute values, there is a strong deformation retraction of [Formula: see text] onto [Formula: see text].