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In this note, we study the delooping of spaces and maps in homotopy type theory. We show that in some cases, spaces have a unique delooping, and give a simple description of the delooping in these cases. We explain why some maps, such as group homomorphisms, have a unique delooping. We discuss some applications to Eilenberg–MacLane spaces and cohomology. [ABSTRACT FROM AUTHOR]

For dimensions n ≥ 3 and k ∈ { 2 , ... , n } , we show that the space of metrics of k -positive Ricci curvature on the sphere S n has the structure of an H -space with a homotopy commutative, homotopy associative product operation. We further show, using the theory of operads and results of Boardman, Vogt and May, that the path component of this space containing the round metric is weakly homotopy equivalent to an n -fold loop space. [ABSTRACT FROM AUTHOR]

W. Goldman and V. Turaev defined a Lie bialgebra structure on the Z -module generated by free homotopy classes of loops of an oriented surface (i.e. the conjugacy classes of its fundamental group). We generalize this construction to a much larger space of equivalence classes of curves by replacing homotopies by thin homotopies, following the combinatorial approach of M. Chas. As an application we use properties of the generalized bracket to give a geometric proof of a conjecture by Chas in the original setting of full homotopy classes, namely a characterization of homotopy classes of simple curves in terms of the Goldman–Turaev bracket. [ABSTRACT FROM AUTHOR]

We construct a differential graded algebra (DGA) modelling certain A ∞ algebras associated with a finite group G with cyclic Sylow subgroups, namely H∗BG and H ∗ Ω B G p ∧ . We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander–Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups. [ABSTRACT FROM AUTHOR]

We describe two major string topology operations, the Chas–Sullivan product and the Goresky–Hingston coproduct, from geometric and algebraic perspectives. The geometric construction uses Thom–Pontrjagin intersection theory while the algebraic construction is phrased in terms of Hochschild homology. We give computations of products and coproducts on lens spaces via geometric intersection, and deduce that the coproduct distinguishes 3-dimensional lens spaces. Algebraically, we describe the structure these operations define together on the Tate–Hochschild complex. We use rational homotopy theory methods to sketch the equivalence between the geometric and algebraic definitions for simply-connected manifolds and real coefficients, emphasizing the role of configuration spaces. Finally, we study invariance properties of the operations, both algebraically and geometrically. [ABSTRACT FROM AUTHOR]

We provide a simple condition on rational cohomology for the total space of a pullback fibration over a connected sum to have the rational homotopy type of a connected sum after looping. This takes inspiration from a recent work of Jeffrey and Selick, in which they study pullback fibrations of this type but under stronger hypotheses compared to our result. [ABSTRACT FROM AUTHOR]

Let X be a closed subspace of a metric space M. It is well known that, under mild hypotheses, one can estimate the Betti numbers of X from a finite set P ⊂ M of points approximating X. In this paper, we show that one can also use P to estimate much more detailed topological properties of X. We achieve this by proving the stability of A ∞ -persistent homology. In its most general case, this stability means that given a continuous function f : Y → R on a topological space Y, small perturbations in the function f imply at most small perturbations in the family of A ∞ -barcodes. This work can be viewed as a proof of the stability of cup-product and generalized-Massey-products persistence. The technical key of this paper consists of figuring out a setting which makes A ∞ -persistence functorial. [ABSTRACT FROM AUTHOR]

In this paper, considering the kth shape loop space ... p k (X, x), for an HPol∗-expansion p : (X, x) → ((Xλ, xλ), [pλλ′ ], Λ) of a pointed topological space (X, x), first we prove that ... k commutes with the product under some conditions and then we show that ... p k (X, x) ≅= lim← ... p k (Xi, xi), for a pro-discrete space (X, x) = lim← (Xi, xi) of compact polyhedra. Finally, we conclude that these spaces are metric, second countable and separable. [ABSTRACT FROM AUTHOR]

We explain our previous results about Hochschild actions (2007/2008) pertaining, in particular, to the coproduct, which appeared in a different form in a work by Goresky and Hingston (2009), and provide a fresh look at the results. We recall the general action, specialize to the aforementioned coproduct and prove that the assumption of commutativity, made for convenience in a previous article (2008), is not needed. We give detailed background material on loop spaces, Hochschild complexes and dualizations, and discuss details and extensions of these techniques which work for all operations given in two previous articles (2007/2008). With respect to loop spaces, we show that the coproduct is well defined modulo constant loops and going one step further that in the case of a graded Gorenstein Frobenius algebra, the coproduct is well defined on the reduced normalized Hochschild complex. We discuss several other aspects such as "time reversal" duality and several homotopies of operations induced by it. This provides a cohomology operation which is a homotopy of the anti-symmetrization of the coproduct. The obstruction again vanishes on the reduced normalized Hochschild complex if the Frobenius algebra is graded Gorenstein. Further structures such as "animation", the BV structure, a coloring for operations on chains and cochains, and a Gerstenhaber double bracket are briefly treated. [ABSTRACT FROM AUTHOR]

We offer short and conceptual re-proofs of some conjectures of Voevodsky's on the slice filtration. The original proofs were due to Marc Levine using the homotopy coniveau tower. Our new proofs use very different methods, namely, recent development in motivic infinite loop space theory together with the birational geometry of Hilbert schemes. [ABSTRACT FROM AUTHOR]

Earlier results show that the N = 1 / 2 supersymmetric path integral J g on a closed even dimensional Riemannian spin manifold (X , g) can be constructed in a mathematically rigorous way via Chen differential forms and techniques from noncommutative geometry, if one considers J g as a current on the loop space LX , that is, as a linear form on differential forms on LX. This construction admits a Duistermaat-Heckman localization formula. In this note, fixing a topological spin structure on X , we prove that any smooth family g • = (g t) t ∈ [ 0 , 1 ] of Riemannian metrics on X canonically induces a Chern-Simons current C g • which fits into a transgression formula for the supersymmetric path integral. In particular, this result entails that the supersymmetric path integral induces a differential topological invariant on X , which essentially stems from the A ˆ -genus of X. [ABSTRACT FROM AUTHOR]

The new secondary cohomology operations are constructed. These operations together with the Adams operations are intended to calculate the mod p cohomology algebra of loop spaces. In particular, the kernel of the loop suspension map is explicitly described. [ABSTRACT FROM AUTHOR]

We 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. [ABSTRACT FROM AUTHOR]

A reactive system’s slow dynamic behavior is approximated well by evolution on manifolds of dimension lower than that of the full composition space. This work addresses the construction of one-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady spatially homogeneous closed reactive systems. Additionally, the relation between the systems’ slow dynamics, described by the constructed manifolds, and thermodynamics is clarified. It is shown that other than identifying the equilibrium state, traditional equilibrium thermodynamic potentials provide no guidance in constructing the systems’ actual slow invariant manifolds. The construction technique is based on analyzing the composition space of the reactive system. The system’s finite and infinite equilibria are calculated using a homotopy continuation method. The slow invariant manifolds are constructed by calculating attractive heteroclinic orbits which connect appropriate equilibria to the unique stable physical equilibrium point. Application of the method to several realistic reactive systems, including a detailed hydrogen-air kinetics model, reveals that constructing the actual slow invariant manifolds can be computationally efficient and algorithmically easy. [ABSTRACT FROM AUTHOR]

Discusses the use of derivative coupling to determine the number of closely spaced conical intersections in the closed loop. Advantages of the method over the geometric phase theorem; Establishment of the locus of the seam of conical intersection of the 2[sup 2] E[sup '] state of sodium [sub 3].

To any topological operad O, we introduce a cofibrant replacement in the category of bimodules over itself such that for every map η : O → O ′ of operads, the corresponding model Bimod O h (O ; O ′) of derived mapping space of bimodules is an algebra over the one dimensional little cubes operad C 1 . We also build an explicit weak equivalence of C 1 -algebras from the loop space Ω Operad h (O ; O ′) to Bimod O h (O ; O ′) . [ABSTRACT FROM AUTHOR]

Abstract The concept of homotopy has been a staple of quasigroup theory since Albert's work on isotopies in the early 1940s. In this paper, we introduce and study the dual concept of a higher homotopy. [ABSTRACT FROM AUTHOR]

Simplicial complexes K, in relation to their Alexander dual Kb , can be classified as self-dual (K = Kb), sub-dual (K ⊆ Kb), super-dual (K ⊇ Kb), or transcendent (neither sub-dual nor super-dual). We explore a connection between sub-dual and self-dual complexes providing a new insight into combinatorial structure of self-dual complexes. The root operator (Definition 4.3) in Section 4 associates with each self-dual complex K a sub-dual complex √ K on a smaller number of vertices. We study the operation of minimal restructuring of self-dual complexes and the properties of the associated neighborhood graph, defined on the set of all self-dual complexes. Some of the operations and relations, introduced in the paper, were originally developed as a tool for computer-based experiments and enumeration of self-dual complexes. [ABSTRACT FROM AUTHOR]

In this paper we consider the based loop space ΩM on a simply connected manifold M. We first prove, only by means of the rational homotopy theory, that the rational homotopy type of ΩM is determined by the second Betti number b2(M). We further consider the problem of computation of the rational Pontryagin homology ring H*(ΩM) when b2(M) ≤ 3. We prove that H*(ΩM) is up to degree 5 generated by the elements of degree 1 for b2(M) = 3. [ABSTRACT FROM AUTHOR]

Let P be an n-gon with n ≥ 3 {n\geq 3}. There is a formal combinatorial A ∞ {A_{\infty}} -coalgebra structure on cellular chains C * ⁢ (P) {C_{*}(P)} with non-vanishing higher order structure when n ≥ 5 {n\geq 5}. If X g {X_{g}} is a closed compact surface of genus g ≥ 2 {g\geq 2} and P g {P_{g}} is a polygonal decomposition, the quotient map q : P g → X g {q\colon P_{g}\to X_{g}} projects the formal A ∞ {A_{\infty}} -coalgebra structure on C * ⁢ (P g) {C_{*}(P_{g})} to a quotient structure on C * ⁢ (X g) {C_{*}(X_{g})} , which persists to homology H ∗ ⁢ (X g ; ℤ 2) {H_{\ast}(X_{g};\mathbb{Z}_{2})} , whose operations are determined by the quotient map q, and whose higher order structure is non-trivial if and only if X g {X_{g}} is orientable with g ≥ 2 {g\geq 2} or unorientable with g ≥ 3 {g\geq 3}. But whether or not the A ∞ {A_{\infty}} -coalgebra structure on homology observed here is topologically invariant is an open question. [ABSTRACT FROM AUTHOR]

Abstract We study some Borsuk–Ulam type results for the loop space of an euclidean sphere without loops equal to their inverses. [ABSTRACT FROM AUTHOR]

Quasi-elliptic cohomology is a variant of Tate K-theory. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. In this paper we show how this theory is equipped with power operations. We also prove that the Tate K-theory of symmetric groups modulo a certain transfer ideal classify the finite subgroups of the Tate curve. [ABSTRACT FROM AUTHOR]

We study the canonical Poisson structure on the loop space of the super-double-twisted-torus and its quantization. As a consequence we obtain a rigorous construction of mirror symmetry as an intertwiner of the |$N=2$| super-conformal structures on the super-symmetric sigma-models on the Kodaira-Thurston nilmanifold and a gerby torus of complex dimension 2. As an application we are able to identify global moduli of equivariant generalized complex structures on these target spaces with moduli of equivariant orthogonal complex structures on the doubled geometry. [ABSTRACT FROM AUTHOR]

This work aims to experimentally investigate the manipulation of a turbulent boundary layer over a flat plate using a proportional-derivative (PD) controller. The control action is generated by an array of two flush-mounted piezo-ceramic actuators. Two different schemes are examined, i.e., feed-forward and feedback PD controls, with a view to suppressing the viscous-scaled near-wall cycle of high-speed events in the near-wall region and hence reducing skin friction drag. It has been found that the use of the feed-forward PD scheme may reduce the local maximum drag reduction by up to 33% at 14 wall units downstream of the actuator array, exceeding the open-loop control result (30%) as well as our previously reported combined feed-forward and feedback scheme (28%) [Z. X. Qiao, Y. Zhou, and Z. Wu, "Turbulent boundary layer under the control of different schemes," Proc. R. Soc. A 473, 20170038 (2017)], and furthermore, this significantly cuts down the required input energy by 27%, compared to the open-loop control. On the other hand, the feedback PD scheme achieves the same control performance as the open-loop control, that is, producing a local maximum drag reduction of 30% without any saving in the input energy. The underlying control mechanism behind these control schemes is proposed based on the analyses of the hot-wire data measured with and without control. [ABSTRACT FROM AUTHOR]

Abstract Quasi-elliptic cohomology is a variant of elliptic cohomology theories. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. Thus, the constructions on it can be made in a neat way. This theory reflects the geometric nature of the Tate curve. In this paper we provide a systematic introduction of its construction and definition. [ABSTRACT FROM AUTHOR]

Using our earlier proposal for Ramond-Ramond fields in an H-flux on loop space (Han et al. in Commun Math Phys 337(1):127-150, 2015. arXiv:1405.1320), we extend the Hori isomorphism in Bouwknegt et al. (Commun Math Phys 249:383-415, 2004. arXiv:hep-th/0306062; Phys Rev Lett 92:181601, 2004. arXiv:hep-th/0312052) from invariant differential forms, to invariant exotic differential forms such that the momentum and winding numbers are exchanged, filling in a gap in the literature. We also extend the compatibility of the action of invariant exact Courant algebroids on the T-duality isomorphism in Cavalcanti and Gualtieri (in: CRM proceedings of lecture notes, vol 50, pp 341-365, American Mathematical Society, Providence, 2010 ), to the T-duality isomorphism on exotic invariant differential forms. [ABSTRACT FROM AUTHOR]

One of the control decisions in the operation of an automated guided vehicle (AGV) system is to determine the home locations of idle vehicles. In this paper, the problems of selecting home location of a vehicle when idle in a single loop AGV network is presented. As the number of unit loads to be picked up at each workstation dynamically changes over time, the optimum home location of vehicles may also change. Based on the objective of minimizing the expected response time of a vehicle, models are constructed. Example problems are given to illustrate the use of the solution algorithms. The results of the system response times obtained using the dynamic dwell point models are compared with those of other dwell point rules. [ABSTRACT FROM AUTHOR]

This paper is about a correspondence between monoidal structures in categories and n-fold loop spaces. We developed a new syntactical technique whose role is to substitute the coherence results, which were the main ingredients in the proof that the Segal-Thomason bar construction provides an appropriate simplicial space. The results we present here enable more common categories to enter this delooping machine. For example, such as the category of finite sets with two monoidal structures brought by the disjoint union and Cartesian product. [ABSTRACT FROM AUTHOR]

Control loop Performance Assessment (CPA) plays an important role in system operations. Stochastic statistical CPA index, such as a minimum variance controller (MVC)-based CPA index, is one of the most widely used CPA indices. In this paper, a new minimum entropy controller (MEC)-based CPA method of linear non-Gaussian systems is proposed. In this method, probability density function (PDF) and rational entropy (RE) are respectively used to describe the characteristics and the uncertainty of random variables. To better estimate the performance benchmark, an improved EDA algorithm, which is used to estimate the system parameters and noise PDF, is given. The effectiveness of the proposed method is illustrated through case studies on an ARMAX system. [ABSTRACT FROM AUTHOR]

Most bilateral teleoperation researches focus on the system stability within time-delays. However, practical teleoperation tasks require high performances besides system stability, such as convergence rate and accuracy. This paper investigates bilateral teleoperation controller design with transient performances. To ensure the transient performances and system stability simultaneously, an adaptive non-singular fast terminal mode controller is proposed to achieve practical finite-time stability considering system uncertainties and time delays. In addition, a novel switching scheme is introduced, in which way the singularity problem of conventional terminal sliding manifold is avoided. Finally, numerical simulations demonstrate the effectiveness and validity of the proposed method. [ABSTRACT FROM AUTHOR]

Let G k denote the gauge group of the principal S p ( 3 ) -bundle over S 4 with first symplectic Pontryagin class k ∈ H 4 S 4 = Z . We show that when localised at an odd prime there is a homotopy equivalence G k ≃ G l if and only if ( 21 , k ) = ( 21 , l ) . We also give bounds on the number of 2-local homotopy types amongst the G k . Our results follow from a thorough examination of the commutator map c : S p ( 1 ) ∧ S p ( 3 ) → S p ( 3 ) and we determine its order to be either 4 ⋅ 3 ⋅ 7 = 84 , 8 ⋅ 3 ⋅ 7 = 168 or 16 ⋅ 3 ⋅ 7 = 336 . In particular its order at odd primes is completely determined. [ABSTRACT FROM AUTHOR]

The pull back of a flat bundle E→X along the evaluation map π:LX→X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E. This construction naturally generalizes to flat Z-graded connections on X. Our main result is that the restriction of this holonomy automorphism to the based loop space Ω*X of X provides an A∞ quasi-equivalence between the dg category of flat Z-graded connections on X and the dg category of representations of C∙(Ω*X), the dg algebra of singular chains on Ω*X. [ABSTRACT FROM AUTHOR]

We use Richter's 2-primary proof of Gray's conjecture to give a homotopy decomposition of the fibre Ω³S17{2} of the H-space squaring map on the triple loop space of the 17-sphere. This induces a splitting of the mod-2 homotopy groups π*(S17; Z=2Z) in terms of the integral homotopy groups of the fibre of the double suspension E²: S²n-1 → Ω²S2n+1 and refines a result of Cohen and Selick, who gave similar decompositions for S5 and S9. We relate these decompositions to various Whitehead products in the homotopy groups of mod-2 Moore spaces and Stiefel manifolds to show that the Whitehead square [i2n; i2n] of the inclusion of the bottom cell of the Moore space P2n+1(2) is divisible by 2 if and only if 2n = 2; 4; 8 or 16. [ABSTRACT FROM AUTHOR]

The existence of definite organizational goals is a longstanding and central premise in organization and management research, yet a re-examination of this body of knowledge is timely and long overdue. Many important aspects of organizational goals have received very fragmented attention, and there has been little prior attempt to synthesize and compare the effects of these different goals on firm behavior and performance. The authors present a review of existing theoretical and empirical evidence on organizational goals, and develop an analytical framework emphasizing the variety of organizational goals, their attributes, antecedents and outcomes, the role of context and feedback loops. Drawing on this framework, the authors set out an agenda for further research aimed at advancing current understanding of organizational goals and implications for firm behavior and performance. [ABSTRACT FROM AUTHOR]