Loop Space Research Articles

Performance of Network-Based RTK GNSS for the Cadastral Survey in Thailand

In Thailand, a network of GNSS Continuously Operating Reference Station (CORS) has been operating as a Network-based Real Time Kinematic (NRTK) GNSS positioning solutions since 2008. The Department of Lands designs a Thai-NRTK system mainly for GNSS cadastral survey using the Virtual Reference Station (VRS) as its positioning concept. The current Thai-NRTK consists of 114 GNSS CORSs covering only some parts of Thailand. The CORS spacing ranges from 25.1 km to198.6 km. According to previous studies, the performance of NRTK-GPS positioning determination in Thailand is strongly depending on the dimension of the triangular CORS locations; known as a “loop”. This research aims to examine the horizontal position performance of this Thai-NRTK GNSS positioning solutions as a quality indicator for GNSS cadastral surveying. The test was carried out with quite a large number of test samples (total of 2,122 points) and where every loop is separated into 4 test groups with differing the station distances starting from the loop spacing of: 30 km to 50 km, 50 km to 70 km, 70 km to 90 km and 90 km to 110 km, respectively. The positioning performance test is conducted in two main folds; namely, positioning accuracy and time to resolve its solutions. On positioning accuracy, the test results show that the spacing of every tested loop achieves the accuracy in accordance with the cadastral surveying requirements. This proves that the Thai-NRTK GNSS positioning network and technique is well-designed and suitable for cadastral positioning within its coverage areas. On time to resolve fixed solution, the test results indicate that the ‘time to first fixed’ and the ‘rate of the horizontal position jump’ increase with longer loop distance; whereby only loop length of 30 km to 50 km has 100% success ‘rate of ambiguity-fixing’ solutions; therefore, in order to achieve a high reliable performance positioning from the Thai-NRTK GNSS solutions for cadastral survey in Thailand, it is recommended that the reference stations shall be kept within 50 km.

A reinforcement-learning-based approach to enhance exhaustive protein loop sampling.

Loop portions in proteins are involved in many molecular interaction processes. They often exhibit a high degree of flexibility, which can be essential for their function. However, molecular modeling approaches usually represent loops using a single conformation. Although this conformation may correspond to a (meta-)stable state, it does not always provide a realistic representation. In this paper, we propose a method to exhaustively sample the conformational space of protein loops. It exploits structural information encoded in a large library of three-residue fragments, and enforces loop-closure using a closed-form inverse kinematics solver. A novel reinforcement-learning-based approach is applied to accelerate sampling while preserving diversity. The performance of our method is showcased on benchmark datasets involving 9-, 12- and 15-residue loops. In addition, more detailed results presented for streptavidin illustrate the ability of the method to exhaustively sample the conformational space of loops presenting several meta-stable conformations. We are developing a software package called MoMA (for Molecular Motion Algorithms), which includes modeling tools and algorithms to sample conformations and transition paths of biomolecules, including the application described in this work. The binaries can be provided upon request and a web application will also be implemented in the short future. Supplementary data are available at Bioinformatics online.

Formality of derived intersections and the orbifold HKR isomorphism

We study when the derived intersection of two smooth subvarieties of a smooth variety is formal. As a consequence we obtain a derived base change theorem for non-transversal intersections. We also obtain applications to the study of the derived fixed locus of a finite group action and argue that for a global quotient orbifold the exponential map is an isomorphism between the Lie algebra of the free loop space and the loop space itself. This allows us to give new proofs of the HKR decomposition of orbifold Hochschild (co)homology into twisted sectors.

Asymptotic Behavior of Memristive Circuits.

The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence that also purely memristive circuits can be employed for computational purposes. In the present paper we show that a polynomial Lyapunov function in the memory parameters exists for the case of DC controlled memristors. Such a Lyapunov function can be asymptotically approximated with binary variables, and mapped to quadratic combinatorial optimization problems. This also shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we show numerically that the distribution of the matrix elements of the projectors can be roughly approximated with a Gaussian distribution, and that it scales with the inverse square root of the number of elements. This provides an approximated but direct connection with the physics of disordered system and, in particular, of mean field spin glasses. Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit. We estimate the number of stationary points of the approximate Lyapunov function and provide a scaling formula as an upper bound in terms of the circuit topology only.

On an Algebra Structure on the Hochschild Homology of Simply Connected Topological Space

In this note, we use the pairing induced by the interchange map in conjunction with the strongly homotopy commutative algebra structure to define products on the Eilenberg–Moore differential Tor and give a simplified proof of an improved outcome of Jones’s result due to Ndombol and Thomas. As a result, we establish an isomorphism of graded algebras between the Hochschild homology and the free loop space cohomology of a simply connected topological space.

Koszul A∞-algebras and free loop space homology

AbstractWe introduce a notion of Koszul A∞-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A∞-algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.

Adiabatic Limit in Yang–Mills Equations in R⁴

Our goal is to present an approach to the proof of the harmonic spheres conjecture based on the adiabatic limit construction. This construction allows to associate with an arbitrary Yang–Mills G-field on the Euclidean 4-dimensional space a harmonic map of the Riemann sphere to the loop space of the group G

The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form $$S^{2n+1}/ \Gamma $$

Let $$M=S^{2n+1}/ \Gamma $$ , $$\Gamma $$ is a finite group which acts freely and isometrically on the $$(2n+1)$$ -sphere and therefore M is diffeomorphic to a compact space form. In this paper, we first investigate Katok’s famous example about irreversible Finsler metrics on the spheres to study the topological structure of the contractible component of the free loop space on the compact space form M, then we apply the result to establish the resonance identity for homologically visible contractible minimal closed geodesics on every Finsler compact space form (M, F) when there exist only finitely many distinct contractible minimal closed geodesics on (M, F). As its applications, using this identity and the enhanced common index jump theorem for symplectic paths proved by Duan et al. (Calc Var PDEs 55(6):55–145, 2016), we show that there exist at least $$2n+2$$ distinct closed geodesics on every compact space form $$S^{2n+1}/ \Gamma $$ with a bumpy irreversible Finsler metric F under some natural curvature condition, which is the optimal lower bound due to Katok’s example.

Infinite loop spaces and positive scalar curvature in the presence of a fundamental group

This is a continuation of our previous work with Botvinnik on the nontriviality of the secondary index invariant on spaces of metrics of positive scalar curvature, in which we take the fundamental group of the manifolds into account. We show that the secondary index invariant associated to the vanishing of the Rosenberg index can be highly nontrivial, for positive scalar curvature Spin manifolds with torsionfree fundamental groups which satisfy the Baum--Connes conjecture. For example, we produce a compact Spin 6-manifold such that its space of positive scalar curvature metrics has each rational homotopy group infinite dimensional. At a more technical level, we introduce the notion of "stable metrics" and prove a basic existence theorem for them, which generalises the Gromov--Lawson surgery technique, and we also give a method for rounding corners of manifold with positive scalar curvature metrics.

Homotopy properties of horizontal loop spaces and applications to closed sub-Riemannian geodesics

Given a manifold M M and a proper sub-bundle Δ ⊂ T M \Delta \subset TM , we investigate homotopy properties of the horizontal free loop space Λ \Lambda , i.e., the space of absolutely continuous maps γ : S 1 → M \gamma :S^1\to M whose velocities are constrained to Δ \Delta (for example: legendrian knots in a contact manifold). In the first part of the paper we prove that the base-point map F : Λ → M F:\Lambda \to M (the map associating to every loop its base-point) is a Hurewicz fibration for the W 1 , 2 W^{1,2} topology on Λ \Lambda . Using this result we show that, even if the space Λ \Lambda might have deep singularities (for example: constant loops form a singular manifold homeomorphic to M M ), its homotopy can be controlled nicely. In particular we prove that Λ \Lambda (with the W 1 , 2 W^{1,2} topology) has the homotopy type of a CW-complex, that its inclusion in the standard free loop space (i.e., the space of loops with no non-holonomic constraint) is a homotopy equivalence, and consequently that its homotopy groups can be computed as π k ( Λ ) ≃ π k ( M ) ⋉ π k + 1 ( M ) \pi _k(\Lambda )\simeq \pi _k(M) \ltimes \pi _{k+1}(M) for all k ≥ 0. k\geq 0. In the second part of the paper we address the problem of the existence of closed sub-Riemannian geodesics. In the general case we prove that if ( M , Δ ) (M, \Delta ) is a compact sub-Riemannian manifold, each non-trivial homotopy class in π 1 ( M ) \pi _1(M) can be represented by a closed sub-Riemannian geodesic. In the contact case, we prove a min-max result generalizing the celebrated Lyusternik-Fet theorem: if ( M , Δ ) (M, \Delta ) is a compact, contact manifold, then every sub-Riemannian metric on Δ \Delta carries at least one closed sub-Riemannian geodesic. This result is based on a combination of the above topological results with the delicate study of an analogue of a Palais-Smale condition in the vicinity of abnormal loops (singular points of Λ \Lambda ).

On shape loop spaces

In this paper, considering the kth shape loop space Ωˇkp(X,x), for an HPol⁎-expansion p:(X,x)→((Xλ,xλ),[pλλ′],Λ) of a pointed topological space (X,x), first we show that Ωˇkp(X,x) is an H-group for every topological space (X,x). Then we prove that Ωˇkc:Top⁎⟶Top⁎ is a functor, for all k∈N0, where c is the Čech HPol⁎-expansion of spaces. Moreover, with some assumptions, we show that if X is πk-shape injective, then X is k-homotopically Hausdorff. Finally, we show that the long exact sequence of shape groups (coarse shape groups) of a pointed pair of certain spaces, is also exact as topological groups.

Optimization of the magnetic field homogeneity of circular and conical coil pairs.

The magnetic field generated by a pair of coaxial circular loops is analyzed in order to find the optimum separation between the loops for each value of a prescribed homogeneity. For the maximum heterogeneities considered here, 0.1%, 1%, and 10%, the optimum loop spacing is obtained following a graphical procedure so that the length of the homogeneous field region along the axis is the longest possible. This study is extended to regions near the axis and to the entire region surrounding the center, calculating the loop separation that produces the largest homogeneous volume. The field homogeneity of a pair of "Helmholtz" coaxial conical coils is also investigated to obtain the optimum spacing between the conical coils; the volumes calculated with the desired homogeneity are compared with those obtained with a single pair of optimum coaxial loops. A new arrangement is hereby proposed based on double conical coils with optimum separation between the coils and optimum aperture of the cones. Some of the proposed arrangements are carried out in the laboratory where the experimental results are in good agreement with the theoretical predictions.

Homotopy groups of certain highly connected manifolds via loop space homology

For $n\geq 2$ we consider $(n-1)$-connected closed manifolds of dimension at most $(3n-2)$. We prove that away from a finite set of primes, the $p$-local homotopy groups of $M$ are determined by the dimension of the space of indecomposable elements in the cohomology ring $H^\ast(M; \mathbb{Q})$. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. This generalizes some of the results of our earlier work [1].

Multiplicative structure in the stable splitting of [formula omitted

The space of based loops in SLn(C), also known as the affine Grassmannian of SLn(C), admits an E2 or fusion product. Work of Mitchell and Richter proves that this based loop space stably splits as an infinite wedge sum. We prove that the Mitchell–Richter splitting is coherently multiplicative, but not E2. Nonetheless, we show that the splitting becomes E2 after base-change to complex cobordism. Our proof of the A∞ splitting involves on the one hand an analysis of the multiplicative properties of Weiss calculus, and on the other a use of Beilinson–Drinfeld Grassmannians to verify a conjecture of Mahowald and Richter. Other results are obtained by explicit, obstruction-theoretic computations.

Poisson structures on loop spaces of [formula omitted] and an [formula omitted]-matrix associated with the universal elliptic curve

We construct a family of Poisson structures of hydrodynamic type on the loop space of ℂPn−1. This family is parametrized by the moduli space of elliptic curves or, in other words, by the modular parameter τ. This family can be lifted to a homogeneous Poisson structure on the loop space of ℂn but in order to do that we need to upgrade the modular parameter τ to an additional field τ(x) with Poisson brackets {τ(x),τ(y)}=0,{τ(x),za(y)}=2πiza(y)δ′(x−y) where z1,…,zn are coordinates on ℂn. These homogeneous Poisson structures can be written in terms of an elliptic r-matrix of hydrodynamic type.

English

English

Filtered finiteness of the image of the unstable Hurewicz homomorphism with applications to bordism of immersions

AbstractAfter recent work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem [M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, Ann. Math. (2), 184 (2016), 1–262], as well as Adams’ solution of the Hopf invariant one problem [J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. (2), 72 (1960), 20–104], an immediate consequence of Curtis conjecture is that the set of spherical classes in H∗Q0S0 is finite. Similarly, Eccles conjecture, when specialized to X=Sn with n> 0, together with Adams’ Hopf invariant one theorem, implies that the set of spherical classes in H∗QSn is finite. We prove a filtered version of the above finiteness properties. We show that if X is an arbitrary CW-complex of finite type such that for some n, HiX≃0 for any i>n, then the image of the composition π∗ΩlΣl+2X→π∗QΣ2X→H∗QΣ2X is finite; the finiteness remains valid if we formally replace X with S−1. As an application, we provide a lower bound on the dimension of the sphere of origin on the potential classes of π∗QSn which are detected by homology. We derive a filtered finiteness property for the image of certain transfer maps ΣdimgBG+→QS0 in homology. As an application to bordism theory, we show that for any codimension k framed immersion f:M↬ℝn+k which extends to an embedding M→ℝd×ℝn+k, if n is very large with respect to d and k then the manifold M as well as its self-intersection manifolds are boundaries. Some results of this paper extend results of Hadi [Spherical classes in some finite loop spaces of spheres. Topol. Appl., 224 (2017), 1–18] and offer corrections to some minor computational mistakes, hence providing corrected upper bounds on the dimension of spherical classes H∗ΩlSn+l. All of our results are obtained at the prime p = 2.

E2 structures and derived Koszul duality in string topology

We construct an equivalence of $E_{2}$ algebras between two models for the Thom spectrum of the free loop space that are related by derived Koszul duality. To do this, we describe the functoriality and invariance properties of topological Hochschild cohomology.

H-GROUP STRUCTURE ON INTUITIONISTIC FUZZY TOPOLOGICAL SPACE

In this study we investigate some properties of the pointed intuitionistic fuzzy topological spaces. Then some properties of intuitionistic fuzzy loop spaces are investigated Finally it is shown that an intuitionistic fuzzy loop space is an H-group.

Homotopy types of gauge groups related to S3-bundles over S4

Let Ml,m be the total space of the S3-bundle over S4 classified by the element lσ+mρ∈π4(SO(4)), l,m∈Z. In this paper we study the homotopy theory of gauge groups of principal G-bundles over manifolds Ml,m when G is a simply connected simple compact Lie group such that π6(G)=0. That is, G is one of the following groups: SU(n)(n≥4), Sp(n)(n≥2), Spin(n)(n≥5), F4, E6, E7, E8. If the integral homology of Ml,m is torsion-free, we describe the homotopy type of the gauge groups over Ml,m as products of recognisable spaces. For any manifold Ml,m with non-torsion-free homology, we give a p-local homotopy decomposition, for a prime p≥5, of the loop space of the gauge groups.