## $For a non-trivial homology class of lowest dimension in the space of loops based at a point$p$or in the free loop space one can define a critical length${\sf crl}_p\left(M,g\right)$resp. Critical values of homology classes of loops and positive curvature ## We prove that the string topology bracket on the$S^1$-equivariant homology$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $of the free loop space of X preserves the Hodge decomposition of$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $, making it a bigraded Lie algebra. Hodge decomposition of string topology ## Loop Space sentence examples within loop space maintainer ## ABSTRACT Objective of this study to was to evaluate the effect of the zirconia band and loop space maintainers on the salivary level of lactobacillus and streptococcus mutans. The Effect Of Zirconia Band And Loop Space Maintainer On Salivary Level Of Lactobacillus And Streptococcus Mutans. ## The band and loop space maintainer is one of the most widely used space maintainer, however depending on the individual cases and anatomy of the supporting structures the traditional band and loop space maintainers can be modified to meet the individual requirements. Band and Loop Space Maintainers in Children: A Case Series 10.1515/ms-2021-0019 ## In this paper, by reviewing the concept of homotopy groups and quotient maps, we find under which conditions the map q × q is a quotient map, where q: Ωn(X, x0) → πn(X, x0), is the natural quotient map from the nth loop space of (X, x0), Ωn(X, x0), with compact-open topology to the quasitopological nth homotopy group πn(X, x0). On first countable quasitopological homotopy groups 10.21474/ijar01/13513 ## In this clinical scenario, we have modified the band and loop spacemaintainer with a pontic to make it a functional space maintainer. BAND AND PONTIC: A MODIFIED BAND AND LOOP SPACE MAINTAINER - CLINICAL INNOVATION 10.1017/S0013091521000274 ## Then, for the loop space of the associated$\mathbb {S}^{2m-1}_{(p)}$-projective space$\mathbb {S}^{2m-1}_{(p)}P(n-1)$, with$m,n\ge 2$and$m\mid p-1$, we derive that$\mbox {nil}\ \Omega (\mathbb {S}^{2m-1}_{(p)}P (n-1))\le 3$. Homotopy nilpotency of localized spheres and projective spaces 10.21608/AJDSM.2020.51738.1140 ## ABSTRACT Objective of this study to was to evaluate the effect of the zirconia band and loop space maintainers on the salivary level of lactobacillus and streptococcus mutans. The Effect Of Zirconia Band And Loop Space Maintainer On Salivary Level Of Lactobacillus And Streptococcus Mutans. 10.1016/J.JSSE.2021.07.005 ## This aims to improve operator visualisation of the surgical field, whilst ensuring safety through maintaining a sterile surgical site with clear plastic and preventing contamination of the closed-loop spacecraft atmosphere. Microgravity surgical workstation design 10.1007/S00229-021-01273-Y ## We study the homotopy nilpotency of the loop spaces $$\Omega (G_{n,m}({\mathbb {K}}))$$ , $$\Omega (F_{n;n_1,\ldots ,n_k}({\mathbb {K}}))$$ , and $$\Omega (V_{n,m}({\mathbb {K}}))$$ of Grassmann $$G_{n,m}({\mathbb {K}})$$ , flag $$F_{n;n_1,\ldots ,n_k}({\mathbb {K}})$$ and Stiefel $$V_{n,m}({\mathbb {K}})$$ manifolds. Homotopy nilpotency of some homogeneous spaces 10.4153/S000843952100028X ## We show that the loop space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www. On homotopy nilpotency of loop spaces of Moore spaces 10.2140/akt.2021.6.425 ## In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod$p$homology of$\Omega(BG^\wedge_p)$, when$G$is a finite group,$BG^\wedge_p$is the$p$-completion of its classifying space, and$\Omega(BG^\wedge_p)$is the loop space of$BG^\wedge_p$. Loop space homology of a small category 10.1109/aero50100.2021.9438342 ## While we were unable to conduct an on-board experiment due to the premature loss of ASTERIA, our effort proved the feasibility of on-board model-based fault management, demonstrating reliable and accurate diagnosis using captured data, and further supporting a closed-loop spacecraft autonomy demonstration including autonomous navigation in off-nominal conditions. On-Board Model Based Fault Diagnosis for CubeSat Attitude Control Subsystem: Flight Data Results 10.36348/sjodr.2021.v06i02.004 ## The band and loop space maintainer is one of the most widely used space maintainer, however depending on the individual cases and anatomy of the supporting structures the traditional band and loop space maintainers can be modified to meet the individual requirements. Band and Loop Space Maintainers in Children: A Case Series 10.4310/jdg/1631124316 ##$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define a critical length ${\sf crl}_p\left(M,g\right)$ resp.

Critical values of homology classes of loops and positive curvature

10.4310/cjm.2021.v9.n2.a3

## We prove a recognition principle for motivic infinite P1-loop spaces over an infinite perfect field.

Motivic infinite loop spaces

10.4103/tdj.tdj_24_20

## Aims This work aimed to assess the salivary and urinary levels of nickel and chromium ions in children with stainless steel crowns and band and loop space maintainers.

Metal ions release in children with stainless steel crowns and banded space maintainers

10.47750/CIBG.2021.27.02.304

## This study aims to determine the prevalence of band and loop space maintainers among children who are 6-10 years old.

Band and Loop Space Maintainer Among Children Between the Age Group Of 6-10 Years - A Retrospective Study

10.1017/fms.2021.26

## We prove that the string topology bracket on the $S^1$-equivariant homology ${\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}})$ of the free loop space of X preserves the Hodge decomposition of ${\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}})$, making it a bigraded Lie algebra.

Hodge decomposition of string topology

10.17796/1053-4625-45.3.1

## Band and loop space maintainers are indicated whenever there is premature loss of primary molar.

3D Printed Band and Loop Space Maintainer: A Digital Game Changer in Preventive Orthodontics.

10.1007/s10958-021-05499-9

## An introduction of a quasi-scalar product, an orthonormal system, and applications in physics (path integral, loop space, functional derivative) are proposed.

Notes on Functional Integration

10.14260/jemds/2021/512

## Hence, the objective of this review was to compare the efficiency between fiber reinforced composite (FRC) space maintainer and the conventional band & loop space maintainer.

Fiber Reinforced Composite Space Maintainers Versus Band and Loop Space Maintainers – A Systematic Review

10.1177/09596518211013178

## To achieve the finite-time convenience of the closed-loop space unmanned system, the fractional control laws have been designed, and an important lemma has been utilized to design the update laws of the adaptive parameters.

Super-twisting disturbance observer–based fuzzy adaptive finite-time control for a class of space unmanned systems with time-varying output constraints

10.1007/s00209-021-02838-z

## This proposes a new $$A_\infty$$ A ∞ model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.

The Milnor-Moore theorem for $$L_\infty$$ algebras in rational homotopy theory

10.1007/s40590-021-00311-9

## In this note, we obtain two families of algebraic numbers which can appear as the radii of convergence of Poincaré series of loop spaces of simply connected finite CW-complexes.

Algebraic equation and the radius of convergence of Poincaré series

10.1007/s11253-021-01898-y

## It is shown that the nth quasitopological homotopy group of a topological space is isomorphic to the (n − 1)th quasitopological homotopy group of its loop space.

Some Results in Quasitopological Homotopy Groups

10.1093/QMATH/HAAB022

## A discussion about how such forms and their constructions and cohomology relate to constructions for diffusion measures on path and loop spaces is also included.

Infinite-Dimensional Degree Theory and Ramer’S Finite Co-Dimensional Differential Forms

10.1016/J.GEOMPHYS.2021.104170

## In the first case, we consider a spin manifold M as the set of fixed points of an S 1 -action on a spin manifold X , and in the second case we consider the spin manifold M as the set of fixed points of an S 1 -action on the loop space of M.

An analytical approach to the equivariant index and Witten genus on spin manifolds

10.1112/S0010437X21007363

## We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space.

A construction of complex analytic elliptic cohomology from double free loop spaces

10.1007/S10489-020-02158-5

## Upcoming space missions are requiring a higher degree of on-board autonomy operations to increase quality science return, to minimize closed-loop space-ground decision making, and to enable new scenarios.

On-board autonomy operations for OPS-SAT experiment

10.15406/paij.2019.03.00152

## We present several clarifying comments on the loop space-self avoid string representation for Q.

On loop space-self avoiding string representations for Q.C.D(SU(∞))

10.5455/IJMRCR.DENTAL-FRANKEL-NEGATIVE-BEHAVIOUR

## Pulpectomy of multiple teeth was performed along with the fabrication of a band and loop space maintainer and aesthetic space maintainer.

Dental intervention using GA for managing a child with Frankel’s definitely negative behavior -

## Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit.

Asymptotic Behavior of Memristive Circuits

## Starting from a dynamical systems formulation of the motion of parameterized loops in a charged particle's phase space, I identify a slow manifold in loop space.

Guiding center dynamics as motion on a slow manifold in loop space

10.1016/j.aim.2019.03.022

## Work of Mitchell and Richter proves that this based loop space stably splits as an infinite wedge sum.

Multiplicative structure in the stable splitting of ΩSLn(C)

10.1007/s00013-020-01485-w

## We prove that generalized loop spaces of Hartogs manifolds are Hilbert–Hartogs.

Loop spaces as Hilbert–Hartogs manifolds

10.1016/j.aim.2019.106860

## This suggests the existence of putative rotation-equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera.

Projective elliptic genera and elliptic pseudodifferential genera

10.4310/hha.2021.v23.n1.a2

## We compute the integral cohomology ring of the classifying spaces of gauge groups of principal U(n)-bundles over the 2-sphere by generalizing the operation for free loop spaces, called the free double suspension.

Cohomology of the classifying spaces of $U(n)$-gauge groups over the 2-sphere

10.1134/s199508021909018x

## These manifolds are also closely related to string theory because they can be realized as the spaces of complex structures on loop spaces.

On Kähler Geometry of Infinite-dimensional Complex Manifolds Diff+(S1)/S1 and Diff+(S1)/Möb(S1)

10.1007/s40879-019-00396-7

## We verify that for a finite simplicial complex $X$ and for piecewise linear loops on $X$, the "thin" loop space is a topological group of the same homotopy type as the space of continuous loops.

Thin Loop Groups

10.1155/2019/4312049

## The band and loop space maintainer is used in majority of patients requiring single tooth space maintenance in both primary and mixed dentitions.

Functional Band and Loop Space Maintainers in Children

## In the first chapter, we show how the conjecture in the case of Grassmannians arises from Givental’s loop space mirror heuristics.

Quiver flag varieties and mirror symmetry

10.1016/J.TOPOL.2019.04.011

## In this paper, considering the kth shape loop space Ω ˇ k p ( X , x ) , for an HPol ⁎ -expansion p : ( X , x ) → ( ( X λ , x λ ) , [ p λ λ ′ ] , Λ ) of a pointed topological space ( X , x ) , first we show that Ω ˇ k p ( X , x ) is an H-group for every topological space ( X , x ).

On shape loop spaces

10.1007/978-3-030-22999-3_17

## Upcoming space missions are requiring a higher degree of on-board autonomy operations to increase quality science return, to minimize close-loop space-ground decision making, and to enable new scenarios.

On Board Autonomy Operations for OPS-SAT Experiment

10.1016/j.indag.2020.08.002

## As a consequence, we deduce that the bar complex of the original de Rham complex of a simply-connected diffeological space is quasi-isomorphic to the singular de Rham complex of the diffeological free loop space provided the factor map for the underlying diffeological space is a quasi-isomorphism.

Simplicial cochain algebras for diffeological spaces

10.1017/s001309152000019x

## For primes $p \ge 5$, work of Selick shows that $S^{2n+1}\{p\}$ admits a nontrivial loop space decomposition if and only if $n=1$ or $p$.

The fibre of the degree $3$ map, Anick spaces and the double suspension.

10.33773/JUM.506510

## Then some properties of intuitionistic fuzzy  loop spaces are investigated Finally it is shown that an intuitionistic fuzzy loop  space is an H-group.

H-GROUP STRUCTURE ON INTUITIONISTIC FUZZY TOPOLOGICAL SPACE

10.1007/978-981-13-5742-8_8

## Chen introduced such a space as a differentiable space in his study of a loop space to employ the idea of iterated path integrals [2, 3, 4, 5].

Mayer–vietoris sequence for differentiable/diffeological spaces

10.1007/978-981-13-5742-8_2

## The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.

The Homotopy Type of the Loops on $$(n-1)$$-Connected $$(2n+1)$$-Manifolds

10.1017/S1474748019000720

## In joint work with Elmanto, Hoyois, Khan and Sosnilo, we computed infinite $\mathbb{P}^1$-loop spaces of motivic Thom spectra, using the technique of framed correspondences.

The unit map of the algebraic special linear cobordism spectrum

10.1016/j.geomphys.2019.02.012

## We construct a family of Poisson structures of hydrodynamic type on the loop space of ℂ P n − 1.

Poisson structures on loop spaces of ℂPn and an r-matrix associated with the universal elliptic curve